This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Strong differential subordination properties for analytic functions involving the Komatu integral operator

Nak Eun Cho

Author Affiliations

Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea

Boundary Value Problems 2013, 2013:44  doi:10.1186/1687-2770-2013-44


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/44


Received:27 November 2012
Accepted:21 January 2013
Published:4 March 2013

© 2013 Cho; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of the present paper is to investigate some strong differential subordination and superordination implications involving the Komatu integral operator which are obtained by considering suitable classes of admissible functions. The sandwich-type theorems for these operators are also considered.

MSC: 30C80, 30C45, 30A20.

Keywords:
strong differential subordination; strong differential superordination; univalent function; convex function; Komatu integral operator

1 Introduction

Let ℋ denote the class of analytic functions in the open unit disk <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M1">View MathML</a>. For a positive integer n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M2">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M3">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M4">View MathML</a>. We also denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M5">View MathML</a> the subclass of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M6">View MathML</a> with the usual normalization <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M7">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M9">View MathML</a> be members of ℋ. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8">View MathML</a> is said to be subordinate to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M9">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M9">View MathML</a> is said to be superordinate to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8">View MathML</a>, if there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M14">View MathML</a> analytic in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M17">View MathML</a>, and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M18">View MathML</a>. In such a case, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M19">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M20">View MathML</a>. If the function F is univalent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M19">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M24">View MathML</a> (cf.[1]).

Following Komatu [2], we introduce the integral operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M25">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M26">View MathML</a>

(1.1)

where the symbol Γ stands for the gamma function. We also note that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M27">View MathML</a> defined by (1.1) can be expressed by the series expansion as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M28">View MathML</a>

(1.2)

Obviously, we have, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M29">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M30">View MathML</a>

Moreover, from (1.2), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M31">View MathML</a>

(1.3)

In particular, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M32">View MathML</a> is closely related to the multiplier transformation studied earlier by Flett [3]. Various interesting properties of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M33">View MathML</a> have been studied by Jung et al.[4] and Liu [5].

To prove our results, we need the following definitions and theorems considered by Antonimo [6,7] and Oros [8,9].

Definition 1.1 ([6], cf.[7,8])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M34">View MathML</a> be analytic in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M35">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8">View MathML</a> be analytic and univalent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>. Then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M34">View MathML</a> is said to be strongly subordinate to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8">View MathML</a> is said to be strongly superordinate to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M34">View MathML</a>, written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M42">View MathML</a>, if for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M34">View MathML</a> as the function of z is subordinate to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M8">View MathML</a>. We note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M42">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M48">View MathML</a>.

Definition 1.2 ([8], cf.[1])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M49">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M50">View MathML</a> be univalent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52">View MathML</a> is analytic in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a> and satisfies the (second-order) differential subordination

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M54">View MathML</a>

(1.4)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52">View MathML</a> is called a solution of the strong differential subordination. The univalent function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M56">View MathML</a> is called a dominant of the solutions of the strong differential subordination, or more simply a dominant, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M57">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52">View MathML</a> satisfying (1.4). A dominant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M59">View MathML</a> that satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M60">View MathML</a> for all dominants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M56">View MathML</a> of (1.4) is said to be the best dominant.

Recently, Oros [9] introduced the following strong differential superordinations as the dual concept of strong differential subordinations.

Definition 1.3 ([9], cf.[10])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M62">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M50">View MathML</a> be analytic in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M66">View MathML</a> are univalent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M43">View MathML</a> and satisfy the (second-order) strong differential superordination

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M69">View MathML</a>

(1.5)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52">View MathML</a> is called a solution of the strong differential superordination. An analytic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M56">View MathML</a> is called a subordinant of the solutions of the strong differential superordination, or more simply a subordinant, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M72">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52">View MathML</a> satisfying (1.5). A univalent subordinant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M59">View MathML</a> that satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M75">View MathML</a> for all subordinants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M56">View MathML</a> of (1.5) is said to be the best subordinant.

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M77">View MathML</a> the class of functions q that are analytic and injective on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M78">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M79">View MathML</a>

and are such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M80">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M81">View MathML</a>. Further, let the subclass of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M82">View MathML</a> for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M83">View MathML</a> be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M85">View MathML</a>.

Definition 1.4 ([8])

Let Ω be a set in ℂ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M86">View MathML</a> and n be a positive integer. The class of admissible functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M87">View MathML</a> consists of those functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M88">View MathML</a> that satisfy the admissibility condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M89">View MathML</a>

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M91">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M92">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M96">View MathML</a>. We write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M97">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M98">View MathML</a>.

Definition 1.5 ([9])

Let Ω be a set in ℂ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M99">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M100">View MathML</a>. The class of admissible functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M101">View MathML</a> consists of those functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M102">View MathML</a> that satisfy the admissibility condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M103">View MathML</a>

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M105">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M107">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M111">View MathML</a>. We write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M112">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M113">View MathML</a>.

For the above two classes of admissible functions, Oros and Oros proved the following theorems.

Theorem 1.1 ([8])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M114">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M83">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M116">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M117">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M57">View MathML</a>.

Theorem 1.2 ([9])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M119">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M83">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M121">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M122">View MathML</a>

is univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M43">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M125">View MathML</a>

implies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M72">View MathML</a>.

In the present paper, making use of the differential subordination and superordination results of Oros and Oros [8,9], we determine certain classes of admissible functions and obtain some subordination and superordination implications of multivalent functions associated with the Komatu integral operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M127">View MathML</a> defined by (1.1). Additionally, new differential sandwich-type theorems are obtained. We remark in passing that some interesting developments on differential subordination and superordination for various operators in connection with the Komatu integral operator were obtained by Ali et al.[11-14] and Cho et al.[15].

2 Subordination results

Firstly, we begin by proving the subordination theorem involving the integral operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M128">View MathML</a> defined by (1.1). For this purpose, we need the following class of admissible functions.

Definition 2.1 Let Ω be a set in ℂ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M131">View MathML</a>. The class of admissible functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M132">View MathML</a> consists of those functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M133">View MathML</a> that satisfy the admissibility condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M134">View MathML</a>

whenever

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M135">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M136">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M139">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M140">View MathML</a>.

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M141">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M143">View MathML</a>

(2.1)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M144">View MathML</a>

Proof Define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M52">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M147">View MathML</a>

(2.2)

From (2.2) with the relation (1.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M148">View MathML</a>

(2.3)

Further computations show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M149">View MathML</a>

(2.4)

Define the transformation from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M150">View MathML</a> to ℂ by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M151">View MathML</a>

(2.5)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M152">View MathML</a>

(2.6)

Using equations (2.2), (2.3) and (2.4), from (2.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M153">View MathML</a>

(2.7)

Hence, (2.1) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M154">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M155">View MathML</a>

and so the admissibility condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M156">View MathML</a> is equivalent to the admissibility condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M157">View MathML</a>. Therefore, by Theorem 1.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M158">View MathML</a> or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M159">View MathML</a>

which evidently completes the proof of Theorem 2.1. □

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M160">View MathML</a> is a simply connected domain, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M161">View MathML</a> for some conformal mapping h of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a> onto Ω. In this case, the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M163">View MathML</a> is written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M164">View MathML</a>. The following result is an immediate consequence of Theorem 2.1.

Theorem 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M165">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M167">View MathML</a>

(2.8)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M168">View MathML</a>

Our next result is an extension of Theorem 2.1 to the case where the behavior of q on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M169">View MathML</a> is not known.

Corollary 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M170">View MathML</a>andqbe univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M172">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M173">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M174">View MathML</a>where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M175">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M177">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M178">View MathML</a>

Proof Theorem 2.1 yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M179">View MathML</a>. The result is now deduced from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M180">View MathML</a>. □

Theorem 2.4Lethandqbe univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M182">View MathML</a>and set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M175">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M184">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M185">View MathML</a>satisfy one of the following conditions:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M186">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M174">View MathML</a>, or

(2) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M188">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M189">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M190">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>satisfies (2.8), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M192">View MathML</a>

Proof The proof is similar to that of [[1], Theorem 2.3d] and so is omitted. □

The next theorem yields the best dominant of the differential subordination (2.7).

Theorem 2.5Lethbe univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M185">View MathML</a>. Suppose that the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M195">View MathML</a>

(2.9)

has a solutionqwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M182">View MathML</a>and satisfies one of the following conditions:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M197">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M165">View MathML</a>,

(2) qis univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M200">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M201">View MathML</a>, or

(3) qis univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M203">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M204">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M190">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>satisfies (2.8) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M207">View MathML</a>

is analytic in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M209">View MathML</a>

andqis the best dominant.

Proof Following the same arguments as in [[1], Theorem 2.3e], we deduce that q is a dominant from Theorem 2.2 and Theorem 2.4. Since q satisfies (2.9), it is also a solution of (2.8) and therefore q will be dominated by all dominants. Hence, q is the best dominant. □

In the particular case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M211">View MathML</a>, and in view of Definition 2.1, the class of admissible functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M212">View MathML</a>, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M213">View MathML</a>, is described below.

Definition 2.2 Let Ω be a set in ℂ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M211">View MathML</a>. The class of admissible functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M217">View MathML</a> consists of those functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M133">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M219">View MathML</a>

(2.10)

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M140">View MathML</a>.

Corollary 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M225">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M227">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M228">View MathML</a>

In the special case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M229">View MathML</a>, the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M230">View MathML</a> is simply denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M231">View MathML</a>.

Corollary 2.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M232">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M234">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M235">View MathML</a>

Corollary 2.8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M211">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M238">View MathML</a>be an analytic function in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M239">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M240">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M241">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M243">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M244">View MathML</a>

Proof This follows from Corollary 2.6 by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M245">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M246">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M247">View MathML</a>. To use Corollary 2.6, we need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M248">View MathML</a>, that is, the admissible condition (2.10) is satisfied. This follows since

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M140">View MathML</a>. Hence, by Corollary 2.6, we deduce the required results. □

3 Superordination and sandwich-type results

The dual problem of differential subordination, that is, differential superordination of the Komatu integral operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M128">View MathML</a> defined by (1.1), is investigated in this section. For this purpose, the class of admissible functions is given in the following definition.

Definition 3.1 Let Ω be a set in ℂ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M256">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M257">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M131">View MathML</a>. The class of admissible functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M260">View MathML</a> consists of those functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M261">View MathML</a> that satisfy the admissibility condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M262">View MathML</a>

whenever

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M263">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M264">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M267">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M268">View MathML</a>.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M269">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M271">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M272">View MathML</a>

is univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M274">View MathML</a>

(3.1)

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M275">View MathML</a>

Proof From (2.7) and (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M276">View MathML</a>

From (2.5), we see that the admissibility condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M277">View MathML</a> is equivalent to the admissibility condition for ψ as given in Definition 1.2. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M278">View MathML</a>, and by Theorem 1.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M279">View MathML</a> or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M280">View MathML</a>

which evidently completes the proof of Theorem 3.1. □

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M281">View MathML</a> is a simply connected domain, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M161">View MathML</a> for some conformal mapping h of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a> onto Ω. In this case, the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M284">View MathML</a> is written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M285">View MathML</a>. Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 3.1.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M286">View MathML</a>, hbe analytic in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M288">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M290">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M291">View MathML</a>

is univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M293">View MathML</a>

(3.2)

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M294">View MathML</a>

Theorem 3.1 and Theorem 3.2 can only be used to obtain subordinants of differential superordination of the form (3.1) or (3.2). The following theorem proves the existence of the best subordinant of (3.2) for certain ϕ.

Theorem 3.3Lethbe analytic in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M296">View MathML</a>. Suppose that the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M297">View MathML</a>

has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M197">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M301">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M302">View MathML</a>

is univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M304">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M305">View MathML</a>

andqis the best subordinant.

Proof The proof is similar to that of Theorem 2.5 and so is omitted. □

Combining Theorem 2.2 and Theorem 3.2, we obtain the following sandwich-type theorem.

Theorem 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M306">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M307">View MathML</a>be analytic functions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M309">View MathML</a>be a univalent function in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M311">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M312">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M313">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M315">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M316">View MathML</a>

is univalent in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M15">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M318">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/44/mathml/M319">View MathML</a>

Competing interests

The author declares that they have no competing interests.

Authors’ contributions

The author worked on the results and he read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619).

References

  1. Miller, SS, Mocanu, PT: Differential Subordination, Theory and Application, Marcel Dekker, New York (2000)

  2. Komatu, Y: Distortion Theorems in Relation to Linear Integral Operators, Kluwer Academic, Dordrecht (1996)

  3. Flett, TM: The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl.. 38, 746–765 (1972). Publisher Full Text OpenURL

  4. Jung, IB, Kim, YC, Srivastava, HM: The Hardy space of analytic functions associated with certain one-parameter families of integral operators. J. Math. Anal. Appl.. 176, 138–147 (1993). Publisher Full Text OpenURL

  5. Liu, JL: A linear operator and strongly starlike functions. J. Math. Soc. Jpn.. 54, 975–981 (2002). Publisher Full Text OpenURL

  6. Antonino, JA: Strong differential subordination to Briot-Bouquet differential equations. J. Differ. Equ.. 114, 101–105 (1994). Publisher Full Text OpenURL

  7. Antonino, JA: Strong differential subordination and applications to univalency conditions. J. Korean Math. Soc.. 43, 311–322 (2006)

  8. Oros, GI, Oros, G: Strong differential subordination. Turk. J. Math.. 33, 249–257 (2009)

  9. Oros, GI: Strong differential superordination. Acta Univ. Apulensis, Mat.-Inform.. 19, 101–106 (2009)

  10. Miller, SS, Mocanu, PT: Subordinants of differential superordinations. Complex Var. Theory Appl.. 48, 815–826 (2003). Publisher Full Text OpenURL

  11. Ali, RM, Ravichandran, V, Seenivasagan, N: Subordination and superordination of the Liu-Srivastava operator on meromorphic functions. Bull. Malays. Math. Soc.. 31, 193–207 (2008)

  12. Ali, RM, Ravichandran, V, Seenivasagan, N: Differential subordination and superordination of analytic functions defined by the multiplier transformation. Math. Inequal. Appl.. 12, 123–139 (2009)

  13. Ali, RM, Ravichandran, V, Seenivasagan, N: Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava linear operator. J. Franklin Inst.. 347, 1762–1781 (2010). Publisher Full Text OpenURL

  14. Ali, RM, Ravichandran, V, Seenivasagan, N: On subordination and superordination of the multiplier transformation of meromorphic functions. Bull. Malays. Math. Soc.. 33, 311–324 (2010)

  15. Cho, NE, Kwon, OS, Srivastava, HM: Strong differential subordination and superordination for multivalently meromorphic functions involving the Liu-Srivastava operator. Integral Transforms Spec. Funct.. 21, 589–601 (2010). Publisher Full Text OpenURL