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

Open Access Research

A general solution of the Fekete-Szegö problem

Jacek Dziok

Author Affiliations

Institute of Mathematics, University of Rzeszów, Rzeszów, 35-310, Poland

Boundary Value Problems 2013, 2013:98  doi:10.1186/1687-2770-2013-98

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/98


Received:24 January 2013
Accepted:9 April 2013
Published:22 April 2013

© 2013 Dziok; 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

In the paper we introduce general classes of analytic functions defined by the Hadamard product. The Fekete-Szegö problem is completely solved in these classes of functions. Some consequences of the main results for new or well-known classes of functions are also pointed out.

MSC: 30C45, 30C50, 30C55.

Keywords:
analytic functions; Fekete-Szegö problem; subordination; Hadamard product

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M1">View MathML</a> denote the class of functions which are analytic in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M2">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M3">View MathML</a> denote the class of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M4">View MathML</a> normalized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M5">View MathML</a>. Each function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6">View MathML</a> can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M7">View MathML</a>

(1)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M8">View MathML</a>, we denote the class of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6">View MathML</a>, which are univalent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M10">View MathML</a>.

A typical problem in geometric function theory is to study a functional made up of combinations of the coefficients of the original function. Usually, there is a parameter over which the extremal value of the functional is needed. The paper deals with one important functional of this type: the Fekete-Szegö functional. The classical Fekete-Szegö functional is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M11">View MathML</a>

and it is derived from the Fekete-Szegö inequality. The problem of maximizing the absolute value of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M12">View MathML</a> in subclasses of normalized functions is called the Fekete-Szegö problem. The mathematicians who introduced the functional, M. Fekete and G. Szegö [1], were able to bound the classical functional in the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M8">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M14">View MathML</a>. Later Pfluger [2] used Jenkin’s method to show that this result holds for complex μ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M15">View MathML</a>. Keogh and Merkes [3] obtained the solution of the Fekete-Szegö problem for the class of close-to-convex functions. Ma and Minda [4,5] gave a complete answer to the Fekete-Szegö problem for the classes of strongly close-to-convex functions and strongly starlike functions. In the literature, there exists a large number of results about inequalities for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M16">View MathML</a> corresponding to various subclasses of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M3">View MathML</a> (see, for instance, [1-23]).

In the paper, we consider the classes of functions which generalize these subclasses of functions.

We say that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M18">View MathML</a> is subordinate to a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M19">View MathML</a> , and write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M20">View MathML</a> (or simply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M21">View MathML</a>), if and only if there exists a function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M22">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M23">View MathML</a>. In particular, if G is univalent in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M24">View MathML</a> we have the following equivalence:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M25">View MathML</a>

For functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M26">View MathML</a> of the forms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M27">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M28">View MathML</a> we denote the Hadamard product (or convolution) of f and g, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M29">View MathML</a>

Let α be complex parameter and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M32">View MathML</a> be of the form

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M34">View MathML</a> we denote the class of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M36">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M37">View MathML</a>

Moreover, let us put

It is clear that the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M39">View MathML</a> contains functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M41">View MathML</a>

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M42">View MathML</a> the class of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M6">View MathML</a> for which there exist a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M44">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M45">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M46">View MathML</a>) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M47">View MathML</a>

Moreover, let us denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M48">View MathML</a>.

In particular, the classes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M49">View MathML</a>

are the well-known classes of α-convex Mocanu functions [24], starlike functions and convex functions, respectively. The class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M50">View MathML</a> is the well-known class of close-to convex functions with argument <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M51">View MathML</a>.

The object of the paper is to solve the Fekete-Szegö problem in the defined classes of functions. Moreover, we find sharp bounds for the second and third coefficient in these classes. Some remarks depicting consequences of the main results are also mentioned.

2 The main results

The following lemmas will be required in our present investigation.

Lemma 1[3]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M53">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M54">View MathML</a>), then

The result is sharp. The functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M56">View MathML</a>

are the extremal functions.

Theorem 1Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M57">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M58">View MathML</a>, then

(2)

(3)

(4)

where

(5)

(6)

The results are sharp.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M58">View MathML</a>. Then there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M66">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M54">View MathML</a>), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M68">View MathML</a>

(7)

It is easy to verify that

(8)

(9)

where

Thus, by (7), we have

(10)

(11)

which by Lemma 1 gives sharp estimation (2). Let μ be a complex number. Then, by (10) and (11) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M74">View MathML</a>

where γ is defined by (6). Thus, by Lemma 1, we have (4). Let functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M75">View MathML</a> satisfy the conditions

Then the functions belong to the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M77">View MathML</a> and they realize the equality in the estimation (4). Thus, the results are sharp. Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M78">View MathML</a> in (4) we get the sharp estimation (3). □

Theorem 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M79">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M80">View MathML</a>, then

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M82">View MathML</a>

The results are sharp.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M83">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M84">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M85">View MathML</a>

the results follow from Theorem 1. □

If we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M86">View MathML</a> in Theorem 1, then we obtain the following theorem.

Theorem 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M87">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M88">View MathML</a>). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M89">View MathML</a>, then

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M91">View MathML</a>

The results are sharp.

Theorem 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M93">View MathML</a>, then

(12)

(13)

(14)

where

(15)

(16)

The results (12) and (13) are sharp for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M99">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M100">View MathML</a>, respectively.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M101">View MathML</a>. Then there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M102">View MathML</a> and functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M103">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M104">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M105">View MathML</a>

(17)

Thus, by (8), we have

(18)

(19)

and by Lemma 1, we obtain the sharp estimation (14). Let μ be a complex number. Then, by (18), (19) and Lemma 1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M108">View MathML</a>

(20)

or equivalently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M109">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M112">View MathML</a>, D are defined by (15) and (16). Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M113">View MathML</a>

(22)

and, in consequence, by Lemma 1 we have (12). It is easy to verify that the equality in (22) is attained by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M115">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M117">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M119">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M121">View MathML</a>. Therefore, we consider functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M75">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M123">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M124">View MathML</a>

respectively. Then the functions belong to the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M125">View MathML</a> and they realize the equality in the estimation (12) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M126">View MathML</a>. Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M78">View MathML</a> in (12), we get the sharp estimation (13). □

The following theorem gives the complete sharp estimation of the Fekete-Szegö functional in the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M128">View MathML</a>.

Theorem 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M130">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M131">View MathML</a>

(23)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M112">View MathML</a>, Dare defined by (15) and (16). The result is sharp.

Proof From Theorem 4, we have sharp estimation (23) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M126">View MathML</a>. Let now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M137">View MathML</a>. Then, by (20) and Lemma 1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M138">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M139">View MathML</a>

Simply calculations give that the function v attains a maximum in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M140">View MathML</a> at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M141">View MathML</a>. Thus, we have (23) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M137">View MathML</a>. Moreover, the equality in (20) is attained by choosing the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M145">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M146">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M148">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M150">View MathML</a>. Therefore, the result is sharp for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M137">View MathML</a>.

Next, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M154">View MathML</a>. Then, by (20) and Lemma 1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M155">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M156">View MathML</a>

Since the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M157">View MathML</a> attains a maximum in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M158">View MathML</a> at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M159">View MathML</a>, we have the estimation (23) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M154">View MathML</a>. The equality in (20) is attained by choosing the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M163">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M164">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M168">View MathML</a>.

Finally, let us assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M169">View MathML</a>. Then, by (20) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M170">View MathML</a>

(24)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M171">View MathML</a>

Since F is the continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M172">View MathML</a>, by (24) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M173">View MathML</a>

(25)

where K is the set of critical points of the function F in T. It is easy to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M174">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M175">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M176">View MathML</a>

Moreover, we have

Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M178">View MathML</a>

which by (25) gives (23) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M169">View MathML</a>. The equality in (24) is attained by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M114">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M115">View MathML</a>. Therefore, the result is sharp and the proof is completed. □

Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M182">View MathML</a> in Theorem 5 we obtain the following theorem.

Theorem 6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M93">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M185">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M188">View MathML</a>, Dare defined by (15) and (16). The result is sharp.

3 Applications

If we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M86">View MathML</a> in Theorem 2, then we obtain the following two corollaries.

Corollary 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M191">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M192">View MathML</a>, then

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M194">View MathML</a>

The results are sharp.

Corollary 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M191">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M196">View MathML</a>, then

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M198">View MathML</a>

The results are sharp.

Choosing the function p in Theorems 1-6, we can obtain several new results.

Let a, b be complex number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M200">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M201">View MathML</a>

It is clear, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M202">View MathML</a>

Thus, by Theorems 1-3 and 5, we obtain the following four corollaries.

Corollary 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M203">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M88">View MathML</a>). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M205">View MathML</a>, then

where

The results are sharp.

Corollary 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M79">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M209">View MathML</a>, then

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M211">View MathML</a>

The results are sharp.

Corollary 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M212">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M213">View MathML</a>). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M214">View MathML</a>, then

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M216">View MathML</a>

The results are sharp.

Corollary 6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M218">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M219">View MathML</a>

where

The result is sharp.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M221">View MathML</a> and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M222">View MathML</a>

It is easy to verify, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M223">View MathML</a>

Thus, by Theorems 1-3 and 5, we obtain the following four corollaries.

Corollary 7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M203">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M88">View MathML</a>). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M226">View MathML</a>, then

where

The results are sharp.

Corollary 8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M79">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M230">View MathML</a>, then

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M232">View MathML</a>

The results are sharp.

Corollary 9Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M212">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M88">View MathML</a>). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M235">View MathML</a>, then

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M237">View MathML</a>

The results are sharp.

Corollary 10Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M92">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M239">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M240">View MathML</a>

where

The result is sharp.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M243">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M244">View MathML</a> is the convex domain contained in the right half plane, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M245">View MathML</a>. More precisely, it is the elliptic domain for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M246">View MathML</a>, the hyperbolic domain for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M247">View MathML</a> and the parabolic domain for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M248">View MathML</a>.

Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M249">View MathML</a> the univalent function, which maps the unit disc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M10">View MathML</a> onto the conic domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M244">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M252">View MathML</a>. Obviously, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M249">View MathML</a> is convex in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M10">View MathML</a>. It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M255">View MathML</a> if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M256">View MathML</a>

The following lemma gives coefficients estimates for the function.

Lemma 2[13]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M257">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M258">View MathML</a>). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M259">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M260">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M261">View MathML</a>is the complete elliptic integral of first kind.

Using Lemma 1 in Theorems 1-5 we obtain the solutions of the Fekete-Szegö problem for the classes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M265">View MathML</a>.

Remark 1 The classes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/98/mathml/M268">View MathML</a> reduced to well-known subclasses by judicious choices of the parameters; see, for example [1-28]. In particular, they generalize several well-known classes defined by linear operators, which were investigated in earlier works. Also, the obtained results generalize several results obtained in these classes of functions.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

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