SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Boundary fractional differential equation in a complex domain

Rabha W Ibrahim1* and Jay M Jahangiri2

Author Affiliations

1 Institute of Mathematical Sciences, University Malaya, Kuala Lumpur, 50603, Malaysia

2 Institute of Mathematical Sciences, Kent State University, Burton, OH, 44021-9500, USA

For all author emails, please log on.

Boundary Value Problems 2014, 2014:66  doi:10.1186/1687-2770-2014-66

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/66


Received:2 December 2013
Accepted:11 March 2014
Published:24 March 2014

© 2014 Ibrahim and Jahangiri; 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 credited.

Abstract

We discuss univalent solutions of boundary fractional differential equations in a complex domain. The fractional operators are taken in the sense of the Srivastava-Owa calculus in the unit disk. The existence of subsolutions and supersolutions (maximal and minimal) is established. The existence of a unique univalent solution is imposed. Applications are constructed by making use of a transformation formula for fractional derivatives as well as generalized fractional derivatives.

1 Introduction

Fractional calculus is the most significant branch of mathematical analysis that transacts with the potential of covering real number powers or complex number powers of the differentiation operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M1">View MathML</a>. This concept was harnessed in geometric function theory (GFT). It was applied to derive different types of differential and integral operators mapping the class of univalent functions and its subclasses into themselves. Hohlov [1,2] imposed sufficient conditions that guaranteed such mappings for the operators defined by means of the Hadamard product (or convolution) with Gauss hypergeometric functions. This was further extended by Kiryakova and Saigo [3] and Kiryakova [4,5] to the operators of the generalized fractional calculus (GFC) consisting of product functions of the Gaussian function, generalized hypergeometric functions, G-functions, Wright functions and Fox-Wright generalized functions as well as rendering integral representations by means of Fox H-functions and the Meijer G-function. These techniques can be used to display sufficient conditions that guarantee mapping of univalent functions (or, respectively, of convex functions) into univalent functions. For example, for the case of Dziok-Srivastava operator see [6], and for an extension to the Wright functions see [7] which is concerned with the Srivastava-Wright operator. With the help of operators introduced in [6] and [7] one can establish univalence criteria for a large number of operators in GFT and GFC and for many of their special cases such as operators of the classical fractional calculus. Srivastava and Owa [8] generalized the definitions of fractional operators as follows.

Definition 1.1 For the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2">View MathML</a> analytic in a simply-connected region of the complex z-plane ℂ containing the origin and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M3">View MathML</a>, the fractional derivative of order α is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M4">View MathML</a>

where the multiplicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M5">View MathML</a> is removed by requiring <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M6">View MathML</a> to be real when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M7">View MathML</a>. Moreover, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M8">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M9">View MathML</a>.

Definition 1.2 For the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2">View MathML</a> analytic in a simply-connected region of the complex z-plane ℂ containing the origin and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M11">View MathML</a>, the fractional integral of order α is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M12">View MathML</a>

where the multiplicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M13">View MathML</a> is removed by requiring <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M6">View MathML</a> to be real when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M7">View MathML</a>.

Remark 1.1 From the above two definitions, we observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M16">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M17">View MathML</a>

Later, the first author [9] modified these Srivastava-Owa operators into two fractional parameters. For a wealth of references on applications of Srivastava-Owa operators, see [10-19].

In this paper, we study univalent solutions of boundary fractional differential equations in a complex domain. The fractional operators are considered in the sense of the Srivastava-Owa [8] differential operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M18">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M19">View MathML</a> is the open unit disk and f is analytic in U satisfying the Riemann mapping conditions. The existence of subsolutions and supersolutions (minimal and maximal) is established. The existence of a unique univalent solution is introduced. Applications are also constructed by making use of some transformation formula for fractional derivatives. Equation (1) is a generalization of Beurling problem.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M20">View MathML</a> be the set of analytic functions f on the unit disk U normalized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M22">View MathML</a>. And let be the subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M20">View MathML</a> normalized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M26">View MathML</a>. We denote by the set of all univalent functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M28">View MathML</a>.

Definition 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M29">View MathML</a> be a positive, continuous and bounded function and the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M30">View MathML</a>

Then every function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M31">View MathML</a> is a subsolution for Ψ. If f is univalent, then it is called a univalent subsolution for Ψ.

Definition 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M29">View MathML</a> be a positive, continuous and bounded function and the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M33">View MathML</a>

Then every function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M34">View MathML</a> is a supersolution for Ψ. If f is univalent, then it is called a univalent supersolution for Ψ.

In this work, we utilize the generalized sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M35">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M36">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M38">View MathML</a>, the above sets reduce to [20]. The next result shows some properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M39">View MathML</a>. This is ultimately Lemma 2.3 from [20], therefore we omit the proof.

Lemma 2.1Let Ψ be a positive, continuous and bounded function on ℂ.

1. Any subsolution for Ψ has a (Lipschitz) continuous extension to the closed unit disk<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40">View MathML</a>. The set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M41">View MathML</a>is uniformly bounded on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40">View MathML</a>and equicontinuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40">View MathML</a>.

2. A function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M44">View MathML</a>with a continuous extension toUis a subsolution for Ψ if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M45">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M46">View MathML</a>is the Poisson kernel.

3. If a sequence of subsolutions from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M39">View MathML</a>converges locally uniformly inUto a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M48">View MathML</a> (algebra of a holomorphic function that satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M50">View MathML</a>), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M51">View MathML</a>.

4. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M52">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M53">View MathML</a>be a positive, continuous and bounded function with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M54">View MathML</a>. Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M55">View MathML</a>sufficiently close to 1, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M57">View MathML</a>, is a subsolution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M41">View MathML</a>.

Lemma 2.2 [[20], Lemma 2.7, Lemma 3.7]

Let Ψ be a positive, continuous and bounded function on ℂ. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M60">View MathML</a>are two subsolutions for Ψ (univalent supersolutions for Ψ). Then the upper of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M59">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M60">View MathML</a>is also a subsolution for Ψ (a univalent supersolution for Ψ).

Lemma 2.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M63">View MathML</a>is a solution to problem (1), thenfhas 0 as its unique critical point.

Lemma 2.3 is a generalization of the result found in [21]. Thence, we cancel the proof.

Next result shows some properties of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M64">View MathML</a>, which basically is a generalization of [[20], Lemma 3.3]. So we skip the proof.

Lemma 2.4Let Ψ be a positive, continuous and bounded function on ℂ.

1. Any univalent supersolutiongfor Ψ satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M65">View MathML</a>.

2. A bounded univalent function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M44">View MathML</a>belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M64">View MathML</a>if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M68">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M46">View MathML</a>is the Poisson kernel.

3. If a uniformly bounded sequence of univalent supersolutions for Ψ converges locally uniformly inU, then the limit functionfis again a univalent supersolution for Ψ.

4. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M70">View MathML</a>be a positive, continuous and bounded function with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M71">View MathML</a>, and letgbe a univalent supersolution for Ψ. Ifgis bounded, then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M72">View MathML</a>sufficiently close to 1, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M74">View MathML</a>, is a univalent supersolution for Φ.

3 Main results

Our aim is to establish the largest univalent solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M75">View MathML</a> and the smallest univalent solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M76">View MathML</a>. We are able to state and prove the following theorem.

Theorem 3.1Let Ψ be a positive continuous function on ℂ. Then there exists a unique univalent function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M77">View MathML</a> (algebra unit disk), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M79">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M81">View MathML</a>. Furthermore, the maximal subsolution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82">View MathML</a>is a solution.

Proof By Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M41">View MathML</a> is non-empty and bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M85">View MathML</a>). Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M86">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M87">View MathML</a>

Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M88">View MathML</a> is the upper of f and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M90">View MathML</a>. In view of Lemma 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M91">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M92">View MathML</a>. On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M93">View MathML</a>

which involves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M94">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M95">View MathML</a>. Then h is a well-defined holomorphic function on U such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M96">View MathML</a>. By letting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M97">View MathML</a>

and for sufficiently small values of α, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M98">View MathML</a>

Letting the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M99">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M100">View MathML</a>

we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M101">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M102">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M103">View MathML</a>

Therefore, by virtue of the principle of subordination, this yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M104">View MathML</a>

and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M105">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M106">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M107">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M108">View MathML</a>.

Now we proceed to show that the maximal subsolution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82">View MathML</a> is a solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M41">View MathML</a>. Define a function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M111">View MathML</a>

where φ is the harmonic function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M112">View MathML</a> whose boundary values are logΨ (see [[22], p.402]). From the definition of Φ, we may conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82">View MathML</a> is the maximal subsolution of Φ. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M114">View MathML</a> be a solution for (1) having a unique critical point at 0 (Lemma 2.3). Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M115">View MathML</a> and the two functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M117">View MathML</a> are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40">View MathML</a>, harmonic on U and coincide on ∂U. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M119">View MathML</a>

on U and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M120">View MathML</a>

In addition, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M121">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M122">View MathML</a>

is identically equal to zero. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82">View MathML</a> is a solution. This completes the proof. □

Remark 3.1 Note that Theorem 3.1 can be introduced for the boundary problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M124">View MathML</a>

where f is analytic in U satisfying the Riemann mapping conditions.

As a consequence of our theorem, we have the following.

Corollary 3.1Let Ψ be a positive continuous function on ℂ. Then there exists a univalent function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M125">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M127">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M128">View MathML</a>is a solution to (1).

Corollary 3.2LetDbe a simply-connected region inand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M129">View MathML</a>be an analytic function on the open unit disk. If Ψ is a positive convex function, then there exists a unique univalent function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M77">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M132">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M133">View MathML</a>is a solution to (1).

Corollary 3.3Let Ψ be a positive continuous sublinear function on ℂ; i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M134">View MathML</a>

Then there exists a univalent function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M77">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M132">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M133">View MathML</a>is a solution to (1).

Next, we discuss the boundary problem for some functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M139">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M140">View MathML</a>. From [[23], Theorem 1.3], for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M141">View MathML</a>, we define the fractional transform

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M142">View MathML</a>

Now, in a manner similar to Theorem 3.1, we have the following theorem.

Theorem 3.2Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M143">View MathML</a>

(2)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M144">View MathML</a> , and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2">View MathML</a>is analytic in a simply-connected region<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M146">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M147">View MathML</a>and Ψ is a continuous function on ℂ, then there exists a unique univalent function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M132">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M151">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M152">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M153">View MathML</a>.

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M154">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M156">View MathML</a>. Also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M157">View MathML</a> is non-empty and bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M40">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M159">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M160">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M161">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M82">View MathML</a> is as in Theorem 3.1. Next, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M163">View MathML</a> is the upper of P and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M164">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M165">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M166">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M167">View MathML</a>. On the other hand, we may have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M168">View MathML</a>

which includes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M169">View MathML</a>. Now the proof is complete by proceeding with a similar manner to that of the first part of the proof of Theorem 3.1. □

As a consequence of Theorem 3.2 above, we have the following.

Corollary 3.4Let Ψ be a positive continuous function on ℂ. Then there exists a univalent function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M125">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M127">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M173">View MathML</a>solves problem (2).

Tremblay [24] studied a fractional calculus operator defined in terms of the Riemann-Liouville fractional differential operator. We extend this operator in the complex plane to involve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M174">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M175">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M176">View MathML</a>

Consequently, for the class consisting of analytic functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M178">View MathML</a> that are univalent in U, we have the following upper bound of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M179">View MathML</a>.

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M180">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M181">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M182">View MathML</a>

(3)

whereFis the hypergeometric function. The equality holds true for the Koebe function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M183">View MathML</a>

Proof By De Branges’ theorem [25] (also known as Bieberbach conjecture, e.g., see Duren [26]), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M184">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M185">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M186">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M187">View MathML</a> is the Pochhammer symbol defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M188">View MathML</a>

Finally, by letting the Koebe function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M189">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2">View MathML</a> in (3), we can show that the result is sharp. Hence the proof. □

Moreover, we can prove the following theorem.

Theorem 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M180">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M193">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M194">View MathML</a>, then we have the sharp bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M195">View MathML</a>

(4)

whereFis a hypergeometric function.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M192">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M193">View MathML</a>. Then we may use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M199">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M200">View MathML</a>, then we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M201">View MathML</a>. Otherwise, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M202">View MathML</a>. From the above two cases, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M194">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M205">View MathML</a>

Now, by applying the last assertion on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M206">View MathML</a>, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M207">View MathML</a>

 □

We obtain the following two corollaries by making use the above operator and, respectively, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M209">View MathML</a>.

Corollary 3.5Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M210">View MathML</a>

(5)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2">View MathML</a>is analytic in a simply-connected region<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M146">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M214">View MathML</a>and Ψ is a continuous function on ℂ, then there exists a unique univalent function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M217">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M220">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M221">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M222">View MathML</a>.

Corollary 3.6Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M223">View MathML</a>

(6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M2">View MathML</a>is analytic in a simply-connected region<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M146">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M213">View MathML</a>and Ψ is a continuous function on ℂ, then there exists a unique univalent function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M229">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M232">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M233">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M234">View MathML</a>.

In the following example we demonstrate that, in view of Theorem 3.1, the above boundary problems have univalent solutions in the unit disk with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M235">View MathML</a>.

Example 3.1 A computation implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M236">View MathML</a>

and for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M237">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M238">View MathML</a>

Thus, we have the boundary problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M239">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M240">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M242">View MathML</a> as well as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M243">View MathML</a>, which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M245">View MathML</a>.

By using the same technique as in the first part of Theorem 3.1, together with Lemma 2.2 and Lemma 2.4, we conclude the following.

Theorem 3.5Let Ψ be a positive, continuous and bounded function. Then there exists a unique function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M246">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M247">View MathML</a>

Then the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M248">View MathML</a>mapsUconformally onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M249">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M250">View MathML</a>

We call the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/66/mathml/M248">View MathML</a> of Theorem 3.5 the minimal univalent supersolution for Ψ.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors jointly worked on deriving the results and approved the final manuscript.

Acknowledgements

This work is supported by University of Malaya High Impact Research Grant no vote UM.C/625/HIR/MOHE/SC/13/2 from the Ministry of Higher Education Malaysia. The authors also would like to thank the referees for giving some suggestions for improving the work.

References

  1. Hohlov, Y: Convolutional operators preserving univalent functions. Ukr. Math. J.. 37, 220–226 (1985)

  2. Hohlov, Y: Convolutional operators preserving univalent functions. Pliska Stud. Math. Bulgar.. 10, 87–92 (1989)

  3. Kiryakova, V, Saigo, M: Criteria for generalized fractional integrals to preserve univalency of analytic functions. C. R. Acad. Bulgare Sci.. 58, 1127–1134 (2005)

  4. Kiryakova, V: Criteria for univalence of the Dziok-Srivastava and the Srivastava-Wright operators in the class A. Appl. Math. Comput.. 218, 883–892 (2011). Publisher Full Text OpenURL

  5. Kiryakova, V: Generalized Fractional Calculus and Applications, Longman, Harlow (1994)

  6. Dziok, J, Srivastava, HM: Classes of analytic functions associated with the generalized hypergeometric function. Appl. Math. Comput.. 103, 1–13 (1999)

  7. Srivastava, HM: Some Fox-Wright generalized hypergeometric functions and associated families of convolution operators. Appl. Anal. Discrete Math.. 1, 56–71 (2007). Publisher Full Text OpenURL

  8. Srivastava, HM, Owa, S: Univalent Functions, Fractional Calculus, and Their Applications, Halsted, New York (1989)

  9. Ibrahim, RW: On generalized Srivastava-Owa fractional operators in the unit disk. Adv. Differ. Equ.. 2011, Article ID 55 (2011)

  10. Ibrahim, RW: On generalized Hyers-Ulam stability of admissible functions. Abstr. Appl. Anal.. 2012, Article ID 749084 (2012)

  11. Ibrahim, RW: Fractional complex transforms for fractional differential equations. Adv. Differ. Equ.. 2012, Article ID 98 (2012)

  12. Ibrahim, RW: On holomorphic solution for space- and time-fractional telegraph equations in complex domain. J. Funct. Spaces Appl.. 2012, Article ID 703681 (2012)

  13. Srivastava, HM, Darus, M, Ibrahim, RW: Classes of analytic functions with fractional powers defined by means of a certain linear operator. Integral Transforms Spec. Funct.. 22, 17–28 (2011). Publisher Full Text OpenURL

  14. Ibrahim, RW, Jalab, HA: Time-space fractional heat equation in the unit disk. Abstr. Appl. Anal.. 2013, Article ID 364042 (2013)

  15. Sokół, J, Piejko, K: On the Dziok-Srivastava operator under multivalent analytic functions. Appl. Math. Comput.. 177, 839–843 (2006). Publisher Full Text OpenURL

  16. Piejko, K, Sokół, J: Subclasses of meromorphic functions associated with the Cho-Kwon-Srivastava operator. J. Math. Anal. Appl.. 337, 1261–1266 (2008). Publisher Full Text OpenURL

  17. Sokół, J: On some applications of the Dziok-Srivastava operator. Appl. Math. Comput.. 201, 774–780 (2008). Publisher Full Text OpenURL

  18. Sivasubramanian, S, Sokół, J: Hypergeometric transforms in certain classes of analytic functions. Math. Comput. Model.. 54, 3076–3082 (2011). Publisher Full Text OpenURL

  19. Hussain, S, Sokół, J: On a class of analytic functions related to conic domains and associated with Carlson-Shaffer operator. Acta Math. Sci. Ser. B. 32, 1399–1407 (2012). Publisher Full Text OpenURL

  20. Bauer, F, Kraus, D, Roth, O, Wegert, E: Beurling’s free boundary value problem in conformal geometry. arXiv:0906.3139v2 [math.CV]

  21. Černe, M, Zajec, M: Boundary differential relations for holomorphic functions on the disc. Proc. Am. Math. Soc.. 139, 473–484 (2011). Publisher Full Text OpenURL

  22. Gamelin, TW: Complex Analysis, Springer, New York (2001)

  23. Gaboury, S, Tremblay, R: A note on some new series of special functions. Integral Transforms Spec. Funct. (2013). Publisher Full Text OpenURL

  24. Tremblay, R: Une contribution a la theorie de la derivee fractionnaire. PhD thesis, Laval University, Quebec, Canada (1974)

  25. De Branges, L: A proof of the Bieberbach conjecture. Acta Math.. 145, 137–152 (1985)

  26. Duren, PL: Univalent Functions, Springer, New York (1983)