Majorization and Fekete-Szeg¨o Problems for Multivalent Meromorphic Functions Associated with the Mittag-Leffler Function

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(1. School of Mathematics and Statistics, Ningxia University, Yinchuan 750000, China; 2. School of Advanced Sciences, Vellore Institute of Technology, Deemed to be University, Vellore 632014, India;3. School of Mathematics and Computer Science, Ningxia Normal University, Guyuan 756099, China)

Abstract: The main object of this present paper is to investigate the problem of majorization of certain class of multivalent meromorphic functions of complex order involving Mittag-Leffler function. Meanwhile,for this subclass the corresponding coefficient estimates and some Fekete-Szeg¨o type inequalities are obtained. Moreover we point out some new or known consequences of our main results.

Keywords: Meromorphic function; Majorization problem; Hadamard product (convolution); Mittag-Leffler function; Fekete-Szeg¨o inequality

§1. Introduction

Denote byAthe class of functions whose elements are of the following form

which are analytic in the open unit disk Δ={z ∈C:|z|＜1}and normalized by the conditionsf(0)=0=f′(0)-1.

Letfandgbe analytic functions in the unit disc Δ={z ∈C:|z|＜1}.Due to MacGregor[16](also see [20]), we say thatfis majorized bygin Δ and write

where one denotes byS*(α,λ) the class ofλ-Spiral-like function of orderαinvestigated by Libera [14] and byS*(λ) Spiral-like functions introduced by ˘Spaˇcek [26] (see [23]).

LetPbe the class of all analytic functionsℓ(z) of the following form

satisfyingℜℓ(z)＞0 andℓ(0)=1.

The Mittag-Leffler function arises naturally in the solutions of fractional order differential and integral equations, and especially in the investigations of fractional generalization of kinetic equation, random walks, L´evy flights, super-diffusive transport and in the study of complex systems. Several properties of Mittag-Leffler function and generalized Mittag-Leffler function can be found,e.g. in[3,4,8–10,13]. Observe that Mittag-Leffler function Eα,β(z)does not belong to the classA. Therefore, it is natural to consider the following normalization of Mittag-Leffler functions as below :

which holds for complex parametersα, βandz ∈C.There has been a growing focus on Mittag-Leffler-type functions in recent years based on the growth of possibilities for their application for probability, applied problems, statistical and distribution theory, among others. In most of our work related to Mittag-Leffler functions, we study the geometric properties, such as the convexity, close-to-convexity and starlikeness. Recent studies on the Mittag-Leffler functionEα,β(z) can be seen in [21]. In fact, the function given by (1.6) is not within the class Σ. Based on the above reason, this special function is then normalized as follows:

§2. A majorization problem for the class τ,A,B)

§3. Fekete-Szeg¨o inequalities for the function class (τ,A,B)

§4. Concluding remarks

Up to this step we have introduced a new meromorphic function subclass which is related to the Mittag-Leffler function and obtain the results on majorization and Fekete-Szeg¨o problems.As for further research we could discuss sufficient and necessary conditions in relation to this subclass. Besides, linear combinations, distortion theory and another properties can be also explored.