Journal of Complex Analysis
Volume 2017 (2017), Article ID 6564705, 4 pages
https://doi.org/10.1155/2017/6564705

Research Article

Sufficient Condition for Strongly Starlikeness of Normalized Mittag-Leffler Function

Deepak Bansal¹ and Sudhananda Maharana²

¹Department of Mathematics, College of Engineering and Technology, Bikaner, Rajasthan 334004, India
²Department of Mathematics, Central University of Rajasthan, NH-8, Bandar Sindri, Kishangarh, Dist. Ajmer, Rajasthan 305801, India

Correspondence should be addressed to Deepak Bansal; deepakbansal_79@yahoo.com

Received 7 August 2017; Accepted 10 October 2017; Published 5 November 2017

Academic Editor: Pranay Goswami

Copyright © 2017 Deepak Bansal and Sudhananda Maharana. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In the present investigation a sufficient condition is obtained for normalized Mittag-Leffler function to be starlike and strongly starlike in the open unit disk. Results obtained are new and their usefulness is depicted by deducing several interesting corollaries and examples.

1. Introduction

Let denote the class of analytic functions in the open unit disk: and denote the subclass of , which are normalized by the condition and have representation of the following form: A function is said to be univalent in a domain if it is one to one in . A function is called starlike, denoted by if is univalent in and is a starlike domain with respect to the origin. For a given , a function is called starlike function of order , class of such function denoted by , if and only if , . Further, a function is called a convex function, if , . It is well known that . Further, let , , be the class of strongly starlike functions of order defined by Note that . For more details about these classes, one can refer to [1].

If , then the function is said to be subordinate to , written as , if there exists a Schwarz function with and () such that In particular, if is univalent in , then we have the following equivalence: The function was introduced by Mittag-Leffler (see [2]) in connection with the summing of divergent series. An important generalization, was introduced by Wiman [3, 4]. The Mittag-Leffler function has a close connection to differential equations of fractional order and integral equations of Abel type, such equations are becoming more and more popular in modelling natural and technical process [5–10]. The Mittag-Leffler function arises naturally in the solution of fractional order differential and integral equations and especially in the investigations of fractional generalization of kinetic equation, random walks, Lévy flights, and superdiffusive transport and in the study of complex systems. These functions interpolate between a purely exponential law and power-law like behavior of phenomena governed by ordinary kinetic equations and their fractional counterparts [6–8, 11]. The most essential properties of these entire functions, investigated by many mathematicians, can be found in [12, 13].

Observe that Mittag-Leffler function does not belong to the family Thus, it is natural to consider the following normalization of Mittag-Leffler function in : Whilst formula (7) holds for complex-valued , , in this paper we shall restrict our attention to the case of real-valued , . It is easy to see that satisfy the following relation:

Several researchers studied families of analytic functions involving special functions to find conditions such that the images of functions under have certain geometric properties like univalency, starlikeness, or convexity. In this context, many results are available in the literature regarding the hypergeometric functions [14], Bessel functions [15–17], Wright function [18], and Mittag-Leffler functions [19]. In this paper, we study some more properties of normalized Mittag-Leffler function .

2. Strongly Starlikeness

To prove results in this section, we shall need the following lemma.

Lemma 1 (Hallenbeck and Ruscheweyh [20]). Let be convex and univalent in with . Let be analytic in , and in . Then, for all , we have

Further, it is easy to see that, under the hypothesis , the inequality, , holds for all , which is equivalent to

Theorem 2. If and , then , where for

Proof. Using inequality (10), we get Note that, under the hypothesis, From (12), we conclude that which gives Using Lemma 1, for and with , we get Consequently, Now from (15) and (17), we conclude that that is, , for given in (11).

Corollary 3. Let and . If and where , then .

Proof. If we put from (19) to (11), we obtain .

Putting in Corollary 3, we get

Corollary 4. If and , where is positive root of , then .

Example 5. If and then From (11), we get and thus, from Theorem 2, we have

3. A Nonlinear Differential Equation

In order to prove results in this section, we required the following known results.

Lemma 6 (see [9]). Let . Suppose that the function satisfies the condition for all , , and . Also, let be an analytic function of the following form: such that for all . Then , where .

Lemma 7 (see [21]). Let be of form (2) which satisfies the inequality . Then is starlike in .

Theorem 8. For all , let the analytic function in satisfy the following inequality: and also let be the (unique) solution of the initial value problem for higher-order linear differential equation given by where (). Then the inequality holds.

Proof. Define by Clearly, function belongs to (23), and then by the related implicit function it immediately follows that and also denotes and by respectively. Then, clearly, Further, for any we have which gives Therefore, in view of Lemma 6, it follows that This completes the proof.

By taking in the above theorem we get the following.

Corollary 9. Let the analytic function in satisfy inequality (24), and also let be the (unique) solution of the initial value problem for third-order linear differential equation given by Then the inequality holds.

Corollary 10. If the analytic function in satisfies the following inequality and the function is the (unique) solution of the initial value problem for third-order linear differential equation given by (34), then .

Proof. By taking , in Corollary 9 (or in Theorem 8 with ), and then using Lemma 7, the proof can easily be obtained.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg, Germany, and Tokyo, 1983.

G. M. Mittag-Leffler, “Sur la nouvelle fonction Eα(x),” Comptes Rendus de l’Academie des Sciences Paris, vol. 137, pp. 554–558, 1903.

A. Wiman, “Über den Fundamental satz in der Theorie der Funcktionen, Eα(x),” Acta Mathematica, vol. 29, no. 1, pp. 191–201, 1905.

A. Wiman, “Über die Nullstellun der Funktionen Eα(x),” Acta Mathematica, vol. 29, no. 1, pp. 217–234, 1905a.

R. Gorenflo and S. Vessella, “Abel Integral Equations,” in Analysis and Applications, Springer-Verlag, Berlin, Heidelberg, Germany, New York, USA, and Tokyo, Japan, 1991.

R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, Singapore, Malaysia, 2000.

K. R. Lang, Astrophysical Formulae, vol. 1 of Radiation, Gas Processes and High-Energy Astrophysics, Revised edition, Springer-Verlag, New York, USA, 3rd edition, 1999a.

K. R. Lang, “Astrophysical Formulae,” in Space, Time, Matter and Cosmology, Springer-Verlag, New York, USA, 1999b.

S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Dekker, New York, NY, USA, 2000.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, USA, 1993.

R. K. Saxena, A. M. Mathai, and H. J. Haubold, “On fractional kinetic equations,” Astrophysics and Space Science, vol. 282, no. 1, pp. 281–287, 2002.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. 3, McGraw-Hill, New York, USA, 1955.

M. M. Dzherbashyan, Integral Transforms and Representations of Functions in the Complex Plane, Moscow, Nauka, 1966, in Russian.

S. S. Miller and P. T. Mocanu, “Univalence of Gaussian and confluent hypergeometric functions,” Proceedings of the American Mathematical Society, vol. 110, no. 2, pp. 333–342, 1990.

Á. Baricz and S. Ponnusamy, “Starlikeness and convexity of generalized Bessel functions,” Integral Transforms and Special Functions, vol. 21, no. 9-10, pp. 641–653, 2010.

Á. Baricz, P. A. Kupán, and R. Szász, “The radius of starlikeness of normalized Bessel functions of the first kind,” Proceedings of the American Mathematical Society, vol. 142, no. 6, pp. 2019–2025, 2014.

J. K. Prajapat, “Certain geometric properties of normalized Bessel functions,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2133–2139, 2011.

J. K. Prajapat, “Certain geometric properties of the Wright function,” Integral Transforms and Special Functions, vol. 26, no. 3, pp. 203–212, 2015.

D. Bansal and J. K. Prajapat, “Certain geometric properties of the Mittag-Leffler functions,” Complex Variables and Elliptic Equations. An International Journal, vol. 61, no. 3, pp. 338–350, 2016.

D. J. Hallenbeck and S. Ruscheweyh, “Subordination by convex functions,” Proceedings of the American Mathematical Society, vol. 52, pp. 191–195, 1975.

N. Tuneski, “On some simple sufficient conditions for univalence,” Mathematica Bohemica, vol. 126, pp. 229–236, 2001.