Some properties of analytic functions involving the Mittag-Leffler function
Abstract
The Mittag-Leffler function was defined by Swedish mathematican Magnus Gustav Mittag-Leffler in 1903. Later, researchers generalized this function by including different parameters. In 2015, Bansal and Prajabat normalized the Mittag-Leffler function and get several sufficient conditions so that the Mittag-Leffler function has certain geometric properties such as univalency, starlikeness, convexity and close-to-convexity in the open unit disc. After this research paper, the Mittag-Leffler function became popular in the studies of univalent functions theory. In this current study, we define a new class of analytic functions involving the Mittag-Leffler function denoted by〖 S〗_(α,β)^γ (k,A,B). We also introduce a subclass of this function class, which is involving negative coefficients. We introduce coefficient estimates, growth and distortion theorems for this function class. Moreover, we obtain integral mean inequalities for this class. We also conclude that for special values of parameters, the classes introduced in this paper are reduced to the several function classes which are defined by researchers.
Keywords
Analytic functions , Coefficient inequality , Starlike functions
Kaynakça
- Bansal, D. and Prajapat, J. K. (2016). Certain geometric properties of the Mittag-Leffler functions. Complex Variables and Elliptic Equations, 61(3), 338-350. https://doi.org/10.1080/17476933.2015.1079628
- Bharati, R., Parvatham, R. and Swaminathan, A. (1997). On subclasses of uniformly convex functions and corresponding class of starlike functions. Tamkang Journal of Mathematics, 28(1), 17-32.
- Duren, P. L. (1983). Univalent Functions. Springer, 259, XIV- 384.
- Goodman, A.W. (1991). On Uniformly Starlike Functions. Journal of Mathematical Analysis and Applications, 155(2), 364-370.
- Gorenflo, R., Kilbas, A.A., Mainardi, F. and Rogosin, S. (2014). Mittag-Leffler functions, related topics and applications. Springer, XIV- 443. https://doi.org/10.1007/978-3-662-43930-2
- Janowski, W. (1973). Some Extremal problems for certain families of analytic functions I. Annales Polonici Mathematici, 28, 297–326. https://doi.org/10.4064/ap-28-3-297-326
- Kanas, S. and Wisniowska, A. (2000). Conic domains and starlike functions. Revue Roumaine des Mathematiques Pures et Appliquees, 45(4), 647-657.
- Littlewood, J. E. (1925). On ınequalities in the theory of functions. Proceedings of the London Mathematical Society, 23(2), 481-519. https://doi.org/10.1112/plms/s2-23.1.481
- Mittag-Leffler, G. (1903). Sur la Nouvelle Fonction Eα(x). Comptes rendus de l'Académie des sciences Paris, 137, 554-558.
- Prabhakar, T. R. (1971). A singular ıntegral equation with a generalized Mittag-Leffler function in the Kernel. Yokohama Mathematical Journal, 19, 7-15.