Anuja Prajapati | Searcharxiv

arXiv2020

Radii problems for the generalized Mittag-Leffler functions

In this paper our aim is to investigate the radii of $η-$uniformly convexity, $α-$convexity, $η-$parabolic starlikeness and strong starlikeness of order $ρ$ of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic. The basic tool of this study is the Mittag-Leffler function in series. Also we have shown that the obtained radii are the smallest positive roots of some functional equations.

math.CV↗

arXiv2019

Radii of starlikeness and convexity of generalized Mittag-Leffler functions

In this paper our aim is to find the radii of starlikeness and convexity of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre-Pólya class and a result of H. Kumar and M.A. Pathan on the reality of the zeros of generalized Mittag-Leffler functions, which origins goes back to Dzhrbashyan, Ostrovskiĭ and Peresyolkova, play important roles in this paper. Moreover, the interlacing properties of the zeros of Mittag-Leffler function and its derivative is also useful in the proof of the main results. By using the Euler-Rayleigh inequalities for the real zeros of the generalized Mittag-Leffler function, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero.

math.CV↗