Radii of starlikeness and convexity of regular Coulomb wave functions

Abstract

In this paper our aim is to determine the radii of univalence, starlikeness and convexity of the normalized regular Coulomb wave functions for two different kinds of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for regular Coulomb wave functions, and properties of zeros of the regular Coulomb wave functions and their derivatives. Moreover, by using the technique of differential subordinations we present some conditions on the parameters of the regular Coulomb wave function in order to have a starlike normalized form. In addition, by using the Euler-Rayleigh inequalities we obtain some tight bounds for the radii of starlikeness of the normalized regular Coulomb wave functions. Some open problems for the zeros of the regular Coulomb wave functions are also stated which may be of interest for further research.

Lemniscate and Exponential Starlikeness of Regular Coulomb Wave Functions

“…For these investigations the readers are referred to the papers [2][3][4][5][6]8,[11][12][13][14][15]19,[21][22][23]26,27,29,32] and the references therein. Also, the authors studied some geometric properties of regular Coulomb wave function in [9,10], while the author investigated the zeros of regular Coulomb wave functions and their derivatives in [16]. Motivated by the above works our main aim is to investigate the lemniscate and exponential starlikeness of regular Coulomb wave function by using differential subordination method.…”

“…The equation (1.4) has two linearly independent solutions that are called regular and irregular Coulomb wave functions. The regular Coulomb wave function F L,η (z) is defined by (see [10])…”

“…and 1 F 1 denotes the Kummer confluent hypergeometric function. It is known from [10] and [31] that the regular Coulomb wave function F L,η (z) has the following Weierstrassian canonical product representation:…”

Radii of starlikeness and convexity of Wright functions

“…Special functions are indispensable in many branches of mathematics and applied mathematics. Geometric properties of some special functions were examined by many authors (see [1,2,3,4,5,7,8,9,10,11,20,21,22,23,24,25]). However, its origins can be traced to Brown [12] (see also [13,14]), to Kreyszig and Todd [19] and to Wilf [26].…”

“…Recently, there has been a vivid interest on geometric properties of special functions such as Bessel, Struve, Lommel functions of the first kind and regular Coulomb wave functions. The first author and his collaborators examined in details the determination of the radii of starlikeness and convexity of some normalized forms of these special functions, see [1,2,3,4,5,7,8,9,10,11] and the references therein. Moreover, one of the most important things which we have learned in these studies is that the radii of univalence, starlikeness and convexity are obtained as solutions of some transcendental equations and the obtained radii satisfy some interesting inequalities.…”