The radius of starlikeness of normalized Bessel functions of the first kind

Abstract

In this note our aim is to determine the radius of starlikeness of the normalized Bessel functions of the first kind for three different kinds of normalization. The key tool in the proof of our main result is the Mittag-Leffler expansion for Bessel functions of the first kind and the fact that, according to Ismail and Muldoon [IM2], the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind.File: starlikeness.tex, printed: 2018-011-05, 0.51 2010 Mathematics Subject Classification. 30C45, 33C10. Key words and phrases. Bessel and modified Bessel functions of the first kind; univalent, starlike and convex functions.A.

The Radius of $$\alpha $$ α -Convexity of Normalized Bessel Functions of the First Kind

“…The case α = 0 was proved already in [4]. Combining where β ν,n stands for the nth positive zero of the Dini function z → (2 − ν)J ν (z) + z J ν (z), it follows that…”

“…We also mention that the univalence, starlikeness and convexity of Bessel functions of the first kind were studied extensively in several papers. We refer to [1][2][3][4][5][6][7][8][9]11,15,16] and to the references therein.…”

Close-to-Convexity of Some Special Functions and Their Derivatives

“…These problems remain open and are subject of further research. As we can see in the case of the Bessel functions of the first kind [5,8,19] these kind of problems are not easy to handle, since require a lot of information about the zeros of Bessel functions. But, the zeros of Struve and Lommel functions are not much studied; for example, there is no formula yet for their derivative with respect to the order, which would be an useful source in the study of the geometric properties of Struve and Lommel functions.…”

“…However, many important problems of Bessel functions, like determining the radius of starlikeness, and the radius of convexity, or finding the optimal parameter for which the normalized Bessel function of the first kind will be starlike, convex, or close-to-convex, have not been studied in details or have not been solved completely. Some of these problems have been studied later in the papers [1,2,3,5,6,8,19,20], however, there are still some open problems in this direction. For example, there is no information about the close-to-convexity or univalence of the derivatives of Bessel functions, or other special functions.…”

Sufficient Condition for Strongly Starlikeness of Normalized Mittag-Leffler Function

“…In this context, many results are available in the literature regarding the hypergeometric functions [14], Bessel functions [15][16][17], Wright function [18], and Mittag-Leffler functions [19]. In this paper, we study some more properties of normalized Mittag-Leffler function E , .…”