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

Abstract

In this paper we determine the radius of convexity for three kind of normalized Bessel functions of the first kind. In the mentioned cases the normalized Bessel functions are starlike-univalent and convex-univalent, respectively, on the determined disks. The key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivative, and the fact that the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind. Moreover, we find the optimal parameters for which these normalized Bessel functions are convex in the open unit disk. In addition, we disprove a conjecture of Baricz and Ponnusamy concerning the convexity of the Bessel function of the first kind.Comment: 16 pages; an error has been correcte

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

“…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.…”

“…We note that the results of Theorem 1 are sharp. Moreover, it is worth to mention that Szász [19,Theorem 6] deduced already the starlikeness of f ν , while Baricz and Szász [8,Theorem 6] deduced already the convexity of f ν , however, our approach is much easier and as we can see below is applicable also for Struve and Lommel functions. Moreover, in the above theorems we have also information on the closeto-convexity or convexity of the derivatives of the Bessel, Struve and Lommel functions, respectively.…”

“…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.…”

Monotonicity properties of some Dini functions

“…Recently, there has been a vivid interest on geometric properties of Bessel functions, like univalence, starlikeness, convexity, close-to-convexity in the open unit disk. For more details we refer to the paper [2] and to the references therein. In [2], in order to determine the radius of convexity of a normalized Bessel function, Baricz and Szász proved a Mittag-Leffler expansion for a special combination of Bessel functions, called sometimes as Dini function.…”

“…For more details we refer to the paper [2] and to the references therein. In [2], in order to determine the radius of convexity of a normalized Bessel function, Baricz and Szász proved a Mittag-Leffler expansion for a special combination of Bessel functions, called sometimes as Dini function. In this note our aim is to deduce an infinite product representation for this Dini function and to apply this result in order to generate some absolute monotonicity properties of some quotients of Dini functions.…”