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.

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

Abstract. In this paper our aim is to deduce some sufficient (and necessary) conditions for the closeto-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of the first kind, which can be expressed in terms of the hypergeometric function 1 F 2 . The key tool in our proofs is a result of Shah and Trimble about transcendental entire functions with univalent derivatives. Moreover, a known result of Pólya on entire functions, the infinite product representations and some results on zeros of Bessel, Struve and Lommel functions of the first kind are used in order to achieve the main results of the paper. Introduction and Main ResultsSpecial functions play an important role in pure and applied mathematics. Bessel functions of the first kind are among of the special functions which were studied by many authors from many different points of view. The geometric properties, like univalence, starlikeness, spirallikeness and convexity were studied already in the sixties by Brown [9,10,11], and also by Kreyszig and Todd [13]. 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. In this paper we make a contribution to the subject by showing some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of the first kind, which can be expressed in terms of the hypergeometric function 1 F 2 . In order to prove our main results we use a result of Shah and Trimble [17, Theorem 2] about transcendental entire functions with univalent derivatives. We use also a well-known result of Pólya on entire functions, and the Weierstrass product representations and some results on zeros of Bessel, Struve and Lommel functions of the first kind are used in order to achieve the main results of the paper. The paper is organized as follows. In this section we recall the result of Shah and Trimble together with the definitions of Bessel, Struve and Lommel functions. Moreover, at the end of this section we present the main results of this paper. Section 2 contains the proofs of these results.The Bessel function of the first kind J ν , the Struve function of the first kind H ν , and the Lommel function of the first kind s µ,ν , are particular solutions of the Bessel differential equation [15, p. 217] (1.1)and the inhomogeneous Bessel differential equation [15,...

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The radii of α-convexity are deduced for three different kinds of normalized Bessel functions of the first kind and it is shown that these radii are between the radii of starlikeness and convexity, when α ∈ [0, 1], and they are decreasing with respect to the parameter α. The results presented in this paper unify some recent results on the radii of starlikeness and convexity for normalized Bessel functions of the first kind.The key tools in the proofs are some interlacing properties of the zeros of some Dini functions and the zeros of Bessel functions of the first kind.