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

Abstract

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,...

Close‐to‐convexity of normalized Dini functions

“…By using Lemma and the idea of the proof of Theorem our aim is to present the following interesting sharp result. We note that some similar results were proved for Bessel functions of the first kind in , , , , but by using different approaches. Theorem The function is starlike and all of its derivatives are close‐to‐convex (and hence univalent) in if and only if where is the unique root of the transcendent equation on (0, ∞).…”

“…By using Lemma 1.1 and the idea of the proof of Theorem 1.2 our aim is to present the following interesting sharp result. We note that some similar results were proved for Bessel functions of the first kind in [4], [7], [8], [18], but by using different approaches.…”

“…Now, consider the function defined by where stands for the Bessel function of the first kind (see [, p. 217] or ). Very recently by using a result of Shah and Trimble [, Theorem 2] in the authors proved that the function and all of its derivatives are convex in if and only if where is the unique root of the equation on We note that in view of Alexander's duality theorem the first part of the above result is equivalent to the fact that the function …”

“…where J ν stands for the Bessel function of the first kind (see [16, p. 217] or [20]). Very recently by using a result of Shah and Trimble [17,Theorem 2] in [8] the authors proved that the function f ν and all of its derivatives are convex in D if and only if ν ≥ ν , where ν −0.1438 . .…”

Hardy Space of Lommel Functions

“…Furthermore, very recently Baricz and Szász [6] considered the starlikeness and close-toconvexity of the derivatives of a normalized form of s µ− . In this section, we get some geometric properties of the function h µ,v (z) which normalized Lommel functions of the first kind s µ,v (z).…”