Radii of starlikeness and convexity of generalized Struve functions

Abstract

The main purpose of this paper is to determine the radii of starlikeness and convexity of the generalized k−Bessel functions for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre-Pólya class plays an crucial role in this paper. Moreover, the interlacing properties of the zeros of k−Bessel function and its derivative is also useful in the proof of the main results. By making use of the Euler-Rayleigh inequalities for the real zeros of the generalized k−Bessel function, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero.

On some properties of generalized Struve function

“…In this section, we are going to discuss some properties like monotonicity and log-concavity of the generalized Struve function by using its product representation which is known as Hadamard product or Weierstrassian decomposition. The generalized Struve function has the following series representation (see [12]):…”

“…for q 2 N; p; b; c 2 C and > 0. The author studied some geometric properties such as starlikeness and convexity of generalized Struve function in [12]. Also, the author showed that the zeros of the generalized Struve function are all real.…”

On monotonic and logarithmic concavity properties of generalized $k$-Bessel function

“…Comprehensive information about the log-concavity and the log-convexity properties can be found in [13] and its references. In this study, motivated by the some earlier results which are given in [14,15], our main aim is to present some monotonic and log-concavity properties of generalized k-Bessel functions. Moreover, we give some specific examples regarding our obtained result by using the relationships between Bessel-type functions and elementary trigonometric and hyperbolic functions.…”

“…where J ν (x) and I ν (x) denote classical Bessel and modified Bessel functions of the first kind, respectively. In [15], the author studied some geomertric properties such as radii of starlikeness and convexity of generalized k-Bessel function. Also, the author gave an infinite product representation of generalized k-Bessel function by using Hadamard's theorem as follow (see [15, Lemma 1.1]):…”