Geometric and monotonic properties of Ramanujan type entire functions

Abstract

In this paper, our aim is to find the radii of starlikeness and convexity of the Ramanujan type function for three different kinds of normalization by using their Mittag–Leffler expansion in such a way that the resulting functions are analytic in the unit disk of the complex plane. A result of Zhang (Proc Am Math Soc 145:241–250, 2017) on the reality of the zeros of Ramanujan type entire functions play important roles in this paper. Moreover, the interlacing properties of the zeros of Ramanujan type functions and its derivative are also useful in the proof of the main results. In addition, by using the Euler–Rayleigh inequalities, we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the Ramanujan type entire functions. Finally, we give monotonicity and Redheffer-type inequalities for this function.