Bounds for radii of starlikeness and convexity of some special functions

Abstract: In this paper we consider some normalized Bessel, Struve and Lommel functions of the first kind, and by using the Euler-Rayleigh inequalities for the first positive zeros of combination of special functions we obtain tight lower and upper bounds for the radii of starlikeness of these functions. By considering two different normalization of Bessel and Struve functions we give some inequalities for the radii of convexity of the same functions. On the other hand, we show that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. In addition, by using some ideas of Ismail and Muldoon we present some new lower and upper bounds for the zeros of derivatives of some normalized Struve and Lommel functions. The Laguerre-P\'olya class of real entire functions plays an important role in our study.