The radii of starlikeness and convexity of the functions including derivatives of Bessel functions

Abstract

Let Jν(z) denote the Bessel function of the first kind of order ν. In this paper, our aim is to determine the radii of starlikeness and convexity for three kind of normalization of the function $N_ν(z) = az^2 J ′′ _ν (z) + bzJ′_ν(z) + cJ_ν(z)$ in the case where zeros are all real except for a single pair, which are conjugate purely imaginary. The key tools in the proof of our main results are the Mittag–Leffler expansion for function $N_ν(z)$and properties of real and complex zeros of it.