Geometric Properties of function $az^{2}J_{\nu }^{\prime \prime }(z)+bzJ_{\nu }^{\prime}(z)+cJ_{\nu }(z)$

TL;DR

This paper determines the radii of starlikeness and convexity for normalized combinations of Bessel functions, using zeros properties and inequalities to establish bounds and evaluate sums of zeros.

Contribution

It introduces new bounds and properties for the geometric behavior of a class of Bessel-related functions, extending understanding of their starlikeness and convexity.

Findings

Derived radii of starlikeness and convexity for the normalized functions.

Established tight bounds using Euler-Rayleigh inequalities.

Evaluated sums of zeros for the function $N_ u(z)$.

Abstract

In this paper our aim is to find the radii of starlikeness and convexity for three different kind of normalization of the $N_\nu(z)=az^{2}J_{\nu }^{\prime \prime }(z)+bzJ_{\nu }^{\prime}(z)+cJ_{\nu }(z)$ function, where $J_\nu(z)$ is called the Bessel function of the first kind of order $\nu.$ The key tools in the proof of our main results are the Mittag-Leffler expansion for $N_\nu(z)$ function and properties of real zeros of it. 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 normalized $N_\nu(z)$ function. Finally, we evaluate certain multiple sums of the zeros for $N_\nu(z)$ function.