Geometric Properties of function $a{z}^2{J}_{\nu }^{\mathrm{\prime }\mathrm{\prime }}(z)+bz{J}_{\nu }^{\mathrm{\prime }}(z)+c{J}_{\nu }(z)$

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 $\nu \mathrm{.}$ 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.