Geometric properties of functions containing derivatives of Bessel function

Let \(J_\xi (z)\) $J_{\xi }(z)$ denotes the Bessel function of the first kind of order \(\xi .\) $\xi .$ For three different kinds of normalizations of \(N_\xi (z)=az^2J_\xi ^{\prime \prime }(z)+bzJ_\xi ^{\prime }(z)+cJ_\xi (z)\) $N_{\xi }(z)=a{z}^2{J}_{\xi }^{″}(z)+bz{J}_{\xi }^{\prime }(z)+c{J}_{\xi }(z)$ , we find the radius for Ma–Minda classes: \(\mathcal {S}^*\left( \varphi \right)\) ${\mathcal{S}}^{\ast }\left(\phi \right)$ -radii and \(\mathcal {C}\left( \varphi \right)\) $\mathcal{C}\left(\phi \right)$ -radii. We further establish the radii of \(\gamma\) $\gamma$ -spirallike of order \(\alpha\) $\alpha$ and convex \(\gamma\) $\gamma$ -spirallike of order \(\alpha\) $\alpha$ of these normalized functions. As an application of our results, we derive sufficient conditions for the \(N_\xi (z)\) $N_{\xi }(z)$ to be a member of the unified subclasses of starlike and convex functions. The obtained radii are sharp.