Inclusion of Bessel functions in a subclass of spiral functions

In this article, we derived conditions on the order $\nu$ of the classical Bessel functions $J_{\nu }$ that guarantee the inclusion of three distinct normalized forms of $J_{\nu }$ in a subclass of $\alpha$-spirallike functions. The primary goal was to determine the subintervals within $(-\frac{\pi }{2},\frac{\pi }{2})$ where these inclusion conditions are satisfied. A key component in establishing our results was the upper bound of the ratio $J_{\nu +1}(1)/J_{\nu }(1)$. The theoretical findings were validated through numerical experiments and accompanying graphical demonstrations.