Radii of Starlikeness of Spirallike Functions

A normalized analytic function $f$ defined on the unit disk $\mathbb{D}$ is called $\alpha$-spirallike if $f(\mathbb{D})$ is an $\alpha$-spirallike domain, that is, for every $z\in f(\mathbb{D})$, the $\alpha$-spiral $z{e}^{-e^{i\alpha }t}$, $t\ge 0$, is contained in $f(\mathbb{D})$, or equivalently, if $\mathrm{Re}\text{\hspace{0.17em}⁣}\left(e^{i\alpha }z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)\right)>0$ for all $z\in \mathbb{D}$. The 0-spirallike functions are the usual starlike functions. The function $f$ is parabolic starlike, if $\mathrm{Re}\text{\hspace{0.17em}⁣}\left(\frac{z{f}^{\mathrm{\prime }}(z)}{f(z)}\right)>\mid \frac{z{f}^{\mathrm{\prime }}(z)}{f(z)}-1\mid$, and lemniscate starlike if $\mathrm{\mid }(z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z){)}^2-1\mathrm{\mid }<1$ respectively. In this paper, we determine the radii of parabolic starlikeness and lemniscate starlikeness of $\alpha$-spirallike functions. Our approach combines geometric characterizations of the underlying regions with the resultant method for eliminating auxiliary variables, leading to explicit equations for the sharp radii. Some known radius results are recovered as special cases.