  For the standard Ma-Minda class $\mathcal{S}^{*}(\psi)$ of univalent starlike
functions, we derive $\mathcal{S}^{*}(\psi)$-radii for some well-known special
functions. In addition, we obtain the set of extremal functions for the
classical problem
  $$\max_{f\in
\mathcal{S}^{*}(\psi)}^{}\left|\Phi\left(\log{(f(z)/z)}\right)\right| \quad
\text{or} \quad \max_{f\in
\mathcal{S}^{*}(\psi)}^{}\Re\Phi\left(\log{(f(z)/z)}\right),$$ where $\Phi$ is
a non-constant entire function. Moreover, we prove certain results on
convolution and radius estimates for the case when $\psi(\mathbb{D})$ is
starlike.
