A Unified Approach to Sharp Subordination Radii

Let $\phi$ and $\psi$ be univalent functions in the unit disk $\mathbb{D}$ satisfying $\phi (0)=\psi (0)=1$ and $\phi \prec ̸\psi$, and let ${\mathcal{P}}_{\phi }$ denote the class of analytic functions $p$ in $\mathbb{D}$ such that $p\prec \phi$. Under suitable analytic continuation, univalence, and boundary-behavior hypotheses on the branch of ${\phi }^{-1}$ satisfying ${\phi }^{-1}(1)=0$, we determine the sharp ${\mathcal{P}}_{\psi }$-radius of ${\mathcal{P}}_{\phi }$ as $$\mathcal{R}(\phi ,\psi )=\underset{\mathrm{\mid }\zeta \mathrm{\mid }=1}{\min }\mid {\phi }^{-1}(\psi (\zeta ))\mid \mathrm{.}$$ For Ma–Minda functions satisfying these hypotheses, the same radius is sharp for the associated classes $\mathcal{S}\mathcal{T}(\phi )$ and $\mathcal{C}\mathcal{V}(\phi )$ with respect to $\mathcal{S}\mathcal{T}(\psi )$ and $\mathcal{C}\mathcal{V}(\psi )$, respectively. We apply the general result to ${\phi }_{\mathrm{L}}(z)=\sqrt{1+z},{\phi }_{\mathrm{L}\mathrm{u}\mathrm{n}\mathrm{e}}(z)=z+\sqrt{1+z^2},{\phi }_{\mathrm{L}\mathrm{i}\mathrm{m}}(z)={(1+z\mathrm{/}\sqrt{2})}^2,$ for several choices of the Ma–Minda function $\psi$, and obtain parameter-dependent extensions for the corresponding generalized families. The resulting boundary minima are evaluated analytically.