On the Largest Disc Mapped by Sum of Convex and Starlike Functions

Abstract

For a normalized analytic function $f$ defined on the unit disc $𝔻$, let $\varphi (f,f^{\prime },f^{\mathrm{\prime \prime }};z)$ be a function of positive real part in $𝔻$, $\psi (f,f^{\prime },f^{\mathrm{\prime \prime }};z)$ need not have that property in $𝔻$, and $\chi =\varphi +\psi$. For certain choices of $\varphi$ and $\psi$, a sharp radius constant $\rho$ is determined, , so that $\chi (\rho z)/\rho$ maps $𝔻$ onto a specified region in the right half-plane.