Omitted values for some subclasses of univalent mappings

Abstract

We study the range of $\operatorname{Re}\{a_2 f(z)\}$ for normalized analytic functions $f$ in the unit disk belonging to several classes of conformal mappings. As our main contribution, we introduce the class $CC_\alpha$ of completely convex mappings of order $\alpha$, defined by a uniform two-point starlikeness condition, and we estimate the range of $\operatorname{Re}\{a_2f(z)\}$ in terms of $\alpha$, for all $f\in CC_\alpha$ and $z\in \mathbb{D}$, generalizing the classical result of Fournier--Ma--Ruscheweyh, which is recovered for $\alpha=0$. We also determine omitted value sets for convex functions of order $\alpha$, spherically convex mappings, uniformly starlike functions, and Nehari classes $\mathcal{N}_t$. The proofs rely primarily on the Schwarz--Pick lemma applied to auxiliary functions constructed from the two-point kernel $zf'(z)/(f(z)-f(x))$.