Omitted values for some subclasses of univalent mappings

We study the range of $\mathrm{Re}\{a_{2}f(z)\}$ for normalized analytic functions $f$ in the unit disk belonging to several classes of conformal mappings. As our main contribution, we introduce the class $C{C}_{\alpha }$ of completely convex mappings of order $\alpha$, defined by a uniform two-point starlikeness condition, and we estimate the range of $\mathrm{Re}\{a_{2}f(z)\}$ in terms of $\alpha$, for all $f\in C{C}_{\alpha }$ 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 $z{f}^{\mathrm{\prime }}(z)\mathrm{/}(f(z)-f(x))$.