AI 中文总结
本文研究了某些单值映射子类的遗漏值问题，引入了完全凸映射类CC_α，并推广了经典结果。
AI 中文摘要
我们研究了单位圆盘中属于几类共形映射的归一化解析函数\(f\)的\(\text{Re}\{a_2 f(z)\}\)的取值范围。作为主要贡献，我们引入了由一致两点星形性条件定义的\(\alpha\)阶完全凸映射类\(CC_\alpha\)，并针对所有\(f\in CC_\alpha\)和\(z\in \mathbb{D}\)，根据\(\alpha\)估计\(\text{Re}\{a_2f(z)\}\)的取值范围，推广了Fournier - Ma - Ruscheweyh的经典结果（当\(\alpha = 0\)时可恢复该结果）。我们还确定了\(\alpha\)阶凸函数、球面凸映射、一致星形函数和Nehari类\(\mathcal{N}_t\)的遗漏值集。证明主要依赖于应用于由两点核\(zf'(z)/(f(z)-f(x))\)构造的辅助函数的Schwarz - Pick引理。
英文摘要
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 $CC_α$ of completely convex mappings of order $α$, defined by a uniform two-point starlikeness condition, and we estimate the range of $\operatorname{Re}\{a_2f(z)\}$ in terms of $α$, for all $f\in CC_α$ and $z\in \mathbb{D}$, generalizing the classical result of Fournier--Ma--Ruscheweyh, which is recovered for $α=0$. We also determine omitted value sets for convex functions of order $α$, 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 $zf'(z)/(f(z)-f(x))$.
Comments13 pages
