论文标题
通用Zalcman猜想的证明，以实现单价函数的初始系数
Proof of The Generalized Zalcman Conjecture for Initial Coefficients of Univalent Functions
论文作者
论文摘要
令$ \ MATHCAL {S} $表示分析和单价（{\ it i.e。}，一对一个）函数$ f（z）= z+s+sum_ {n = 2}^{\ infty} a_n z^n $在单位disk $ \ mathbb中\ Mathbb {C}：| z | <1 \} $。对于$ f \ in \ nathcal {s} $，ma提出了广义的zalcman猜想，即$$ | a_ | a_ {n} a_ {m} -a_ {n+m-1} | \ le（n-1）（n-1）（m-1）（m-1），\，\，\，\，\，\，\，\，\，\，\，\，\ ge2 n \ ge2 n \ ge2 n \ ge2，\， $ k（z）= z/（1 -z）^2 $及其旋转。在本文中，使用Holomorphic Motion的特性和Krushkal的手术引理\ Cite {Krushkal-1995}，当$ n = 2 $，$ m = 3 $和$ n = 2 $，$ m = 4 $时，我们证明了广义的Zalcman猜想。
Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, Ma proposed the generalized Zalcman conjecture that $$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1),\,\,\,\mbox{ for } n\ge2,\, m\ge 2,$$ with equality only for the Koebe function $k(z) = z/(1 - z)^2$ and its rotations. In this paper using the properties of holomorphic motion and Krushkal's Surgery Lemma \cite{Krushkal-1995}, we prove the generalized Zalcman conjecture when $n=2$, $m=3$ and $n=2$, $m=4$.
