论文标题
Zalcman的猜想，用于某些分析和单价功能
Zalcman Conjecture for certain analytic and univalent functions
论文作者
论文摘要
令$ \ MATHCAL {A} $表示单位磁盘$ \ MATHBB {d} $中的分析函数类别的类别函数，$ f（z）= z+\ sum_ {n = 2}^{\ infty} a_n z^n $ and $ n ostions $ f cling o functions （{\ it，即}，一对一）。在1960年代，L。Zalcman推测$ | a_n^2-a_ {2n-1} | \ le（n-1）^2 $ for $ n \ ge 2 $，这意味着著名的bieberbach猜想$ | a_n | a_n | \ le n $ for $ n \ ge 2 $。对于$ f \ in \ Mathcal {s} $，ma \ cite {ma-b-1999}提出了一个广义的zalcman comenture $$ | a_ {n} a_ {m} a_ {m} -a_ {n+m-1} | \ le（n-1）（n-1）（m-1）$ n \ n \ ge ge 2，m \ ge 2 $ \ ge 2 $ 2 $。令$ \ mathcal {u} $为\ Mathcal {a} $满足$ f \ in \ mathcal {a} $的类别类别$$和$ \ MATHCAL {f} $表示功能类$ f \ in \ Mathcal {a} $满足$ {\ rm re \，}（1-Z）^{2} f'（z）> 0 $ in $ \ \ sathbb {d} $。在本文中，我们证明了Zalcman的猜想和使用Extreme Point理论的类$ \ Mathcal {U} $的Zalcman猜想。我们还证明了Zalcman的猜想和广义的Zalcman猜想，用于$ \ Mathcal {F} $的初始系数。
Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ and $\mathcal{S}$ denote the class of functions $f\in\mathcal{A}$ which are univalent ({\it i.e.}, one-to-one). In 1960s, L. Zalcman conjectured that $|a_n^2-a_{2n-1}|\le (n-1)^2$ for $n\ge 2$, which implies the famous Bieberbach conjecture $|a_n|\le n$ for $n\ge 2$. For $f\in \mathcal{S}$, Ma \cite{Ma-1999} proposed a generalized Zalcman conjecture $$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1) $$ for $n\ge 2, m\ge 2$. Let $\mathcal{U}$ be the class of functions $f\in\mathcal{A}$ satisfying $$ \left|f'(z)\left(\frac{z}{f(z)}\right)^2-1 \right|< 1 \quad\mbox{ for } z\in\mathbb{D}. $$ and $\mathcal{F}$ denote the class of functions $f\in \mathcal{A}$ satisfying ${\rm Re\,}(1-z)^{2}f'(z)>0$ in $\mathbb{D}$. In the present paper, we prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{U}$ using extreme point theory. We also prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{F}$ for the initial coefficients.
