arXiv 2006.077832020-06-14math.CV

一类解析单叶函数的Zalcman猜想

Zalcman Conjecture for certain analytic and univalent functions

Vasudevarao Allu, Abhishek Pandey

备注：: 15 pages

摘要：

令$\mathcal{A}$表示形式为$f(z)= z+\sum_{n=2}^{\infty}a_n z^n$的单位圆盘$\mathbb{D}$中的解析函数类，$\mathcal{S}$表示单叶函数类$f\in\mathcal{A}$（{ \it即}，一对一）。20世纪60年代，L. Zalcman证明了$|a_n^2-a_{2n-1}|\le (n-1)^2$ for $n\ge 2$，这暗示了著名的Bieberbach猜想$|a_n|\le n$ for $n\ge 2$ .对于$f\in \mathcal{S}$，Ma \cite{Ma-1999}对$n\ge 2, m\ge 2$提出了一个推广的Zalcman猜想$$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1) $$.令$\mathcal{U}$是满足$$ \left|f'(z)\left(\frac{z}{f(z)}\right)^2-1 \right|< 1 \quad\mbox{ for } z\in\mathbb{D}. $$的函数$f\in\mathcal{A}$的类，并且$\mathcal{F}$表示在$\mathbb{D}$中满足${\rm Re\,}(1-z)^{2}f'(z)>0$的函数$f\in \mathcal{A}$的类。本文利用极点理论证明了$\mathcal{U}$类的Zalcman猜想和广义Zalcman猜想。我们还证明了Zalcman猜想和广义Zalcman猜想类$\mathcal{F}$的初始系数。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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