论文标题
在广义的Zalcman猜想上
On the generalized Zalcman conjecture
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 \} $。对于1999年的$ f \ in \ Mathcal {s} $，MA提出了$$ | a_ | a_ {n} a_ {m} -a__ {n+m-1} | \ le（n-1）（n-1）（m-1）（m-1），\，\，\，\，\，\，\，\ ge2 koebe函数$ k（z）= z/（1 -z）^2 $及其旋转。在同一篇论文中，ma \ cite {ma-1999}询问$λ$的积极真实值以下不平等能力吗？ \ begin {equination} \ label {conjecture} |λa_na_m-a___ {n+m-1} | \leλnm-n-m+1 \ 1 \，\，\，\，\，\，\，（n \ ge ge 2，\ \ \ \ \ \，m \ ge3）。 \ end {equation}显然，koebe函数$ k（z）= z/（1 -z）^2 $及其旋转。在本文中，我们证明了$λ= 3，n = 2，m = 3 $的不等式（\ ref {sumpture}）。此外，我们为$λ= 2，n = 2，m = 3 $提供了最大化的极端函数（\ ref {undienture}）的几何条件。
Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $ f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, In 1999, Ma proposed the generalized Zalcman conjecture that $$|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 the same paper, Ma \cite{Ma-1999} asked for what positive real values of $λ$ does the following inequality hold? \begin{equation}\label{conjecture} |λa_na_m-a_{n+m-1}|\le λnm -n-m+1 \,\,\,\,\, (n\ge 2, \,m\ge3). \end{equation} Clearly equality holds for the Koebe function $k(z) = z/(1 - z)^2$ and its rotations. In this paper, we prove the inequality (\ref{conjecture}) for $λ=3, n=2, m=3$. Further, we provide a geometric condition on extremal function maximizing (\ref{conjecture}) for $λ=2,n=2, m=3$.
