On the generalized Zalcman conjecture

Abstract

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 $\lambda$ does the following inequality hold? \begin{equation}\label{conjecture} |\lambda a_na_m-a_{n+m-1}|\le \lambda nm -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 $\lambda=3, n=2, m=3$. Further, we provide a geometric condition on extremal function maximizing (\ref{conjecture}) for $\lambda=2,n=2, m=3$.