On the generalized Zalcman conjecture

Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $f(z)=z+{\sum }_{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$. For $f\in \mathcal{S}$, In 1999, Ma proposed the generalized Zalcman conjecture that $$\mathrm{\mid }{a}_n{a}_m-a_{n+m-1}\mathrm{\mid }\le (n-1)(m-1),forn\ge 2,m\ge 2,$$ with equality only for the Koebe function $k(z)=z\mathrm{/}(1-z{)}^2$ 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} |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 $k(z)=z\mathrm{/}(1-z{)}^2$ 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 $\lambda =2,n=2,m=3$.