On the generalized Zalcman conjecture

Let \(\mathcal {S}\) $\mathcal{S}$ denote the class of analytic and univalent (i.e., one-to-one) functions \( f(z)= z+\sum _{n=2}^{\infty }a_n z^n\) $f(z)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$ in the unit disk \(\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}\) $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ . For \(f\in \mathcal {S}\) $f\in \mathcal{S}$ , In 1999, Ma proposed the generalized Zalcman conjecture that \(\begin{aligned}|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1),\,\,\, \text{ for } n\ge 2,\, m\ge 2,\end{aligned}\) $$\begin{array}{c}\hfill |a_n{a}_m-a_{n+m-1}|\le (n-1)(m-1),\text{for}n\ge 2,m\ge 2,\end{array}$$ with equality only for the Koebe function \(k(z) = z/(1 - z)^2\) $k(z)=z/{(1-z)}^2$ and its rotations. In the same paper, Ma (J Math Anal Appl 234:328–339, 1999) asked for what positive real values of \(\lambda \) $\lambda$ does the following inequality hold? 0.1 \(\begin{aligned} |\lambda a_na_m-a_{n+m-1}|\le \lambda nm -n-m+1 \,\,\,\,\, (n\ge 2, \,m\ge 3). \end{aligned}\) $$\begin{array}{c}\hfill |\lambda a_n{a}_m-a_{n+m-1}|\le \lambda nm-n-m+1(n\ge 2,m\ge 3).\end{array}$$ Clearly equality holds for the Koebe function \(k(z) = z/(1 - z)^2\) $k(z)=z/{(1-z)}^2$ and its rotations. In this paper, we prove the inequality (0.1) for \(\lambda =3, n=2, m=3\) $\lambda =3,n=2,m=3$ . Further, we provide a geometric condition on extremal function maximizing (0.1) for \(\lambda =2,n=2, m=3\) $\lambda =2,n=2,m=3$ .