On the generalized Zalcman conjecture

Let \(\mathcal {S}\) denote the class of analytic and univalent (i.e., one-to-one) functions \( f(z)= z+\sum _{n=2}^{\infty }a_n z^n\) 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

$$\begin{aligned}|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1),\,\,\, \text{ for } n\ge 2,\, m\ge 2,\end{aligned}$$

with equality only for the Koebe function \(k(z) = z/(1 - z)^2\) and its rotations. In the same paper, Ma (J Math Anal Appl 234:328–339, 1999) asked for what positive real values of \(\lambda \) does the following inequality hold?

$$\begin{aligned} |\lambda a_na_m-a_{n+m-1}|\le \lambda nm -n-m+1 \,\,\,\,\, (n\ge 2, \,m\ge 3). \end{aligned}$$

(0.1)

Clearly equality holds for the Koebe function \(k(z) = z/(1 - z)^2\) and its rotations. In this paper, we prove the inequality (0.1) for \(\lambda =3, n=2, m=3\). Further, we provide a geometric condition on extremal function maximizing (0.1) for \(\lambda =2,n=2, m=3\).