Summary: 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 \[ |a_na_m-a_{n+m-1}| \le (n-1)(m-1), \quad \text{for } n\ge 2,\, m\ge 2, \] 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? \[ |\lambda a_na_m-a_{n+m-1}| \le \lambda nm -n-m+1 \quad (n\ge 2, \,m\ge 3). \tag{0.1} \] 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 \(\lambda =2\), \(n=2\), \(m=3\).