On the generalized Zalcman conjecture

Let 𝒮 denote the class of analytic and univalent (i.e., one-to-one) functions f(z)= z+∑ _n=2^∞a_n z^n in the unit disk 𝔻={z∈ℂ:|z|<1} . For f∈𝒮 , In 1999, Ma proposed the generalized Zalcman conjecture that |a_na_m-a_n+m-1|≤ (n-1)(m-1), for n≥ 2, m≥ 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 λ does the following inequality hold? 0.1 |λ a_na_m-a_n+m-1|≤λ nm -n-m+1 (n≥ 2, m≥ 3). 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 λ =3, n=2, m=3 . Further, we provide a geometric condition on extremal function maximizing (0.1) for λ =2,n=2, m=3 .