On the generalized Zalcman conjecture

Let S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}$$\end{document} denote the class of analytic and univalent (i.e., one-to-one) functions f(z)=z+∑n=2∞anzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f(z)= z+\sum _{n=2}^{\infty }a_n z^n$$\end{document} in the unit disk D={z∈C:|z|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}$$\end{document}. For f∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \mathcal {S}$$\end{document}, In 1999, Ma proposed the generalized Zalcman conjecture that |anam-an+m-1|≤(n-1)(m-1),forn≥2,m≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1),\,\,\, \text{ for } n\ge 2,\, m\ge 2,\end{aligned}$$\end{document}with equality only for the Koebe function k(z)=z/(1-z)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(z) = z/(1 - z)^2$$\end{document} and its rotations. In the same paper, Ma (J Math Anal Appl 234:328–339, 1999) asked for what positive real values of λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} does the following inequality hold? 0.1|λanam-an+m-1|≤λnm-n-m+1(n≥2,m≥3).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\lambda a_na_m-a_{n+m-1}|\le \lambda nm -n-m+1 \,\,\,\,\, (n\ge 2, \,m\ge 3). \end{aligned}$$\end{document}Clearly equality holds for the Koebe function k(z)=z/(1-z)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k(z) = z/(1 - z)^2$$\end{document} and its rotations. In this paper, we prove the inequality (0.1) for λ=3,n=2,m=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =3, n=2, m=3$$\end{document}. Further, we provide a geometric condition on extremal function maximizing (0.1) for λ=2,n=2,m=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =2,n=2, m=3$$\end{document}.