Sharp Bounds for Sums of Coefficients of Inverses of Convex Functions

Let D denote the open unit disc and f: D → ℂ be holomorphic and injective in D such that f(D) is a convex domain and f(0) = f’(0) − 1 = 0. Let F be the inverse function of f defined in a neighbourhood of the origin and k ∈ ℕ. We consider the Taylor expansions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(F(w))^k=\sum_{n=k}^{\infty} A_{n,k}w^{n}$$\end{document}. We prove that the inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\biggr|\sum_{k=1}^nA_{n,k}\biggr| \leq 2^{n-1}$$\end{document} is valid for any n ∈ ℕ and that equality occurs in this inequality for a fixed n ≥ 2 if and only if f(z) = z/(1 + z).