Computational Methods and Function Theory 7 (2007), No. 1, 105--109
Copyright Heldermann Verlag 2007
Farit G. Avkhadiev
farit.avhadiev@ksu.ru , Kazan State University, Chebotarev Research Institute, 420008 Kazan, Russia.
Karl-Joachim Wirths
kjwirths@tu-bs.de , TU Braunschweig, Institut für Analysis und Algebra, 38106 Braunschweig, Germany.
Let $\mathbb{D}$ denote the open unit disc and $f\colon \mathbb{D}\to \mathbb{C}$ 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 $k\in \mathbb{N}$. We consider the Taylor expansions $$ (F(w))^k=\sum_{n=k}^{\infty} A_{n,k}w^n. $$ We prove that the inequality $$ \left|\sum_{k=1}^nA_{n,k}\right|\leq 2^{n-1} $$ is valid for any $n\in \mathbb{N}$ and that equality occurs in this inequality for a fixed $n\geq 2$ if and only if $f(z)=z/(1+z)$.
Keywords: Taylor coefficients, convex functions, inverse functions, bounded functions.
MSC 2000: 30C50, 30C45, 30D50.
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