JournalscmhVol. 81 , No. 4DOI 10.4171/cmh/74

Sharp inequalities for the coefficients of concave schlicht functions

  • Farit G. Avkhadiev

    Kazan State University, Russian Federation
  • Christian Pommerenke

    Technische Universität Berlin, Germany
  • Karl-Joachim Wirths

    Universität Braunschweig, Germany
Sharp inequalities for the coefficients of concave schlicht functions cover

Abstract

Let DD denote the open unit disc and let f ⁣:DCf\colon D\to \mathbb{C} be holomorphic and injective in DD. We further assume that f(D)f(D) is unbounded and Cf(D)\mathbb{C}\setminus f(D) is a convex domain. In this article, we consider the Taylor coefficients an(f)a_n(f) of the normalized expansion

f(z)=z+n=2an(f)zn,zD,f(z)=z+\sum_{n=2}^{\infty}a_n(f)z^n, z\in D,

and we impose on such functions ff the second normalization f(1)=f(1)=\infty. We call these functions concave schlicht functions, as the image of DD is a concave domain. We prove that the sharp inequalities

an(f)n+12n12,n2,|a_n(f)-\frac{n+1}{2}|\leq\frac{n-1}{2}, n\geq 2,

are valid. This settles a conjecture formulated in [2].