Sharp estimates for integral means for three classes of domains

Abstract

In this paper, the following sharp estimate is proved: ∫ 0 2π|F′(e iθ)| p, dθ ≤ √π2 1+pΓ(1/2+p/2)/Γ(1+p/2)}, p>-1, where F is the conformal mapping of the domain D - = {ζ:|ζ| > 1} onto the exterior of a convex curve, with F'(\infty)=\nomathbreak 1. For p=\nomathbreak 1, this result is due to Pólya and Shiffer. We also obtain several generalizations of this estimate under other geometric assumptions about the structure of the domain F(D -).