Coefficients of univalent functions with restricted maximum modulus

Abstract

Suppose that is univalent in Classical results due essentially to Littlewood and Paley assert that an inequality implies provided a α > ½ In this paper we show that this implication remains true for α ≧.497. In particular, this confirms Szegö's conjecture that coefficients of fourfold symmetric functions satisfy. Tools include Hayman's theorem which asserts that a univalent function cannot be too big at too many different places, and a localized version of an inequality of Clunie and Pommerenke which those authors had used to prove a_n=(n^−.503) for bounded univalent f

AMS (MOS):

• 30C50

^*Research supported in part by the National Science Foundation.

^*Research supported in part by the National Science Foundation.

Notes

^*Research supported in part by the National Science Foundation.