Suppose thatis univalent inClassical results due essentially to Littlewood and Paley assert that an inequalityimpliesprovided 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 an=(n−.503) for bounded univalentf