On Univalent Functions With Real Coefficients

Abstract

1. Various authors have studied the subclass of normalized univalent functions in the unit circle consisting of functions which have real coefflcients in their Taylor expansions about the origin. Many of these have actually operated with the wider class of typically real functions (see [4, p. 4], also [1]). Some authors however have used a special form of Lowner's method (cf. [5]) and recently Singh [6] has used the variational method to discuss some problems for functions with normalizations other than the usual one. The object of the present paper is to treat the class of univalent functions with real coefficients using the usual normalization in a unified manner through the medium of the General Coefficient Theorem [3; 4, p. 51]. In this way we obtain substantially more complete results than have been obtained before. Among others we obtain the exact domain covered by the image of the unit circle under every such function, the region of values of such functions at a point in the unit circle and new bounds involving the derivative of these functions.