Some questions on the univalence of functions of the class Σ

Abstract

In this paper we give an example of two convex functions in | ζ| > 1 whose arithmetic mean is nonconvex. We calculate the radius of convexity of the sum of two convex functions; it is equal to {Mathematical expression}. For functions F(ζ)=ζ+b1/ζ+..., where F′(ζ)=f(ζ)/ζ, if f(ζ) = ζ + a1/ζ+... is univalent |ζ| > 1, then the radius of univalence is the root of the equation 4E· (1/r)/K(1/r)+1/r2=3. © 1976 Plenum Publishing Corporation.