Some questions on the univalence of functions of the class $\Sigma$

E. A. Shirokova

Matematičeskie zametki, Tome 18 (1975) no. 3, pp. 403-410

Résumé

In this paper we give an example of two convex functions in $|\zeta|>1$ whose arithmetic mean is nonconvex. We calculate the radius of convexity of the sum of two convex functions; it is equal to $\sqrt{1+\sqrt2}$. For functions $F(\zeta)=\zeta+b_1/\zeta+\dots$, where $F'(\zeta)=f(\zeta)/\zeta$, if $f(\zeta)=\zeta+a_1/\zeta+\dots$ is univalent $|\zeta|>1$, then the radius of univalence is the root of the equation $4E(1/r)/K(1/r)+1/r^2=3$.