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

E. A. Shirokova

Kazan State University

Abstract: 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 $4E(1/r)/K(1/r)+1/r^2=3$.

UDC: 517.5

Received: 24.09.1973

 English version: Mathematical Notes, 1975, 18:3, 828–832

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