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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 $4E(1/r)/K(1/r)+1/r^2=3$.
Received: 24.09.1973
Citation:
E. A. Shirokova, “Some questions on the univalence of functions of the class $\Sigma$”, Mat. Zametki, 18:3 (1975), 403–410; Math. Notes, 18:3 (1975), 828–832
Linking options:
https://www.mathnet.ru/eng/mzm7668 https://www.mathnet.ru/eng/mzm/v18/i3/p403
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Abstract page: | 158 | Full-text PDF : | 78 | First page: | 1 |
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