Directional convexity of level lines for functions convex in a given direction

Prokhorov, Dmitri; Szynal, Jan

doi:10.1090/S0002-9939-02-06675-3

Directional convexity of level lines for functions convex in a given direction
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by Dmitri V. Prokhorov and Jan Szynal PDF

Proc. Amer. Math. Soc. 131 (2003), 1453-1457 Request permission

Abstract:

Let $K(\varphi )$ be the class of functions $f(z)=z+a_{2}z^{2}+\dots$ which are holomorphic and convex in direction $e^{i\varphi }$ in the unit disk $D$, i.e. the domain $f(D)$ is such that the intersection of $f(D)$ and any straight line $\{w:w=w_{0}+te^{i\varphi },t\in \mathbb {R}\}$ is a connected or empty set. In this note we determine the radius $r_{\psi ,\varphi }$ of the biggest disk $|z|\leq r_{\psi ,\varphi }$ with the property that each function $f\in K(\psi )$ maps this disk onto the convex domain in the direction $e^{i\varphi }$.

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