Univalence criteria and analogs of the John constant

Yong Chan Kim; Toshiyuki Sugawa

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arXiv · 1209.5167

Univalence criteria and analogs of the John constant

Yong Chan Kim·Toshiyuki Sugawa

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Abstract

Let $p(z)=zf'(z)/f(z)$ for a function $f(z)$ analytic on the unit disk $|z|<1$ in the complex plane and normalized by $f(0)=0, f'(0)=1.$ We will provide lower and upper bounds for the best constants $δ_0$ and $δ_1$ such that the conditions $e^{-δ_0/2}<|p(z)|<e^{δ_0/2}$ and $|p(w)/p(z)|<e^{δ_1}$ for $|z|,|w|<1$ respectively imply univalence of $f$ on the unit disk.

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Yong Chan Kim, Toshiyuki Sugawa. 2012-10-23. Univalence criteria and analogs of the John constant. https://arxiv.org/abs/1209.5167

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