Univalence criteria and analogs of the John constant

Abstract

Let $p(z)=z{f}^{\prime }(z)/f(z)$ for a function $f(z)$ analytic on the unit disk $|z|<1$ in the complex plane and normalized by $f(0)=0,f^{\prime }(0)=1.$ We will provide lower and upper bounds for the best constants ${\delta }_0$ and ${\delta }_1$ such that the conditions $e^{-{\delta }_0/2}<|p(z)|<e^{{\delta }_0/2}$ and $|p(w)/p(z)|<e^{{\delta }_1}$ for $|z|,|w|<1$ respectively imply univalence of $f$ on the unit disk.

11 pages; the title has been changed

Cited by in corpus (1)

• Errata and Addenda to Mathematical Constants