Univalence criteria and analogs of the John constant

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 $\delta_0$ 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.