UNIVALENCE CRITERIA AND ANALOGUES OF THE JOHN CONSTANT

Abstract

AbstractLet $p(z)= z{f}^{\prime } (z)/ f(z)$ for a function $f(z)$ analytic on the unit disc $\mid z\mid \lt 1$ in the complex plane and normalised by $f(0)= 0, {f}^{\prime } (0)= 1$. We provide lower and upper bounds for the best constants ${\delta }_{0} $ and ${\delta }_{1} $ such that the conditions ${e}^{- {\delta }_{0} / 2} \lt \mid p(z)\mid \lt {e}^{{\delta }_{0} / 2} $ and $\mid p(w)/ p(z)\mid \lt {e}^{{\delta }_{1} } $ for $\mid z\mid , \mid w\mid \lt 1$ respectively imply univalence of $f$ on the unit disc.