Univalence criteria and analogues of theâ johnâ constant

Let p(z) = z f 0(z)= f (z) for a function f (z) analytic on the unit disc j z j< 1 in the complex plane and normalised by f (0) = 0; f 0(0) = 1. We provide lower and upper bounds for the best constants δ0 and δ1 such that the conditions e. δ0=2 <j p(z) j< e0=2 and j p(w)=p(z) j< e1 for j z j; j w j< 1 respectively imply univalence of f on the unit disc.