DERIVATIVES OF UNIVALENT FUNCTIONS
AND THE HYPERBOLIC METRIC

KOK SENG CHUA

Abstract:

Let f be an analytic and univalent function on a simply connected domain D, and let be the hyperbolic metric on D. We prove the sharp inequality

This can be viewed as a generalization of de Branges's famous result that for function in the class S. Our proof of the above also uses a generalization of K. Löwner's sharp estimate of the coefficients of the inverses of functions in S. We generalize Löwner's result to arbitrary powers of the inverse. We also consider the case when f is convex univalent and when D is convex.