Derivatives of univalent functions and the hyperbolic metric

Let f be an analytic and univalent function on a simply connected domain D, and let lambda(D) be the hyperbolic metric on D. We prove the sharp inequality \fn(w)/f(w)\ less than or equal to n!4(n-1)lambda(D)(w)(n-1), w is an element of D. This can be viewed as a generalization of de Branges's famous result that \a(n)\ less than or equal to n for function in the class S. Our proof of the above also uses a generalization of K. Lowner's sharp estimate of the coefficients of the inverses of functions in S. We generalize Lowner's result to arbitrary powers of the inverse. We also consider the case when f is convex univalent and when D is convex.