Abstract
Let PG(z) be the Poincaré density of the Poincaré metric PG(z)|dz| in a domain G in the complex plane C such that C{\backslash}G contains at least two points. Let δG(z) be the distance of z∈G and the boundary of G in C. It is well known that δG(z)PG(z){≤}1 everywhere, and if G is simply connected further, then 1/4{≤}δG(z)PG(z) everywhere. These inequalities have their roots in the classical and general inequalities of A. J. Macintyre, W. Seidel and J. L. Walsh for holomorphic functions defined in the open unit disk D. We prove sharp inequalities for the derivative of a holomorphic function in D, inequalities which, in particular, generalize the classical ones. Applications to PG will be given.
