The derivative of a holomorphic function and estimates of the Poincaré density

Abstract

Let P_G(z) be the Poincaré density of the Poincaré metric P_G(z)|dz| in a domain G in the complex plane C such that C{\backslash}G contains at least two points. Let δ_G(z) be the distance of z∈G and the boundary of G in C. It is well known that δ_G(z)P_G(z){≤}1 everywhere, and if G is simply connected further, then 1/4{≤}δ_G(z)P_G(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 P_G will be given.