The derivative of a bounded holomorphic function in the disk

Abstract:

Let a nonconstant function $f$ be holomorphic and bounded, $|f| < 1$ in $D:|z| < 1$. We shall estimate ${f^{\ast }}(z) = (1 - |z{|^2})|f’(z)|/(1 - |f(z){|^2})$ at each point $z\epsilon D$ ((1) in Theorem 1). The function $d$ appearing in the estimate concerns the sizes of the schlicht disks on the Riemannian image $\mathcal {F}$ of $D$ by $f$. Boundary properties of $f$ and ${f^{\ast }}$ will be stated in Theorems 2 and 3; use is made of the cluster sets of $d$.