The higher derivations of functions bounded in various senses

Abstract

An extension (Theorem 1) of Schwarz and Pick's lemma motivates us to study the analogues for functions which are bounded in the sense of Bloch, normal, or yoshida. A typical result is that, for a function f holomorphic in D = {|z| < 1} and Bloch, that is, , with the expansion f(w) = c0 + cn (w − z)n + … (n ≧ 1) about 2 ε D, we have (1 − |z|2)n|f(n) (z)|/n! ≦ Anα, where An is an absclute constant; the estimate is sharp.