Norm of the composition operator mapping Bloch space into Hardy or Bergman space

Abstract

Let 1 leq p < infty and alpha > -1. If f is aholomorphic self-map of the open unit disc U of Bbb C withf(0)=0, then the quantity int_U left{frac {vertf'(z)vert}{1 - vert f (z)vert^2 } right}^p (1-|z|)^{alpha+p}dx dy is equivalent to the operator norm of the compositionoperator C_f : {Cal B} rightarrow A^{p,alpha} defined byC_f h = h circ f -h(0) , where {Cal B} and A^{p,alpha}are the Bloch space and the weighted Bergman space on Urespectively.