Composition Operators on the Generalized Bergman Space

Abstract

In this note we prove the following result:
Let ∅: U∅U be analytic and nonconstant and w be convex modulus function. Then the composition operator T_∅(f)=fo∅ on the generalized weighted Bergman space B_w,ω(U) satisfies the inequality.
"–T_∅(f)"–B_w,ω≤C"–f"–B_w,ω

for some C independent of f, where "–f"–B_w,ω=∫w(|f(z)|)ω(z)dm(z).