Lipschitz-type conditions on homogeneous Banach spaces of analytic functions

In this paper we deal with Banach spaces of analytic functions X defined on the unit disk satisfying that R-t f is an element of X for any t > 0 and f is an element of X, where R-t f(z)= f(e(it) z). We study the space of functions in X such that parallel to P-r(Df)parallel to x = O(omega(1-r)/1-r), r -> 1(-) where Df(z) = Sigma(infinity)(n=0) (n + 1)a(n)z(n) and omega is a continuous and non-decreasing weight satisfying certain mild assumptions. The space under consideration is shown to coincide with the subspace of functions in X satisfying any of the following conditions: (a) parallel to R(t)f-f parallel to x = O(w(t)); (b) parallel to P(r)f-f parallel to x = O(omega(1-r)), (c) parallel to Delta(n)f parallel to X = O(omega)(2(-n))), (d) parallel to f - s(n)f parallel to x = O(omega(n(-1))), where P(r)f(z) = f(rz), s(n)f(z) = Sigma(n)(k=0) a(k)z(k) and Delta(n)f = s(2n) f - s(2n-1) f. Our results extend those known for Hardy or Bergman spaces and power weights omega(t) = t(alpha). (C) 2016 Elsevier Inc. All rights reserved.