Modulus of continuity with respect to semigroups of analytic functions and applications

Given a complex Banach space E, a semigroup of analytic functions (phi(t)) and an analytic function F : D -> E we introduce the modulus w(phi) (F,t) = sup (vertical bar z vertical bar 1<1) parallel to F(phi(t) (z)) - F(z)parallel to. We show that if 0 < alpha <= 1 and F belongs to the vector-valued disc algebra A(D, E), the Lipschitz condition M-infinity(F', r) = as r -> 1 is equivalent to w(phi) (F,t) = O(t(alpha)) as t -> 0 for any semigroup of analytic functions (phi(t)), with (phi(t) (0) = 0 and infinitesimal generator g, satisfying that c4 and G belong to H-infinity(D) with sup(0 <= t <=) parallel to phi'parallel to(infinity) <infinity, and in particular is equivalent to the condition parallel to F' -Fr parallel to(A(D,E)) = O ((1 - r)(alpha)) as r -> 1. We apply this result to particular semigroups (phi(t)) and particular spaces of analytic functions E, such as Hardy or Bergman spaces, to recover several known results about Lipschitz type functions. (C) 2015 Elsevier Inc. All rights reserved.