On invariance of some classes of holomorphic functions under integrodifferential operators

The following classes of functions analytic in the unit disk are considered: Nωp - (∫01 ω(1-r)Tp(f, r)dr)1/p+∞) and Ñωp = (f ε ∈ H(D): ∫0- ∫-ππ ω(1-r)(log+|f(reiφ)|)p rdrdφ < + ∞), where T(f, r) = 1/2π ∫-ππ log+|∫(reiφ)|dφ is the Nevanlinna characteristic and ω is a properly changing positive function on (0, 1]. Necessary and sufficient conditions on ω are established under which the classes Nωp and Ñωp are invariant under the operators of differentiation and integration. © 2001 Plenum Publishing Corporation.