On Invariance of Some Classes of Holomorphic Functions Under Integrodifferential Operators

Abstract

The following classes of functions analytic in the unit disk are considered:

$$N_\omega ^p = \left\{ {f \in H(D):\parallel T(f)\parallel _{L_{(w)}^p } = \left( {\int\limits_0^1 {\omega (1 - r)T^p } (f,r)dr} \right)^{1/p} < + \infty } \right\}$$

and

$$\widetilde N_\omega ^p = \left\{ {f \in H(D):\int\limits_0^1 {\int\limits_{ - \pi }^\pi {\omega (1 - r)(\log ^ + \left| {f(re^{i\varphi } )} \right|)^p rdrd\varphi < + \infty } } } \right\}$$

where $T(f,r) = \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {\log ^ + \left| {f(re^{i\varphi } )} \right|d\varphi }$ is the Nevanlinna characteristic and $\omega$ is a properly changing positive function on (0,1]. Necessary and sufficient conditions on $\omega$ are established under which the classes $N_\omega ^p$ and $\widetilde N_w^p$ are invariant under the operators of differentiation and integration. Bibliography: 7 titles.