On the invariance of some classes of holomorphic functions under integral and differential operators

The following classes of functions analytic in the unit disk are considered:
$$ N^p_\omega=\biggl\{f\in H(D):\|T(f)\|_{L^p_{(\omega)}}=\bigl(\int\limits^1_0\omega(1-r)T^p(f,r)dr\bigr)^{1/p}<+\infty\biggr\}, $$

$$ \tilde N^p_\omega=\biggl\{f\in H(D):\int^1_0\,\int^\pi_{-\pi}\omega(1-r)\bigl(\ln^+|f(re^{i\varphi})|\bigr)^p\,rdrd\varphi<+\infty\biggr\}, $$
where $T(f,r)=\frac1{2\pi}\int\limits^\pi_{-\pi}\ln^+|f(re^{i\varphi})|d\varphi$ is the Nevanlinna haracteristic and $\omega$ is a positive function on $(0,1]$. Necessary and sufficient conditions on $\omega$ are established, under which the classes $N^p_\omega$ and $\tilde N^p_\omega$ are invariant under the operators of differentiation and integration.