Majorants in Hardy-Littlewood type theorem for higher order derivatives of analytic functions

Abstract

We study the classes $\mathop{\rm Lip}_\lambda(\mathbb D)$, $\mathscr L_\lambda^k$ and $\mathscr B_\lambda^k,$ $k\in\mathbb N,$ consisting of analytic functions $f$ for which respectively $|f(z_1)-f(z_2)|=O(|z_1-z_2|)$, $|f^{(k)} z)|=O(|\lambda^{(k)}(1-|z|)|)$ and $|f^{(k)}(z)|=O(\lambda(1-|z|)/(1-|z|)^k)$, $z\in\mathbb D$. We investigate a question of embedding such classes and give conditions for equalities $\mathop{\rm Lip}_\lambda(\mathbb D)=\mathscr L_\lambda^k=\mathscr B_\lambda^k$