Majorants in Hardy-Littlewood type theorem for higher order derivatives of analytic functions
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-|...
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| Date: | 2013 |
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| Keywords: | keywords |
| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Інститут математики НАН України
2013
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| Online Access: | https://trim.imath.kiev.ua/index.php/trim/article/view/163 |
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| Journal Title: | Transactions of Institute of Mathematics of NAS of Ukraine |
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Transactions of Institute of Mathematics of NAS of Ukraine| Summary: | 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 $\mathop{\rm Lip}_\lambda(\mathbb D)=\mathscr L_\lambda^k=\mathscr B_\lambda^k$ |
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