Extension of a~theorem by Hardy and Littlewood

S. V. Bykov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 21-33

Résumé

We give the following extension of a theorem by Hardy and Littlewood. Suppose $f$ is a holomorphic function in the unit disk and $$ M_p(r,f)=\Bigl(\frac1{2\pi}\int_{-\pi}^\pi|f(re^{i\theta})|^pd\theta\Bigr)^{\frac1p}=O(\varphi(r)),\quad r\to1-0, $$ where $\varphi$ is a monotone increasing function on $(0,1)$ and $$ \alpha_\varphi=\lim_{r\to1-0}\frac{\varphi'(r)(1-r)}{\varphi(r)}. $$ 1) If $0\leq\alpha_\varphi+\infty$, then $M_p(r,f')=O(\frac{\varphi(r)}{1-r})$, $r\to1-0$; 2) If $\alpha_\varphi=+\infty$, then $M_p(r,f')=O(\varphi'(r))$, $r\to1-0$.