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\DeclareMathOperator{\IM}{Im}
\DeclareMathOperator{\ch}{ch}

\newcommand{\cd}{(\cdot)}
\newcommand*{\lt}{L_2(\mathbb R)}
\newcommand*{\Ht}{\mathcal H_2^\beta}
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\begin{document}
\begin{center}  {\bf On the Hardy-Littlewood-P\'olya Inequality for
Analytic Functions}\\
{K.~Yu.~Osipenko (``MATI" --- Russian State Technological University)}
\end{center}

Hardy, Littlewood, and P\'olya proved in 1934 that for all integer $0<k<r$
and all functions $x\cd\in\lt$ such that the $(r-1)$-st derivative is
locally absolutely continuous on $\mathbb R$ and $x^{(r)}\cd\in\lt$ the
following exact inequality
$$\|x^{(k)}\cd\|_{\lt}\le\|x\cd\|_{\lt}^{1-\frac kr}\|x^{(r)}\cd\|_{\lt}^{
\frac kr}$$
holds. We consider the analogue of this inequality for functions which are
analytic in the strip $S_\beta=\{z\in\mathbb C:|\IM z|<\beta\}$.

Denote by $\Ht$ the Hardy space of functions $f\cd$ analytic in the strip $
S_\beta$ and satisfying the condition
$$\|f\cd\|_{\Ht}=\biggl(\sup_{0\le\eta<\beta}\frac12\int_{\mathbb R}(|f(t+i
\eta)|^2+|f(t-i\eta)|^2)\,dt\biggr)^{1/2}<\infty.$$
The Hardy-Sobolev space $\Hr$ is the space of functions $f\cd$ analytic in
the strip $S_\beta$ for which $f^{(r)}\cd\in\Ht$. Denote by $\mu_{r\beta}(x
)$ the unique solution of the equation $t^r\sqrt{\ch2\beta t}=x$ from the
interval $[0,+\infty)$.

{\bf Theorem.} {\it For all $r,k\in\mathbb N$, $k\le r$, and $\gamma_1,
\gamma_2>0$
$$\sup_{\substack{f\cd\in\Hr\cap\lt\\\|f^{(r)}\cd\|_{\Ht}\le\gamma_1\\\|f
\cd\|_{\lt}\le\gamma_2}}\|f^{(k)}\cd\|_{\lt}=\gamma_2\mu_{r\beta}^k\left(
\frac{\gamma_1}{\gamma_2}\right).$$}

In other words, for all functions $f\cd\in\Hr\cap\lt$ which are not
equivalent to zero the following exact inequality
$$\|f^{(k)}\cd\|_{\lt}\le\|f\cd\|_{\lt}\mu_{r\beta}^k\left(\frac{\|f^{(r)}
\cd\|_{\Ht}}{\|f\cd\|_{\lt}}\right)$$
holds.

\smallskip

{\small The research was carried out with the financial support of the
Russian Foundation for Basic Research (grant nos.\ 05-01-00275 and
05-01-00261), the President Grant for State Support of Leading Scientific
Schools in Russian Federation (grant no.\ NSH-304.2003.1), and the Program
``Universities of Russia" (grant nos.\ UR.03.01.130 and UR.04.02.536))}

\end{document}
