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\begin{document}
\title{The Hardy-Littlewood-P\'olya Inequality and Optimal Recovery of
Derivatives in Hardy Spaces}
\author{K.~Yu.~Osipenko}{``MATI" --- Russian State Technological
University}

\begin{abstract}
In 1934 Hardy, Littlewood, and P\'olya proved that for all integer $0<k<r$
and all functions $x\cd\in\lt$ such that $x^{(r-1)}\cd$ 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 $S_\beta=\{z\in\mathbb C:|\IM z|<\beta\}$. We also
show that this problem is closely connected with the problem of optimal
recovery of derivatives of analytic functions by inaccurate information
about their traces on $\mathbb R$.

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
$$t^r\sqrt{\ch2\beta t}=x$$
which belongs to the interval $[0,+\infty)$.

{\bf Theorem.} {\it For all $r,k\in\mathbb N$, $k\le r$, and all functions
$f\cd\in\Hr\cap\lt$ which are not equivalent to zero the following exact
inequalities
\begin{align*}
\|f^{(k)}\cd\|_{\lt}&\le\|f\cd\|_{\lt}\mu_{r\beta}^k\left(\frac{\|f^{(r)}
\cd\|_{\Ht}}{\|f\cd\|_{\lt}}\right),\\
\|f^{(k)}\cd\|_{\Ht}&\le\|f^{(r)}\cd\|_{\lt}\mu_{r\beta}^{k-r}\left(\frac{
\|f^{(r)}\cd\|_{\Ht}}{\|f\cd\|_{\lt}}\right)
\end{align*}
hold.}

\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{abstract}
\end{document}