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\begin{document}
\title{Hardy-Littlewood-Polya inequality and the Hadamar three-circle theorem}
\author{Osipenko K. (Moscow, Russia)}

\maketitle

The Hadamard three-circle theorem states that if $f(z)$ a holomorphic function on the annulus $r_1\le|z|\le r_2$ and $M(r)=\max_{|z|=r}|f(z)|$, then $\log M(r)$ is a convex function of the $\log r$. The conclusion of the theorem can be restated as
$$M(r)\le M(r_1)^{\frac{\log r_2/r}{\log r_2/r_1}}M(r_2)^{\frac{\log r/r_1}{\log r_2/r_1}}$$
for any three concentric circles of radii $r_1<r<r_2$.

The Hardy-Littlewood-Polya inequality is the following one
$$\|x^{(k)}\cd\|_{\lt}\le\|x\cd\|_{\lt}^{1-\frac kr}\|x^{(r)}\cd\|_{\lt}^{
\frac kr}.$$ 
One may formulate it in the Hadamar three-circle theorem form. Namely, in the following form. $\log\|x^{(k)}\cd\|_{\lt}$ is a convex function of $k$. We use this fact to obtain optimal recovery methods for the $k$-th derivative on the basis of inaccurate information about some other derivatives.
\end{document}