30A10-HadamardThreecircleTheorem.tex

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\pmtitle{Hadamard three-circle theorem}
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\pmtype{Theorem}
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\pmrelated{HardysTheorem}

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\begin{document}
Let $f(z)$ be a complex analytic function on the annulus
$r_1\leq\abs{z}\leq r_3$. Let $M(r)$ be the maximum of
$\abs{f(z)}$ on the circle $\abs{z}=r$. Then $\log M(r)$ is a
convex function of $\log r$. Moreover, if $f(z)$ is not of the form $cz^\lambda$ for some $\lambda$, then $\log M(r)$ is a \PMlinkname{strictly convex}{ConvexFunction} as a function of $\log r$.

The conclusion of the theorem can be restated as
\begin{equation*}
\log\frac{r_3}{r_1} \log M(r_2) \leq \log\frac {r_3}{r_2} \log M(r_1) + 
\log\frac {r_2}{r_1} \log M(r_3)
\end{equation*}
for any three concentric circles of radii $r_1<r_2<r_3$.
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\end{document}