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'Hadamard three-circle theorem'
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| Title of object: |
Hadamard three-circle theorem |
| Canonical Name: |
HadamardThreeCircleTheorem |
| Type: |
Theorem |
| Created on: |
2004-02-20 18:52:19 |
| Modified on: |
2005-06-13 16:05:50 |
| Classification: |
msc:30A10, msc:30C80 |
Revision comment (for changes between this and next version):
| Changes for correction #6767 ('Italic "log" ?'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\newcommand*{\abs}[1]{\left\lvert #1\right\rvert}
\makeatletter
\@ifundefined{bibname}{}{\renewcommand{\bibname}{References}}
\makeatother |
Content:
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) \le 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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