PlanetMath: Hadamard three-circle theorem

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Hadamard three-circle theorem

(Theorem)

Let be a complex analytic function on the annulus $r_1\leq\left\lvert z\right\rvert \leq r_3$ . Let be the maximum of $\left\lvert f(z)\right\rvert$ on the circle $\left\lvert z\right\rvert =r$ . Then $\log M(r)$ is a convex function of $\log r$ . Moreover, if is not of the form $cz^\lambda$ for some $\lambda$ , then $\log M(r)$ is a strictly convex as a function of $\log r$ .

The conclusion of the theorem can be restated as

$\displaystyle \log\frac{r_3}{r_1} \log M(r_2) \leq \log\frac {r_3}{r_2} \log M(r_1) + \log\frac {r_2}{r_1} \log M(r_3)$

for any three concentric circles of radii

"Hadamard three-circle theorem" is owned by bbukh.

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Attachments:: proof of Hadamard three-circle theorem (Proof) by Simone

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Cross-references: radii, concentric circles, conclusion, function, convex function, circle, annulus, complex analytic function
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This is version 4 of Hadamard three-circle theorem, born on 2004-02-20, modified 2005-06-13.
Object id is 5605, canonical name is HadamardThreeCircleTheorem.
Accessed 2713 times total.

Classification:

AMS MSC:	30A10 (Functions of a complex variable :: General properties :: Inequalities in the complex domain)
	30C80 (Functions of a complex variable :: Geometric function theory :: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination)

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