PlanetMath: proof of Hadamard three-circle theorem

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proof of Hadamard three-circle theorem

(Proof)

Let be holomorphic on a closed annulus $0<r_1\le \vert z\vert\le r_2$ . Let

$\displaystyle s=\frac{\log r_1-\log r}{\log r_2-\log r_1}.$

Let $M(r)=M_f(r)=\vert\vert f\vert\vert _r=\max_{\vert z\vert=r}\vert f(z)\vert$ . Then we have to prove that

$\displaystyle \log M(r)\le (1-s)\log M(r_1)+s\log M(r_2).$

For this, let $\alpha$ be a real number; the function $\alpha\log\vert z\vert+\log\vert f(z)\vert$ is harmonic outside the zeros of

. Near the zeros of

the above function has values which are large negative. Hence by the maximum modulus principle this function has its maximum on the boundary of the annulus, specifically on the two circles $\vert z\vert=r_1$ and $\vert z\vert=r_2$ . Therefore

$\displaystyle \alpha\log\vert z\vert+\log\vert f(z)\vert\le\max(\alpha\log r_1+\log M(r_1),\alpha\log r_2+\log M(r_2))$

for all

in the annulus. In particular, we get the inequality

$\displaystyle \alpha\log r+\log M(r)\le\max(\alpha\log r_1+\log M(r_1),\alpha\log r_2+\log M(r_2)).$

Now let $\alpha$ be such that the two values inside the parentheses on the right are equal, that is

$\displaystyle \alpha=\frac{\log M(r_2)-\log M(r_1)}{\log r_1-\log r_2}.$

Then from the previous inequality, we get

$\displaystyle \log M(r)\le\alpha\log r_1+\log M(r_1)-\alpha\log r,$

which upon substituting the value for $\alpha$ gives the result stated in the theorem.

Bibliography

Lang, S. Complex analysis, Fourth edition. Graduate Texts in Mathematics, 103. Springer-Verlag, New York, 1999. xiv+485 pp. ISBN 0-387-98592-1

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Cross-references: right, inequality, circles, annulus, boundary, modulus, negative, near, harmonic, function, real number, closed annulus, holomorphic

This is version 2 of proof of Hadamard three-circle theorem, born on 2006-05-31, modified 2006-09-06.
Object id is 7943, canonical name is ProofOfHadamardThreeCircleTheorem.
Accessed 1357 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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