Hadamard three-circle theorem

In complex analysis, a branch of mathematics, the Hadamard three-circle theorem is a result about the behavior of holomorphic functions.

Statement

Hadamard three-circle theorem: Let

$f(z)$

be a holomorphic function on the annulus

$r_1\le \left|z\right|\le r_3$

. Let

$M(r)$

be the maximum of

$|f(z)|$

on the circle

$|z|=r.$

Then,

$\log M(r)$

is a convex function of the logarithm

$\log (r).$

Moreover, if

$f(z)$

is not of the form

$c{z}^{\lambda }$

for some constants

$\lambda$

and

$c$

, then

$\log M(r)$

is strictly convex as a function of

$\log (r).$

The conclusion of the theorem can be restated as

$\log \left(\frac{r_3}{r_1}\right)\log M(r_2)\le \log \left(\frac{r_3}{r_2}\right)\log M(r_1)+\log \left(\frac{r_2}{r_1}\right)\log M(r_3)$

for any three concentric circles of radii $r_1<r_2<r_3.$

Proof

The three circles theorem follows from the fact that for any real a, the function Re log(z^af(z)) is harmonic between two circles, and therefore takes its maximum value on one of the circles. The theorem follows by choosing the constant a so that this harmonic function has the same maximum value on both circles.

The theorem can also be deduced directly from Hadamard's three-line theorem.

History

A statement and proof for the theorem was given by J.E. Littlewood in 1912, but he attributes it to no one in particular, stating it as a known theorem. Harald Bohr and Edmund Landau attribute the theorem to Jacques Hadamard, writing in 1896; Hadamard published no proof.

See also

• Maximum principle
• Logarithmically convex function
• Hardy's theorem
• Hadamard three-line theorem
• Borel–Carathéodory theorem
• Phragmén–Lindelöf principle

References

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• E. C. Titchmarsh, The theory of the Riemann Zeta-Function, (1951) Oxford at the Clarendon Press, Oxford. (See chapter 14)
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External links

• "proof of Hadamard three-circle theorem"