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 <math>f(z)</math> be a holomorphic function on the annulus <math>r_1\leq\left| z\right| \leq r_3</math>. Let <math>M(r)</math> be the maximum of <math>|f(z)|</math> on the circle <math>|z|=r.</math> Then, <math>\log M(r)</math> is a convex function of the logarithm <math>\log (r).</math> Moreover, if <math>f(z)</math> is not of the form <math>cz^\lambda</math> for some constants <math>\lambda</math> and <math>c</math>, then <math>\log M(r)</math> is strictly convex as a function of <math>\log (r).</math>

The conclusion of the theorem can be restated as

<math>\log\left(\frac{r_3}{r_1}\right)\log M(r_2)\leq

\log\left(\frac{r_3}{r_2}\right)\log M(r_1) +\log\left(\frac{r_2}{r_1}\right)\log M(r_3)</math>

for any three concentric circles of radii <math>r_1<r_2<r_3.</math>

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.<ref>Template:Harvnb</ref>

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.<ref>Template:Harvnb</ref>

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"