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

Let be a holomorphic function on the annulus

Let be the maximum of on the circle Then, is a convex function of the logarithm Moreover, if is not of the form for some constants and , then is strictly convex as a function of

The conclusion of the theorem can be restated as

for any three concentric circles of radii

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.^{[1]}

The three circles theorem follows from the fact that for any real *a*, the function Re log(*z*^{a}*f*(*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.^{[2]}

**^**Edwards 1974, Section 9.3**^**Ullrich 2008

- Edwards, H.M. (1974),
*Riemann's Zeta Function*, Dover Publications, ISBN 0-486-41740-9 - Littlewood, J. E. (1912), "Quelques consequences de l'hypothese que la function ζ(s) de Riemann n'a pas de zeros dans le demi-plan Re(s) > 1/2.",
*Les Comptes rendus de l'Académie des sciences*,**154**: 263–266 - E. C. Titchmarsh,
*The theory of the Riemann Zeta-Function*, (1951) Oxford at the Clarendon Press, Oxford.*(See chapter 14)* - Ullrich, David C. (2008),
*Complex made simple*, Graduate Studies in Mathematics, vol. 97, American Mathematical Society, pp. 386–387, ISBN 0821844792

*This article incorporates material from Hadamard three-circle theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

- "proof of Hadamard three-circle theorem"