Hadamard three-circle theorem

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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 f(z) be a holomorphic function on the annulus

Let M(r) be the maximum of | f(z) | on the circle | z | = r. Then, logM(r) is a convex function of the logarithm log(r). Moreover, if f(z) is not of the form cz^λ for some constants λ and c, then logM(r) is strictly convex as a function of log(r).

The conclusion of the theorem can be restated as

for any three concentric circles of radii r₁ < r₂ < r₃.

[edit] 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. H. Bohr and E. Landau claim the theorem was first given by Jacques Hadamard in 1896, although Hadamard had published no proof.^[1]

[edit] See also

• maximum principle
• logarithmically convex function
• Hardy's theorem

[edit] References

• ^  H.M. Edwards, Riemann's Zeta Function, (1974) Dover Publications, ISBN 0-486-41740-9 (See section 9.3.)
• E. C. Titchmarsh, The theory of the Riemann Zeta-Function, (1951) Oxford at the Clarendon Press, Oxford. (See chapter 14)
• This article incorporates material from Hadamard three-circle theorem on PlanetMath, which is licensed under the GFDL.