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.

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(zaf(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-lines theorem.[2]

See also



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

External links

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