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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.

Contents

Let f ( z ) be a holomorphic function on the annulus

r 1 ≤ | z | ≤ 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 λ for some constants λ 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 ⁡ ( r 3 r 1 ) log ⁡ M ( r 2 ) ≤ log ⁡ ( r 3 r 2 ) log ⁡ M ( r 1 ) + log ⁡ ( r 2 r 1 ) log ⁡ M ( r 3 )

for any three concentric circles of radii r 1 < r 2 < r 3 .

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.

Proof

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.

References

Hadamard three-circle theorem Wikipedia
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