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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.

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

Proof

[1]

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.[2]

See also

Notes

  • ^ H.M. Edwards, Riemann's Zeta Function, (1974) Dover Publications, ISBN 0-486-41740-9 (See section 9.3.)

References