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

$r_{1}\leq \left|z\right|\leq r_{3}.$

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 $cz^{\lambda }$ for some constants $\lambda$ and $c$, then $\log M(r)$ is strictly convex as a function of $\log(r).$

The conclusion of the theorem can be restated as

$\log \left({\frac {r_{3}}{r_{1}}}\right)\log M(r_{2})\leq \log \left({\frac {r_{3}}{r_{2}}}\right)\log M(r_{1})+\log \left({\frac {r_{2}}{r_{1}}}\right)\log M(r_{3})$

for any three concentric circles of radii $r_{1}<r_{2}<r_{3}.$

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

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

Notes

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

References

• E. C. Titchmarsh, The theory of the Riemann Zeta-Function, (1951) Oxford at the Clarendon Press, Oxford. (See chapter 14)
• Hadamard three-circle theorem at PlanetMath.