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 $r_1<r_2<r_3.$

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.^[1]

Proof

The three circles theorem follows from the fact that for any real a, the function Re log(z^af(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]

Notes

↑ Edwards 1974, Section 9.3
↑ Ullrich 2008

References

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

Littlewood, J. E. (1912), "Quelques consequences de l'hypothese que la function ζ(s) de Riemann n'a pas de zeros dans le demi-plan Re(s) > 1/2.", Les Comptes rendus de l'Académie des sciences 154: 263–266
E. C. Titchmarsh, The theory of the Riemann Zeta-Function, (1951) Oxford at the Clarendon Press, Oxford. (See chapter 14)
Ullrich, David C. (2008), Complex made simple, Graduate studies in mathematics 97, American Mathematical Society, pp. 386–387, ISBN 0821844792

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