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 [math]\displaystyle{ f(z) }[/math] be a holomorphic function on the annulus

[math]\displaystyle{ r_1\leq\left| z\right| \leq r_3. }[/math]

Let [math]\displaystyle{ M(r) }[/math] be the maximum of [math]\displaystyle{ |f(z)| }[/math] on the circle [math]\displaystyle{ |z|=r. }[/math] Then, [math]\displaystyle{ \log M(r) }[/math] is a convex function of the logarithm [math]\displaystyle{ \log (r). }[/math] Moreover, if [math]\displaystyle{ f(z) }[/math] is not of the form [math]\displaystyle{ cz^\lambda }[/math] for some constants [math]\displaystyle{ \lambda }[/math] and [math]\displaystyle{ c }[/math], then [math]\displaystyle{ \log M(r) }[/math] is strictly convex as a function of [math]\displaystyle{ \log (r). }[/math]

The conclusion of the theorem can be restated as

[math]\displaystyle{ \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) }[/math]

for any three concentric circles of radii [math]\displaystyle{ r_1\lt r_2\lt r_3. }[/math]

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(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.

See also

• Maximum principle
• Logarithmically convex function
• Hardy's theorem
• Hadamard three-lines theorem
• Borel–Carathéodory theorem
• Phragmén–Lindelöf principle

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 

External links

• "proof of Hadamard three-circle theorem"