Hadamard three-circle theorem

Hadamard three-circle theorem: Let $f(z)$ be a holomorphic function on the annulus $r\_1\backslash leq\backslash left|\; z\backslash right|\; \backslash leq\; r\_3$. Let $M(r)$ be the maximum of $|f(z)|$ on the circle $|z|=r.$ Then, $\backslash log\; M(r)$ is a convex function of the logarithm $\backslash log\; (r).$ Moreover, if $f(z)$ is not of the form $cz^\backslash lambda$ for some constants $\backslash lambda$ and $c$, then $\backslash log\; M(r)$ is strictly convex as a function of $\backslash log\; (r).$

The conclusion of the theorem can be restated as

:$\backslash log\backslash left(\backslash frac\{r\_3\}\{r\_1\}\backslash right)\backslash log\; M(r\_2)\backslash leq$

\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

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-line theorem.{{harvnb|Ullrich|2008}}

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.{{harvnb|Edwards|1974|loc=Section 9.3}}

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

{{reflist}}

• {{citation|first=H.M.|last=Edwards|authorlink=Harold Edwards (mathematician)|title=Riemann's Zeta Function|year=1974|publisher=Dover Publications|isbn=0-486-41740-9}}
• {{Citation | last1=Littlewood | first1=J. E. | title=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. | year=1912 | journal=Les Comptes rendus de l'Académie des sciences | volume=154 | pages=263–266}}
• E. C. Titchmarsh, The theory of the Riemann Zeta-Function, (1951) Oxford at the Clarendon Press, Oxford. (See chapter 14)
• {{citation|title=Complex made simple|volume= 97|series= Graduate Studies in Mathematics|first=David C.|last= Ullrich|publisher=American Mathematical Society|year= 2008|isbn=978-0821844793|pages=386–387}}

{{PlanetMath attribution|id=5605|title=Hadamard three-circle theorem}}

• [https://planetmath.org/proofofhadamardthreecircletheorem "proof of Hadamard three-circle theorem"]

Category:Inequalities (mathematics)

Category:Theorems in complex analysis