In complex analysis, a branch of mathematics, the Hadamard three-circle theorem is a result about the behavior of holomorphic functions.
Statement Hadamard three-circle theorem: Let f ( z ) {\displaystyle f(z)} be a holomorphic function on the annulus r 1 ≤ | z | ≤ r 3 {\displaystyle r_{1}\leq \left|z\right|\leq r_{3}} . Let M ( r ) {\displaystyle M(r)} be the maximum of | f ( z ) | {\displaystyle |f(z)|} on the circle | z | = r . {\displaystyle |z|=r.} Then, log M ( r ) {\displaystyle \log M(r)} is a convex function of the logarithm log ( r ) . {\displaystyle \log(r).} Moreover, if f ( z ) {\displaystyle f(z)} is not of the form c z λ {\displaystyle cz^{\lambda }} for some constants λ {\displaystyle \lambda } and c {\displaystyle c} , then log M ( r ) {\displaystyle \log M(r)} is strictly convex as a function of log ( r ) . {\displaystyle \log(r).}
The conclusion of the theorem can be restated as
log ( r 3 r 1 ) log M ( r 2 ) ≤ log ( r 3 r 2 ) log M ( r 1 ) + log ( r 2 r 1 ) log M ( r 3 ) {\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})}
for any three concentric circles of radii r 1 < r 2 < r 3 . {\displaystyle r_{1}<r_{2}<r_{3}.}
Proof The three circles theorem follows from the fact that for any real a, the function Re log(zaf(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.
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.
See also Maximum principle Logarithmically convex function Hardy's theorem Hadamard three-line theorem Borel–Carathéodory theorem Phragmén–Lindelöf principle
Notes
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, vol. 97, American Mathematical Society, pp. 386–387, ISBN 978-0821844793 This article incorporates material from Hadamard three-circle theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
External links "proof of Hadamard three-circle theorem"