Hadamard three-circle theorem

Hadamard three-circle theorem In complex analysis, une branche des mathématiques, the Hadamard three-circle theorem is a result about the behavior of holomorphic functions.

Laisser {style d'affichage f(z)} be a holomorphic function on the annulus {style d'affichage r_{1}gauche|zright|leq r_{3}.} Laisser {style d'affichage M(r)} be the maximum of {style d'affichage |F(z)|} on the circle {style d'affichage |z|=r.} Alors, {displaystyle log M(r)} is a convex function of the logarithm {displaystyle log(r).} En outre, si {style d'affichage f(z)} n'est pas de la forme {displaystyle cz^{lambda }} for some constants {style d'affichage lambda } et {displaystyle c} , alors {displaystyle log M(r)} is strictly convex as a function of {displaystyle log(r).} The conclusion of the theorem can be restated as {displaystyle log left({frac {r_{3}}{r_{1}}}droit)log M(r_{2})leq log left({frac {r_{3}}{r_{2}}}droit)log M(r_{1})+bûche à gauche({frac {r_{2}}{r_{1}}}droit)log M(r_{3})} for any three concentric circles of radii {style d'affichage r_{1}External links 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(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-lines theorem.[2] See also Maximum principle Logarithmically convex function Hardy's theorem Hadamard three-lines theorem Borel–Carathéodory theorem Phragmén–Lindelöf principle Notes ^ Edwards 1974, Section 9.3 ^ Ullrich 2008 References Edwards, H.M. (1974), Riemann's Zeta Function, Publications de Douvres, 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, Études supérieures en mathématiques, volume. 97, Société mathématique américaine, pp. 386–387, ISBN 0821844792 This article incorporates material from Hadamard three-circle theorem on PlanetMath, qui est sous licence Creative Commons Attribution/Share-Alike License.

Liens externes "proof of Hadamard three-circle theorem" Catégories: InequalitiesTheorems in complex analysis