Hadamard three-circle theorem

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 {displaystyle f(z)} be a holomorphic function on the annulus {displaystyle r_{1}leq left|zright|leq r_{3}.} Let {displaystyle M(r)} be the maximum of {displaystyle |f(z)|} on the circle {displaystyle |z|=r.} Then, {displaystyle log M(r)} is a convex function of the logarithm {displaystyle log(r).} Moreover, if {displaystyle f(z)} is not of the form {displaystyle cz^{lambda }} for some constants {displaystyle lambda } and {displaystyle c} , then {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}}}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 {displaystyle r_{1} 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 0821844792 This article incorporates material from Hadamard three-circle theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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