Hadamard three-lines theorem

Hadamard three-lines theorem In complex analysis, a branch of mathematics, the Hadamard three-lines theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane. The theorem is named after the French mathematician Jacques Hadamard.

Contents 1 Statement 2 Applications 3 See also 4 References Statement Hadamard three-lines theorem — Let {displaystyle f(z)} be a bounded function of {displaystyle z=x+iy} defined on the strip {displaystyle {x+iy:aleq xleq b},} holomorphic in the interior of the strip and continuous on the whole strip. If {displaystyle M(x)=sup _{y}|f(x+iy)|} then {displaystyle log M(x)} is a convex function on {displaystyle [a,b].} In other words, if {displaystyle x=ta+(1-t)b} with {displaystyle 0leq tleq 1,} then {displaystyle M(x)leq M(a)^{t}M(b)^{1-t}.} show Proof Applications The three-line theorem can be used to prove the Hadamard three-circle theorem for a bounded continuous function {displaystyle g(z)} on an annulus {displaystyle {z:rleq |z|leq R},} holomorphic in the interior. Indeed applying the theorem to {displaystyle f(z)=g(e^{z}),} shows that, if {displaystyle m(s)=sup _{|z|=e^{s}}|g(z)|,} then {displaystyle log ,m(s)} is a convex function of {displaystyle s.} The three-line theorem also holds for functions with values in a Banach space and plays an important role in complex interpolation theory. It can be used to prove Hölder's inequality for measurable functions {displaystyle int |gh|leq left(int |g|^{p}right)^{1 over p}cdot left(int |h|^{q}right)^{1 over q},} where {displaystyle {1 over p}+{1 over q}=1,} by considering the function {displaystyle f(z)=int |g|^{pz}|h|^{q(1-z)}.} See also Riesz–Thorin theorem References Hadamard, Jacques (1896), "Sur les fonctions entières" (PDF), Bull. Soc. Math. Fr., 24: 186–187 (the original announcement of the theorem) Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics, Volume 2: Fourier analysis, self-adjointness, Elsevier, pp. 33–34, ISBN 0-12-585002-6 Ullrich, David C. (2008), Complex made simple, Graduate Studies in Mathematics, vol. 97, American Mathematical Society, pp. 386–387, ISBN 0-8218-4479-2 Categories: Convex analysisTheorems in complex analysis