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 f(z) be a holomorphic function on the annulus :r_1\leq\left, z\ \leq r_3. Let M(r) be the maximum of , f(z), on the circle , z, =r. Then, \log M(r) is a convex function of the logarithm \log (r). Moreover, if f(z) is not of the form cz^\lambda for some constants \lambda and c, then \log M(r) is strictly convex as a function of \log (r). The conclusion of the theorem can be restated as :\log\left(\frac\right)\log M(r_2)\leq \log\left(\frac\right)\log M(r_1) +\log\left(\frac\right)\log M(r_3) for any three concentric circles of radii r_1 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Complex Analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having placed first in the entrance examinations both there and at the École Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat and Picard. He obtained his doctorate in 1892 and in the same year was awarded the for his essay on the Riemann zeta function. In 1892 Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters. The following year he took up a lectureship in the University of Bordeaux, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Graduate Studies In Mathematics Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books 1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) 2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) 3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) 4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) 5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) 6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) 7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) 8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) 9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) 10 ''Representations of Finite and Compact Groups'', Barry Simon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Phragmén–Lindelöf Principle In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function f (i.e, , f(z), ) on an unbounded domain $\Omega$ when an additional (usually mild) condition constraining the growth of $, f,$ on $\Omega$ is given. It is a generalization of the maximum modulus principle, which is only applicable to bounded domains. Background In the theory of complex functions, it is known that the modulus (absolute value) of a [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Borel–Carathéodory Theorem In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory. Statement of the theorem Let a function f be analytic on a closed disc of radius ''R'' centered at the origin. Suppose that ''r'' 1, so we may assume ''f'' is nonconstant. First let ''f''(0) = 0. Since Re ''f'' is harmonic, Re ''f''(0) is equal to the average of its values around any circle centered at 0. That is, : \operatorname f(0) = \frac \int_ \operatorname f(z) dz. Since ''f'' is regular and nonconstant, we have that Re ''f'' is also nonconstant. Since Re ''f''(0) = 0, we must have Re f(z) > 0 for some ''z'' on the circle , z, =R, so we may take A>0. Now ''f'' maps into the half-plane ''P'' to the left of the ''x''=''A'' line. Roughly, our goal is to map this half-plane to a disk, apply Schwarz's lemma In mathematics, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hardy's Theorem In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions. Let f be a holomorphic function on the open ball centered at zero and radius R in the complex plane, and assume that f is not a constant function. If one defines :I(r) = \frac \int_0^\! \left, f(r e^) \ \,d\theta for 0< r < R, then this function is strictly increasing and is a convex function of $\log r$ . See also * Maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Logarithmically Convex Function In mathematics, a function ''f'' is logarithmically convex or superconvex if \circ f, the composition of the logarithm with ''f'', is itself a convex function. Definition Let be a convex subset of a real vector space, and let be a function taking non-negative values. Then is: * Logarithmically convex if \circ f is convex, and * Strictly logarithmically convex if \circ f is strictly convex. Here we interpret \log 0 as -\infty. Explicitly, is logarithmically convex if and only if, for all and all , the two following equivalent conditions hold: :\begin \log f(tx_1 + (1 - t)x_2) &\le t\log f(x_1) + (1 - t)\log f(x_2), \\ f(tx_1 + (1 - t)x_2) &\le f(x_1)^tf(x_2)^. \end Similarly, is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all . The above definition permits to be zero, but if is logarithmically convex and vanishes anywhere in , then it vanishes everywhere in the interior of . Equivalent conditions If is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Maximum Principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. In the simplest case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unless is a constant function, the maximum cannot also be achieved anywhere on itself. Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the siz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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. Statement Define F(z) by : F(z)=f(z) M(a)^M(b)^ where , F(z), \leq 1 on the edges of the strip. The result follows once it is shown that the inequality also holds in the interior of the strip. After an affine transformation in the coordinate z, it can be assumed that a = 0 and b = 1. The function : F_n(z) = F(z) e^e^ tends to 0 as , z, tends to infinity and satisfies , F_n, \leq 1 on the boundary of the strip. The maximum modulus principle can therefore be applied to F_n in the strip. So , F_n(z), \leq 1. Because F_n(z) tends to F(z) as n tends to infinity, it follows that , F(z), \leq 1. ∎ Applications The three-line theorem can be used to prove the Hadamard three-circle theorem for a bou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Harmonic Function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as : \nabla^2 f = 0 or :\Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as ''harmonics''. Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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