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List Of Inequalities
This article lists Wikipedia articles about named mathematical inequalities. Inequalities in pure mathematics Analysis * Agmon's inequality * Askey–Gasper inequality * Babenko–Beckner inequality * Bernoulli's inequality * Bernstein's inequality (mathematical analysis) * Bessel's inequality * Bihari–LaSalle inequality * Bohnenblust–Hille inequality * Borell–Brascamp–Lieb inequality * Brezis–Gallouet inequality * Carleman's inequality * Chebyshev–Markov–Stieltjes inequalities * Chebyshev's sum inequality * Clarkson's inequalities * Eilenberg's inequality * Fekete–Szegő inequality * Fenchel's inequality * Friedrichs's inequality * Gagliardo–Nirenberg interpolation inequality * Gårding's inequality * Grothendieck inequality * Grunsky's inequalities * Hanner's inequalities * Hardy's inequality * Hardy–Littlewood inequality * Hardy–Littlewood–Sobolev inequality * Harnack's inequality * Hausdorff–Young inequality * Hermite–Hadamard inequa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b''). The re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fekete–Szegő Inequality
In mathematics, the Fekete–Szegő inequality is an inequality for the coefficients of univalent analytic functions found by , related to the Bieberbach conjecture In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It .... Finding similar estimates for other classes of functions is called the Fekete–Szegő problem. The Fekete–Szegő inequality states that if :f(z)=z+a_2z^2+a_3z^3+\cdots is a univalent analytic function on the unit disk and 0\leq \lambda < 1, then : References * {{DEFAULTSORT:Fekete-Szego inequality Inequalities ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite–Hadamard Inequality
In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ : [''a'', ''b''] → R is convex function, convex, then the following chain of inequalities hold: : f\left( \frac\right) \le \frac\int_a^b f(x)\,dx \le \frac. The inequality has been generalized to higher dimensions: if \Omega \subset \mathbb^n is a bounded, convex domain and f:\Omega \rightarrow \mathbb is a positive convex function, then : \frac \int_\Omega f(x) \, dx \leq \frac \int_ f(y) \, d\sigma(y) where c_n is a constant depending only on the dimension. A corollary on Vandermonde-type integrals Suppose that , and choose distinct values from . Let be convex, and let denote the Volterra operator, "integral starting at " operator; that is, :(If)(x)=\int_a^x. Then : \sum_^n \frac \leq \frac \sum_^n f(x_i) Equality holds for all iff is linear, and for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff–Young Inequality
The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by and extended by . It is now typically understood as a rather direct corollary of the Plancherel theorem, found in 1910, in combination with the Riesz-Thorin theorem, originally discovered by Marcel Riesz in 1927. With this machinery, it readily admits several generalizations, including to multidimensional Fourier series and to the Fourier transform on the real line, Euclidean spaces, as well as more general spaces. With these extensions, it is one of the best-known results of Fourier analysis, appearing in nearly every introductory graduate-level textbook on the subject. The nature of the Hausdorff-Young inequality can be understood with only Riemann integration and infinite series as prerequisite. Given a continuous function , define its "Fourier coefficients" by :c_n=\int_0^1 e^f(x)\,dx for each integer . The Hausdorf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harnack's Inequality
In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. , and generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Such results can be used to show the interior regularity of weak solutions. Perelman's solution of the Poincaré conjecture uses a version of the Harnack inequality, found by , for the Ricci flow. The statement Harnack's inequality applies to a non-negative function ''f'' defined on a closed ball in R''n'' with radius ''R'' and centre ''x''0. It states that, if ''f'' is continuous on the closed ball and harmonic on its interior, then for every point ''x'' with , ''x'' − ''x''0, = ''r'' 0 (depending only on ''K'', \tau, t-\tau, and the coefficients of \mathcal) such that, for each t\in(\tau, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Littlewood–Sobolev Inequality
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev. Sobolev embedding theorem Let denote the Sobolev space consisting of all real-valued functions on whose first weak derivatives are functions in . Here is a non-negative integer and . The first part of the Sobolev embedding theorem states that if , and are two real numbers such that :\frac-\frac = \frac -\frac, then :W^(\mathbf^n)\subseteq W^(\mathbf^n) and the embedding is continuous. In the special case of and , Sobolev embedding gives :W^(\mathbf^n) \subseteq L^(\mathbf^n) where is the Sobolev conjugate of , given byp. (Note that 1/p^*p.) Thus, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Littlewood Inequality
In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n-dimensional Euclidean space \mathbb R^n, then :\int_ f(x)g(x) \, dx \leq \int_ f^*(x)g^*(x) \, dx where f^* and g^* are the symmetric decreasing rearrangements of f and g, respectively. The decreasing rearrangement f^* of f is defined via the property that for all r >0 the two super-level sets :E_f(r)=\left\ \quad and \quad E_(r)=\left\ have the same volume (n-dimensional Lebesgue measure) and E_(r) is a ball in \mathbb R^n centered at x=0, i.e. it has maximal symmetry. Proof The layer cake representation allows us to write the general functions f and g in the form f(x)= \int_0^\infty \chi_ \, dr \quad and \quad g(x)= \int_0^\infty \chi_ \, ds where r \mapsto \chi_ equals 1 for rs the indicator functions x \mapsto \chi_(x) and x \map ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy's Inequality
Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. It states that if a_1, a_2, a_3, \dots is a sequence of non-negative real numbers, then for every real number ''p'' > 1 one has :\sum_^\infty \left (\frac\right )^p\leq\left (\frac\right )^p\sum_^\infty a_n^p. If the right-hand side is finite, equality holds if and only if a_n = 0 for all ''n''. An integral version of Hardy's inequality states the following: if ''f'' is a measurable function with non-negative values, then :\int_0^\infty \left (\frac\int_0^x f(t)\, dt\right)^p\, dx\le\left (\frac\right )^p\int_0^\infty f(x)^p\, dx. If the right-hand side is finite, equality holds if and only if ''f''(''x'') = 0 almost everywhere. Hardy's inequality was first published and proved (at least the discrete version with a worse constant) in 1920 in a note by Hardy. The original formulation was in an integral form slightly different from the above. General one-dimensional version The general weighted one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hanner's Inequalities
In mathematics, Hanner's inequalities are results in the theory of ''L''''p'' spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of ''L''''p'' spaces for ''p'' ∈ (1, +∞) than the approach proposed by James A. Clarkson James Andrew Clarkson (7 February 1906 – 6 June 1970) was an American mathematician and professor of mathematics who specialized in number theory. He is known for proving inequalities in Hölder condition#Hölder spaces, Hölder spaces, and deri ... in 1936. Statement of the inequalities Let ''f'', ''g'' ∈ ''L''''p''(''E''), where ''E'' is any measure space. If ''p'' ∈ , 2 then :\, f+g\, _p^p + \, f-g\, _p^p \geq \big( \, f\, _p + \, g\, _p \big)^p + \big, \, f\, _p-\, g\, _p \big, ^p. The substitutions ''F'' = ''f'' + ''g'' and ''G'' = ''f'' − ''g'' yield the second of Hanner's inequalities: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grunsky's Theorem
In mathematics, Grunsky's theorem, due to the German mathematician Helmut Grunsky, is a result in complex analysis concerning Holomorphic function, holomorphic univalent functions defined on the unit disk in the complex numbers. The theorem states that a univalent function defined on the unit disc, fixing the point 0, maps every disk '', z, '' 1 and λ''i'' are arbitrary complex numbers. Taking ''n'' = 2. with λ''1'' = – λ''2'' = λ, the inequality implies : \left, \log \\le \log . If ''g'' is an odd function and η = – ζ, this yields : \left, \log \ \le . Finally if ''f'' is any normalized univalent function in ''D'', the required inequality for ''f'' follows by taking : g(\zeta)=f(\zeta^)^ with z=\zeta^. Proof of the theorem Let ''f'' be a univalent function on ''D'' with ''f''(0) = 0. By Nevanlinna's criterion, ''f'' is starlike on '', z, '' < ''r'' if and only if : for '', z, '' < ''r''. Equivalently : |
Grothendieck Inequality
In mathematics, the Grothendieck inequality states that there is a universal constant K_G with the following property. If ''M''''ij'' is an ''n'' × ''n'' (real or complex) matrix with : \Big, \sum_ M_ s_i t_j \Big, \le 1 for all (real or complex) numbers ''s''''i'', ''t''''j'' of absolute value at most 1, then : \Big, \sum_ M_ \langle S_i, T_j \rangle \Big, \le K_G for all vectors ''S''''i'', ''T''''j'' in the unit ball ''B''(''H'') of a (real or complex) Hilbert space ''H'', the constant K_G being independent of ''n''. For a fixed Hilbert space of dimension ''d'', the smallest constant that satisfies this property for all ''n'' × ''n'' matrices is called a Grothendieck constant and denoted K_G(d). In fact, there are two Grothendieck constants, K_G^(d) and K_G^(d), depending on whether one works with real or complex numbers, respectively. The Grothendieck inequality and Grothendieck constants are named after Alexander Grothendieck, who proved the existen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gårding's Inequality
In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding. Statement of the inequality Let Ω be a bounded, open domain in ''n''-dimensional Euclidean space and let ''H''''k''(Ω) denote the Sobolev space of ''k''-times weakly differentiable functions ''u'' : Ω → R with weak derivatives in ''L''2. Assume that Ω satisfies the ''k''-extension property, i.e., that there exists a bounded linear operator ''E'' : ''H''''k''(Ω) → ''H''''k''(R''n'') such that (''Eu''), Ω = ''u'' for all ''u'' in ''H''''k''(Ω). Let ''L'' be a linear partial differential operator of even order ''2k'', written in divergence form :(L u)(x) = \sum_ (-1)^ \mathrm^ \left( A_ (x) \mathrm^ u(x) \right), and suppose that ''L'' is uniformly elliptic, i.e., there exists a constant ''θ'' > 0 such that :\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
