In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by Lebedev andMilin ( 1965 ) and Isaak MoiseevichMilin ( 1977 ). It was used in the proof of the Bieberbach conjecture, as it shows that the Milin conjecture implies the Robertson conjecture.
They state that if
for complex numbers and , and is a positive integer, then
See also exponential formula (on exponentiation of power series).
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 was posed by Ludwig Bieberbach and finally proven by Louis de Branges.
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.
In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication.
In mathematics, the Mittag-Leffler function is a special function, a complex function which depends on two complex parameters and . It may be defined by the following series when the real part of is strictly positive:
In mathematics, Grönwall's inequality allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. There are two forms of the lemma, a differential form and an integral form. For the latter there are several variants.
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations.
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series
In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following:
Koebe Quarter Theorem. The image of an injective analytic function from the unit disk onto a subset of the complex plane contains the disk whose center is and whose radius is .
In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper and used in the proof of the Bieberbach conjecture.
Isaak Moiseevich Milin, ; * February 16, 1919, Oster, Ukrainian Soviet Socialist Republic – † November 17, 1992 Saint-Petersburg, Russian Federation) was a prominent Soviet/Russian mathematician, doctor of science in physics and mathematics, senior researcher, specialist in Geometric Theory of Functions of a Complex Variable and Applied Mathematics, engineer-lieutenant-colonel at the Soviet Air Force.
Nikolai Andreevich Lebedev was a Soviet mathematician who worked on complex function theory and geometric function theory. Jointly with Isaak Milin, he proved the Lebedev–Milin inequalities that were used in the proof of the Bieberbach conjecture.
In statistical mechanics, the cluster expansion is a power series expansion of the partition function of a statistical field theory around a model that is a union of non-interacting 0-dimensional field theories. Cluster expansions originated in the work of Mayer & Montroll (1941). Unlike the usual perturbation expansion which usually leads to a divergent asymptotic series, the cluster expansion may converge within a non-trivial region, in particular when the interaction is small and short-ranged.
In mathematics, Montgomery's pair correlation conjecture is a conjecture made by Hugh Montgomery that the pair correlation between pairs of zeros of the Riemann zeta function is
In mathematics, the Fekete–Szegő inequality is an inequality for the coefficients of univalent analytic functions found by Fekete and Szegő, related to the Bieberbach conjecture. Finding similar estimates for other classes of functions is called the Fekete–Szegő problem.
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. Nearly fifteen years earlier, Alexander Gelfond had considered the problem with only integer coefficients to be of "extraordinarily great significance". The result, proved by Alan Baker, subsumed many earlier results in transcendental number theory. Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1.
In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight on the interval . The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.
In complex analysis and geometric function theory, the Grunsky matrices, or Grunsky operators, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single holomorphic function on the unit disk or a pair of holomorphic functions on the unit disk and its complement. The Grunsky inequalities express boundedness properties of these matrices, which in general are contraction operators or in important special cases unitary operators. As Grunsky showed, these inequalities hold if and only if the holomorphic function is univalent. The inequalities are equivalent to the inequalities of Goluzin, discovered in 1947. Roughly speaking, the Grunsky inequalities give information on the coefficients of the logarithm of a univalent function; later generalizations by Milin, starting from the Lebedev–Milin inequality, succeeded in exponentiating the inequalities to obtain inequalities for the coefficients of the univalent function itself. The Grunsky matrix and its associated inequalities were originally formulated in a more general setting of univalent functions between a region bounded by finitely many sufficiently smooth Jordan curves and its complement: the results of Grunsky, Goluzin and Milin generalize to that case.
In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings, Loewner's method was later developed in 1943 by the Russian mathematician Pavel Parfenevich Kufarev (1909–1968). Any family of domains in the complex plane that expands continuously in the sense of Carathéodory to the whole plane leads to a one parameter family of conformal mappings, called a Loewner chain, as well as a two parameter family of holomorphic univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions on the disk with positive real part. The Loewner semigroup generalizes the notion of a univalent semigroup.
In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.
Goodman's conjecture on the coefficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.
