 | This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations .(August 2022) |
Goodman's conjecture on the coefficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.
Let be a -valent function. The conjecture claims the following coefficients hold:
It's known that when , the conjecture is true for functions of the form where is a polynomial and is univalent.
In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f from U onto the open unit disk
The de Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function H(λ,z), where λ is a real parameter and z is a complex variable. More precisely,
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
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 (1916) and finally proven by Louis de Branges (1985).
In complex analysis, the Hardy spacesHp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz, who named them after G. H. Hardy, because of the paper. In real analysis Hardy spaces are certain spaces of distributions on the real line, which are boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the Lp spaces of functional analysis. For 1 ≤ p < ∞ these real Hardy spaces Hp are certain subsets of Lp, while for p < 1 the Lp spaces have some undesirable properties, and the Hardy spaces are much better behaved.
In algebra, given a polynomial
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime. A torus knot is trivial if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.
In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by Lebedev and Milin (1965) and Isaak Moiseevich Milin (1977). It was used in the proof of the Bieberbach conjecture, as it shows that the Milin conjecture implies the Robertson 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 mathematics, and particularly complex dynamics, the escaping set of an entire function ƒ consists of all points that tend to infinity under the repeated application of ƒ. That is, a complex number belongs to the escaping set if and only if the sequence defined by converges to infinity as gets large. The escaping set of is denoted by .
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.
Walter Kurt Hayman FRS was a British mathematician known for contributions to complex analysis. He was a professor at Imperial College London.
Adolph Winkler Goodman was an American mathematician who contributed to number theory, graph theory and to the theory of univalent functions: The conjecture on the coefficients of multivalent functions named after him is considered the most interesting challenge in the area after the Bieberbach conjecture, proved by Louis de Branges in 1985.
In mathematics, the mean value problem was posed by Stephen Smale in 1981. This problem is still open in full generality. The problem asks:
In mathematics, Cohn's theorem states that a nth-degree self-inversive polynomial has as many roots in the open unit disk as the reciprocal polynomial of its derivative. Cohn's theorem is useful for studying the distribution of the roots of self-inversive and self-reciprocal polynomials in the complex plane.
James Allister Jenkins was a Canadian–American mathematician, specializing in complex analysis.
Patrick Brendan Kennedy was an Irish chess champion, and an academic in Mathematics, notable for his work in complex analysis.
