Complex analysis of univalent function
What is a univalent function in complex analysis?
In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective..
What is the area theorem for univalent functions?
By the area theorem for a certain class of univalent functions f(z), zu220.
- B, with B a domain, one usually understands any inequality having the property that equality holds if and only if the area of the complement \xafG of f(B) is zero, and the same applies for a class of systems of univalent functions {fk(z):zu220
- Bk}nk=0,n=1,
What is univalent function in complex analysis?
From Encyclopedia of Mathematics.
A regular or meromorphic function f in a domain B of the extended complex plane \xafC such that f(z1)≠f(z2) whenever z1≠z2, z1,z2u220.
- B, that is, f is a one-to-one mapping from B into \xafC
.Jun 6, 2020
- We say that f is holomorphic on Ω if f is complex differentiable at each point of Ω.
The function f : Ω → C is then called the complex derivative of f or just the derivative.
If there exists a holomorphic function F defined on Ω such that F = f, we say that F is a primitive of f.
In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.ExamplesBasic propertiesComparison with real functions
Study of space and shapes locally given by a convergent power series
Geometric function theory is the study of geometric properties of analytic functions.
A fundamental result in the theory is the Riemann mapping theorem.
In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems.
Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representation as dilations of a univalent holomorphic mapping, or a semigroup of mappings, of the unit disk in the complex numbers into itself.
In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle.
This result, the Herglotz-Riesz representation theorem, was proved independently by Gustav Herglotz and Frigyes Riesz in 1911.
It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part.
Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients.
