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Fekete–Szegő Inequality
In mathematics, the Fekete–Szegő inequality is an inequality for the coefficients of univalent analytic functions In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ... found by , related to the Bieberbach conjecture. 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 * Inequalities (mathematics) {{mathanalysis-stub ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Univalent Function
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. Examples The function f \colon z \mapsto 2z + z^2 is univalent in the open unit disc, as f(z) = f(w) implies that f(z) - f(w) = (z-w)(z+w+2) = 0. As the second factor is non-zero in the open unit disc, z = w so f is injective. Basic properties One can prove that if G and \Omega are two open connected sets in the complex plane, and :f: G \to \Omega is a univalent function such that f(G) = \Omega (that is, f is surjective), then the derivative of f is never zero, f is invertible, and its inverse f^ is also holomorphic. More, one has by the chain rule :(f^)'(f(z)) = \frac for all z in G. Comparison with real functions For real analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Functions
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if for every x_0 in its domain, its Taylor series about x_0 converges to the function in some neighborhood of x_0 . This is stronger than merely being infinitely differentiable at x_0 , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \cdots in which the coefficients a_0, a_1, \dots ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 was posed by and finally proven by . The statement concerns the Taylor coefficients a_n of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that a_0=0 and a_1=1. That is, we consider a function defined on the open unit disk which is holomorphic and injective ('' univalent'') with Taylor series of the form :f(z)=z+\sum_ a_n z^n. Such functions are called ''schlicht''. The theorem then states that : , a_n, \leq n \quad \textn\geq 2. The Koebe function (see below) is a function for which a_n=n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute value of the nth coefficient. Schlicht functions The normalization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |