Functions with a Concave Modulus of Continuity

The modulus of continuity of analytic functions and CR- geometry

1 Introduction Contour and solid modulus of continuity. The one-dimensional case There has been an extensive study of the relation between the " contour " and the " solid " moduli of continuity for analytic functions in planar domains, starting with the following beautiful result of Hardy and Littlewood. For a domain G in C n (n ≥ 1) we denote as usual by A(G) the algebra of analytic functions ...

Majorization of the Modulus of Continuity of Analytic Functions

Let G be a bounded domain in the complex plane, let f be analytic in G and continuous in G, and let μ be a majorant, that is, a non-negative non-decreasing function defined for t ≥ 0 such that μ(2t) ≤ 2μ(t) for all t ≥ 0. Suppose that z1 ∈ ∂G and that |f(z1) − f(z2)| ≤ μ(|z1 − z2|) for all z2 ∈ ∂G. We show that then |f(z1)−f(z2)| ≤ Cμ(|z1−z2|) for all z2 ∈ G where C = 3456. If the assumption is...

The Sugeno fuzzy integral of concave functions

The fuzzy integrals are a kind of fuzzy measures acting on fuzzy sets. They can be viewed as an average membershipvalue of fuzzy sets. The value of the fuzzy integral in a decision making environment where uncertainty is presenthas been well established. Most of the integral inequalities studied in the fuzzy integration context normally considerconditions such as monotonicity or comonotonicity....

Modulus of continuity on parts of the boundary and solid modulus of continuity

Model Selection for Regression with Continuous Kernel Functions Using the Modulus of Continuity

This paper presents a new method of model selection for regression problems using the modulus of continuity. For this purpose, we suggest the prediction risk bounds of regression models using the modulus of continuity which can be interpreted as the complexity of functions. We also present the model selection criterion referred to as the modulus of continuity information criterion (MCIC) which ...