Characterization of holomorphic functions in terms of their moduli

It was shown in (Boche, H. and Pohl, V., 2005, Spectral factorization in the disk algebra. Complex Variables. Theory and Applications, 50, 383–387.) that if the modulus |f| of a function is continuous in the closure of the unit disk, the function f itself needs not to be continuous there, in general. This article shows that if the modulus of continuity of a function is a weak regular majorant, the continuity of the modulus always implies the continuity of the function itself.     

Iteration operators on a set of continuous functions of Baire space

Continuous functions on Baire space are considered. Iteration operators are defined on a set of continuous functions. The idea of a module of continuity of a function is introduced. The condition for the growth of module of continuity φ whose satisfaction guarantees that for any enumerable sequence of integration operators and any natural n there exists (n + 1) argument function with the module of continuity φ which cannot be obtained from n-argument functions with the module of continuity φ using any operator of this sequence is formulated. Examples of iteration operators are given.     

Nonlinear scalarizing functions in set optimization problems

In this paper, we obtain Hölder continuity of the nonlinear scalarizing function for l-type less order relation, which is introduced by Hernández and Rodríguez-Marín (J. Math. Anal. Appl. 2007;325:1–18). Moreover, we introduce the nonlinear scalarizing function for u-type less order relation and establish continuity, convexity and Hölder continuity of the nonlinear scalarizing function for u-type less order relation. As applications, we firstly obtain Lipschitz continuity of solution mapping to the parametric equilibrium problems and then establish Lipschitz continuity of strongly approximate solution mappings for l-type less order relation, u-type less order relation and set less order relation to the parametric set optimization problems by using convexity and Hölder continuity of the nonlinear scalarizing functions.     

Analysis of Symmetric Matrix Valued Functions

For any symmetric function f: ? ⁿ  → ? ⁿ , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.     

Construction of nonlinear regulators in economic growth models

We prove an inequality for the second moduli of continuity of continuous functions. Applying this inequality, we construct a nonnegative nonincreasing continuous function ω on [0, +g8) that vanishes at zero and is such that the function ω(δ)/δ ² decreases on (0, +g8) while ω is not asymptotically (as δ → 0) equivalent to the second modulus of continuity of any continuous function.     

On superpositions of continuous functions defined on the baire space

For continuous functions specified on the Baire space, conditions for the representability of a function of several variables as a superposition of functions of a smaller number of variables are considered. With the use of linear functions of the form (1+α)t, a boundary value of the modulus of continuity separating the positive from the negative solution of the problem is found. For the case in which the problem has a negative solution, a constructive method for obtaining (n+1)-variable continuous functions with modulus of continuity ϕ(t) that are not representable as superposition ofn-variable continuous functions with the same modulus of continuity ϕ(t) is suggested. Translated fromMatermaticheskie Zametki, Vol. 66, No. 5, pp. 696–705, November, 1999.     

Local times of continuous N-parameter strong martingales

Let M be a 4N-integrable, real-valued continuous N-parameter strong martingale. By extending Itô-type formulas for M to a function whose 2Nth derivative is Dirac's δ-distribution, Tanaka-type formulas for M are obtained. They represent local time of M with respect to occupation time scaled by the N-fold product of the Stieltjes measure defined by the quadratic variation of M and its kth derivatives in space, where k ≤ N − 1. Applications of Doob's and Burkholder's inequalities give continuity properties: space time continuity for local time, space continuity for the derivatives. In case N is even, for the continuity of the (N − 1)st derivative an additional condition on M is needed which may have a relation to the existence of local times of M w.r.t. different occupation time scales.     

Some properties of compatibility function based on <Emphasis Type="Italic">T</Emphasis>-operation

In this paper, the concept of compatibility function is proposed for fuzzy value with respect to a kind of the binomial operations which is called t-operation. Besides the continuity of compatibility function on three kinds of metric spaces are discussed, respectively.     

A multidimensional analog of a theorem due to zygmund

Zygmund proved an inequality describing the dependence of the modulus of continuity of the adjoint function on that of the original function lying in the space of 2π-periodic continuous functions. The present article contains estimates of partial moduli of continuity of the adjoint function of several variables in the spaceC. Examples show that these estimates are sharp. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 717–727, May, 1997. Translated by A. M. Chebotarev     

Hölder-continuity of certain generalized convex functions

It is proved that for a rationally s-convex function continuity and local s-Hölder - Continuity are equivalent at each interior point of the domain of definition of the function. Furthermore, it is shown that a rationally s-convex function which is bounded on a nonempty open convex set is s-Hölder-continuous on every compact subset of this set.     

Representation of preference orderings on Lp -spaces by integral functionals: discounting,continuity and TAS utility

This paper presents an axiomatic approach in a continuous time framework for representing preference orderings on L^p -spaces in terms of integral functionals. We show that if preference orderings on L^p -spaces satisfy continuity, separability, sensitivity, substitutability, additivity and lower boundedness, then there exists a utility function for the preference orderings such that the utility function is an integral functional with an upper semicontinuous integrand satisfying the growth condition. Moreover, if the preference orderings satisfy the continuity with respect to the weak topology of L^p -spaces, then the integrand is a concave integrand. As a result, time additive separable (TAS) utility functions with constant discount rates are obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the continuity of minimizers for quasilinear functionals

In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by log log(1/|x|)⁻¹. Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 103–116.     

Some approximation results involving the q‐Szász–Mirakjan–Kantorovich type operators via Dunkl's generalization

The purpose of this paper is to introduce a family of q‐Szász–Mirakjan–Kantorovich type positive linear operators that are generated by Dunkl's generalization of the exponential function. We present approximation properties with the help of well‐known Korovkin's theorem and determine the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and the second‐order modulus of continuity. Furthermore, we obtain the approximation results for bivariate q‐Szász–Mirakjan–Kantorovich type operators that are also generated by the aforementioned Dunkl generalization of the exponential function. Copyright © 2017 John Wiley & Sons, Ltd.     

Modulus of continuity of harmonic functions

LetG be a bounded plane domain, the diameters of whose boundary components have a fixed positive lower bound. Letu be harmonic inG and continuous in the closureG ofG. Suppose that the modulus of continuity ofu on the boundary ofG is majorized by a function of a suitable type. We shall then obtain upper bounds for the modulus of continuity ofu inG. Further, we shall show that in some situations these estimates cannot be essentially improved. We shall also consider the same problem for certain bounded domains in space. Research partially supported by the U.S. National Science Foundation. AMS (1980) Classification. Primary 31A05.     

Extension of functions preserving the modulus of continuity

The problem of the extension of a real-valued function from a subset of a metric space to the entire space is treated. An extension operator preserving the modulus of continuity of a function is proposed and its properties are studied. An application to the problem of the trace of a locally Lipschitz function on a compact subset of a metric space is given. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 236–245, February, 1997. Translated by N. K. Kulman     

A characterization of almost resolvable spaces

A space is said to be resolvable if it has two disjoint dense subsets. It is shown thatX is a Baire space with no resolvable open subsets iff every real function defined onX has a dense set of points of continuity. Thus almost resolvable spaces, as defined by Bolstein, are shown to be characterized as the union of a first category set and a closed resolvable set.     

On a <Emphasis Type="Italic">q</Emphasis>-analogue of Stancu operators

This paper is concerned with a generalization in q-Calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established.     

Nonparametric estimation of compound distributions with applications in insurance

A nonparametric estimator of the distribution functionG of a random sum of independent identically distributed random variables, with distribution functionF, is proposed in the case where the distribution of the number of summands is known and a random sample fromF is available. This estimator is found by evaluating the functional that mapsF ontoG at the empirical distribution function based on the random sample. Strong consistency and asymptotic normality of the resulting estimator in a suitable function space are established using appropriate continuity and differentiability results for the functional. Bootstrap confidence bands are also obtained. Applications to the aggregate claims distribution function and to the probability of ruin in the Poisson risk model are presented.     

The Correction Theorem for Anisotropic Spaces

The following old problem is solved. Given an > 0, a function f:[0,1]ⁿ , and the partial moduli of continuity of this function evaluated in a symmetric space X, find a set I() of measure larger than 1- such that the partial uniform moduli of continuity of f determined for the points of this set admit an unimprovable (with respect to all restrictions to sets of measure larger than 1- ) estimate of partial uniform moduli of continuity and write out this estimate of the uniform partial moduli of continuity.     

A geometrical approach to monotone functions in \mathbb R^n

The paper is concerned with the fine properties of monotone functions on . We study the continuity and differentiability properties of these functions, the approximability properties, the structure of the distributional derivatives and of the weak Jacobians. Moreover, we exhibit an example of a monotone function u which is the gradient of a convex function and whose weak Jacobian Ju is supported on a purely unrectifiable set. Received October 9, 1996; in final form April 21, 1997