Wright-convex Functions Research Articles

Strong geometric derivatives

For a real valued function f, defined on an open interval I, and an arbitrary real number h, we consider the lower and upper limits of2n(f(y+h2n)−f(y)) whenever n tends to infinity and y tends to a fixed element x of I. We consider two families of functions determined by the properties of these limits. The first interesting property is when these lower and upper limits are finite and equal to each other for every real number h and every x in I. The second notable family is determined by the property that both limits are finite and increasing (in a specific sense) with respect to x for every positive number h. These properties are motivated by the families of continuously differentiable functions and convex functions, respectively. However, restrictions of additive mappings belong to these classes as well. Our decomposition theorems establish that these motivating examples generate the whole classes. Namely, every function belonging to the first family can be represented as the sum of a continuously differentiable function and an additive one, while every function taken from the second family turns out to be the sum of a convex function and an additive one. We apply our results in order to give a local and approximate characterization of affine functions and Wright-convex functions, respectively.

On Functions with Monotonic Differences

Abstract Motivated by the Szostok problem on functions with monotonic differences (2005, 2007), we consider a-Wright convex functions as a generalization of Wright convex functions. An application of these results to obtain new proofs of known results as well as new results is presented.

Remarks on Wright-convex functions

AbstractIn the present paper we prove a generalized version of the famous decomposition theorem of Ng. We also focus on the problem posed by Zsolt Páles concerning the Wright-convex functions.

On convexity properties with respect to a Chebyshev system

The main purpose of this paper is to introduce various convexity concepts in terms of a positive Chebyshev system ω and give a systematic investigation of the relations among them. We generalize a celebrated theorem of Bernstein–Doetsch to the setting of ω-Jensen convexity. We also give sufficient conditions for the existence of discontinuous ω-Jensen affine functions. The concept of Wright convexity is extended to the setting of Chebyshev systems, as well, and it turns out to be an intermediate convexity property between ω-convexity and ω-Jensen convexity. For certain Chebyshev systems, we generalize the decomposition theorems of Wright convex and higher-order Wright convex functions obtained by C. T. Ng in 1987 and by Maksa and Páles in 2009, respectively.

An Elementary Proof for the Decomposition Theorem of Wright Convex Functions

Abstract The main goal of this paper is to give a completely elementary proof for the decomposition theorem of Wright convex functions which was discovered by C. T. Ng in 1987. In the proof, we do not use transfinite tools, i.e., variants of Rodé’s theorem, or de Bruijn’s theorem related to functions with continuous differences.

Functions generating (m,M,varvec{Psi })-Schur-convex sums

The notion of (m,M,Psi )-Schur-convexity is introduced and functions generating (m,M,Psi )-Schur-convex sums are investigated. An extension of the Hardy–Littlewood–Pólya majorization theorem is obtained. A counterpart of the result of Ng stating that a function generates (m,M,Psi )-Schur-convex sums if and only if it is (m,M,psi )-Wright-convex is proved and a characterization of (m,M,psi )-Wright-convex functions is given.

An elementary proof for the majorization principle for Wright-convex functions

We give another elementary proof for the majorization principle for Wright-convex functions. This inequality is due to Ng.

On the classes of higher-order Jensen-convex functions and Wright-convex functions, II

Recently Nikodem, Rajba and Wąsowicz compared the classes of n-Wright-convex functions and n-Jensen-convex functions by showing that the first one is a proper subclass of the latter one, whenever n is an odd natural number. Till now the case of even n was an open problem. In this paper the complete solution is given: it is shown that the inclusion is proper for any natural n. The classes of strongly n-Wright-convex and strongly n-Jensen-convex functions are also compared (with the same assertion).

Corrigendum to "Hermite-Hadamard type inequalities for Wright-convex functions of several variables" [Opuscula Math. 35, no. 3 (2015), 411-419

We correct a small mistake made by the authors of the paper [Hermite-Hadamard type inequalities for Wright-convex functions of several variables, Opuscula Math. 35, no. 3 (2015), 411-419].

Characterizations and decomposition of strongly Wright-convex functions of higher order

Motivated by results on strongly convex and strongly Jensen-convex functions by R. Ger and K. Nikodem in (Strongly convex functions of higher order, Nonlinear Anal. 74 (2011), 661-665) we investigate strongly Wright-convex functions of higher order and we prove decomposition and characterization theorems for them. Our decomposition theorem states that a function f is strongly Wright-convex of order n if and only if it is of the form f(x) = g(x)+p(x)+cx n+1 , where g is a (continuous) n-convex function and p is a polynomial function of degree n. This is a counterpart of Ng's decomposition theorem for Wright-convex functions. We also characterize higher order strongly Wright-convex functions via generalized derivatives.

Hermite-Hadamard type inequalities for Wright-convex functions of several variables

We present Hermite-Hadamard type inequalities for Wright-convex, strongly convex and strongly Wright-convex functions of several variables defined on simplices.

Symmetrization, convexity and applications

Based on permutation enumeration of the symmetric group and ‘generalized’ barycentric coordinates on arbitrary convex polytope, we develop a technique to obtain symmetrization procedures for functions that provide a unified framework to derive new Hermite–Hadamard type inequalities. We also present applications of our results to the Wright-convex functions with special emphasis on their key role in convexity. In one dimension, we obtain (up to a positive multiplicative constant) a method of symmetrization recently introduced by Dragomir (2014) [3], and also by El Farissi et al. (2012/2013) [4]. So our approach can be seen as a multivariate generalization of their method.

On the classes of higher-order Jensen-convex functions and Wright-convex functions

The classes of n-Wright-convex functions and n-Jensen-convex functions are compared with each other. It is shown that for any odd natural number n the first one is the proper subclass of the second one. To reach this aim new tools connected with measure theory are developed.

A majorization inequality for Wright-convex functions revisited

We give an elementary proof of a majorization inequality concerning Wright-convex functions.

Remarks on strongly Wright-convex functions

Remarks on strongly Wright-convex functions

Measurability of (M, N)-Wright convex functions

Let \(I \subset {\mathbb{R}}\) be an open interval and \(M, N : I^2 \longrightarrow I\) be means on I. Let \( \varphi : I \longrightarrow {\mathbb{R}}\) be solution of the functional equation $$\varphi(M(x, y)) + \varphi(N(x, y)) = \varphi(x) + \varphi(y), \quad \quad x, y \in I$$ . We give sufficient conditions on \(M, N\) and the function \(\varphi\) such that for every Lebesgue measurable solution \(f : I \longrightarrow \mathbb{R}\) of the functional inequality $$f(M(x, y)) + f(N(x, y)) \leq f(x) + f(y), \quad \quad x, y \in I$$ , the function \(f \circ \varphi^{-1} : \varphi(I) \longrightarrow {\mathbb{R}}\) is convex.

Decomposition of higher-order Wright-convex functions

In this paper we consider higher-order Wright-convex functions and prove that they are representable as the sum of a continuous higher-order convex function and a polynomial function.

ON QUASI CONVEX FUNCTIONS AND HADAMARD'S INEQUALITY

In this paper we establish some inequalities of Hadamard’s type involving Godunova-Levin functions, P-functions, quasi-convex functions, J-quasi-convex functions, Wright-convex functions and Wright-quasi-convex functions.

On quasi convex functions and Hadamardʼs inequality

AbstractIn this paper we establish some inequalities of Hadamardʼs type involving Godunova-Levin functions, P-functions, quasi-convex functions, J-quasi-convex functions, Wright-convex functions and Wright-quasi-convex functions.

Some conditions implying the continuity of $t$-Wright convex functions

Some conditions implying the continuity of $t$-Wright convex functions