Functions generating (m,M,Ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Psi }$$\end{document})-Schur-convex sums

The notion of (m,M,Ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m,M,\Psi )$$\end{document}-Schur-convexity is introduced and functions generating (m,M,Ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m,M,\Psi )$$\end{document}-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,Ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m,M,\Psi )$$\end{document}-Schur-convex sums if and only if it is (m,M,ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m,M,\psi )$$\end{document}-Wright-convex is proved and a characterization of (m,M,ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m,M,\psi )$$\end{document}-Wright-convex functions is given.


Introduction
Let (X, · ) be a real normed space. Assume that D is a convex subset of X and c is a positive constant. A function f : D → R is called: for all x, y ∈ D and t ∈ [0, 1]; -strongly Wright-convex with modulus c if for all x, y ∈ D and t ∈ [0, 1]; -strongly Jensen-convex with modulus c if (1) is assumed only for t = 1 2 , that is The usual concepts of convexity, Wright-convexity and Jensen-convexity correspond to the case c = 0, respectively. The notion of strongly convex functions was introduced by Polyak [22] and they play an important role in optimization theory and mathematical economics. Many properties and applications of them can be found in the literature (see, for instance, [10,15,19,[22][23][24]27]). Let us also mention the paper [18] by the second author which is a survey article devoted to strongly convex functions and related classes of functions.
In [1] the first author introduced the following concepts of (m, ψ)-lower convex, (M, ψ)-upper convex and (m, M, ψ)-convex functions (see also [2][3][4] Let us observe that if f ∈ B(D, m, M, ψ) then f − mψ and Mψ − f are convex and then (M − m)ψ is also convex, implying that M ≥ m whenever ψ is not trivial (i.e. is not the zero function).
If m > 0 and (X, · ) is an inner product space (that is, the norm · in X is induced by an inner product: x = x, x ) the notions of (m, · 2 )lower convexity and strong convexity with modulus m coincide. Namely, in this case the following characterization was proved in [19]: A function f is strongly convex with modulus c if and only if f − c · 2 is convex (for X = R n this result can be also found in [8,Prop. 1.1.2]). However, if (X, · ) is not an inner product space, then the two notions are different. There are functions f ∈ L(D, m, · 2 ) which are not strongly convex with modulus m, as well as there are functions strongly convex with modulus m which do not belong to L(D, m, · 2 ) (see the examples given in [6]).
If M > 0 and f ∈ U(D, M, ψ), then f is a difference of two convex functions. Such functions are called d.c. convex or δ-convex and play an important role in convex analysis (cf. e.g. [26] and the reference therein). Functions from the class U(D, M, · 2 ) with M > 0 were also investigated in [13] under the name approximately concave functions.
In [5] Dragomir and Ionescu introduced the concept of g-convex dominated functions, where g is a given convex function. Namely, a function f is called g-convex dominated, if the functions g + f and g − f are convex. Note that this concept can be obtained as a particular case of (m, M, ψ)-convexity by choosing m = −1, M = 1 and ψ = g. Observe also (cf. [1]), that in the case which is a convenient condition to verify in applications. Let I ⊂ R be an interval and x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ I n , where n ≥ 2. Following I. Schur (cf. e.g. [12,25]) we say that x is majorized by y, and write x y, if there exists a doubly stochastic n × n matrix P (i.e. a matrix containing nonnegative elements with all rows and columns summing up to 1) such that x = y · P . A function F : It is known, by the classical works of Schur [25], Hardy et al. [7] and Karamata [9] that if a function f : I → R is convex then it generates Schur-convex sums, that is the function F : I n → R defined by is Schur-convex. It is also known that the convexity of f is a sufficient but not necessary condition under which F is Schur-convex. A full characterization of functions generating Schur-convex sums was given by Ng [16]. Namely, he proved that a function f : I → R generates Schur-convex sums if and only if it is Wright-convex (cf. also [17]). Recently Nikodem et al. [20] obtained similar results in connection with strong convexity in inner product spaces. Let us also mention the paper by Olbryś [21] in which delta Schur-convex mappings are investigated.
The aim of this paper is to present some generalizations and counterparts of the above mentioned results related to (m, ψ)-lower convexity, (M, ψ)-upper convexity and (m, M, ψ)-convexity. We introduce the notion of (m, M, Ψ)-Schur-convex functions and give a sufficient and necessary condition for a function f to generate (m, M, Ψ)-Schur-convex sums. As a corollary we obtain a counterpart of the classical Hardy-Littlewood-Pólya majorization theorem. Finally we introduce the concept of (m, M, ψ)-Wright-convex functions, prove a representation theorem for them and present an Ng-type characterization of functions generating (m, M, Ψ)-Schur-convex sums. It is worth underlining, that our results concern a few different classes of functions related to convexity and are formulated in vector spaces, that is in a much more general setting than the original ones.

Main results
Let X be a real vector space. Similarly as in the classical case we define majorization in the product space X n . Namely, given two n-tuples x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ X n we say that x is majorized by y, written x y, if (x 1 , . . . , x n ) = (y 1 , . . . , y n ) · P for some doubly stochastic n × n matrix P .
In what follows we will assume that D is a convex subset of a real vector space X, ψ : D → R is a convex function and m, M ∈ R. For any n ≥ 2 define Ψ n : D n → R by We say that a function F : and If only condition (6) [condition (7)] is satisfied, we say that F is (m, Ψ n )-lower It follows from the fact that the function ψ is convex and so it generates Schur-convex sums Ψ n .
Given a function f : D → R and an integer n ≥ 2 we define the function F n : D n → R by Now, let D be a convex subset of a real vector space X, and let m, M ∈ R. Assume that ψ : D → R is a convex function and Ψ n : D n → R is defined by (5). We will prove now that (m, M, ψ)-convex functions generate (m, M, Ψ n )-Schur-convex sums.
As an immediate consequence of the above theorem, we obtain the following counterpart of the classical Hardy-Littlewood-Pólya majorization theorem [7].
We say that a function f : As was shown above in Theorems 1 and 2, if a function f : D → R is (m, M, ψ)-convex, then for every n ≥ 2 the corresponding function F n defined by (8)  Recall also that a subset D of a vector space X is said to be algebraically open if for every x ∈ D and for every y ∈ X there exists ε > 0 such that Proof. To prove (i) assume that f is (m, ψ)-lower Wright-convex and fix an n ≥ 2. Since the function g = f − mψ is Wright-convex, it is of the form g = g 1 + a, where g 1 is convex and a is additive (cf. [11]; here the assumption that D is algebraically open is needed). Therefore it generates Schur-convex sums. Thus, for x = (x 1 , . . . , x n ) y = (y 1 , . . . , y n ), we have g(x 1 ) + · · · + g(x n ) ≤ g(y 1 ) + · · · + g(y n ). Hence which means that that is F n is (m, Ψ n )-lower Schur-convex. Now, assume that for some n ≥ 2 the function F n is (m, Ψ n )-lower Schur-convex. Take y 1 , y 2 ∈ D and t ∈ (0, 1). Put and, if n > 2, take additionally x i = y i = z ∈ D for i = 3, . . . , n. Then x = (x 1 , . . . , x n ) y = (y 1 , . . . , y n ). Therefore, by (6), Hence, for g = f − mψ we get Thus g is Wright-convex, which means that f is (m, ψ)-lower Wright-convex. The proof of part (ii) is similar. Part (iii) follows from (i) and (ii).
Remark 6. In the special case where (X, · ) is an inner product space, ψ = · 2 and m = c > 0, parts (i) of the above Theorems 1, 4, 5 reduce to the results obtained in [20] for strong Schur-convexity. For m = 0 and X = R n they coincide with the Ng theorem [16]. Finally, we give a representation theorem for (m, M, ψ)-Wright-convex functions. It is known (and easy to check) that every convex function is Wrightconvex, and every Wright-convex function is Jensen-convex, but not the converse (some examples can be found in [18]). In [16] Ng proved that a function f defined on a convex subset of R n is Wright-convex if and only if it can be represented in the form f = f 1 + a, where f 1 is a convex function and a is an additive function (see also [18]). Kominek [11] extended that result to functions defined on algebraically open subset of a vector space. An analogous result for strongly Wright-convex functions was obtained in [14]. In the next theorem we give a similar representation for (m, M, ψ)-Wright-convex functions. In the proof we will use the following fact: Then a is an affine function on D.
Going to the limit we get which proves that a is affine on D. Proof. To prove (i) assume first that f is (m, ψ)-lower Wright-convex, that is h = f −mψ is Wright-convex. By the Ng representation theorem [16] (extended by Kominek [11] to functions defined on algebraically open domains), there exist a convex function h 1 : D → R and an additive function a 1 : X → R such that h = h 1 + a 1 on D. Then g 1 = h 1 + mψ belongs to L(D, m, ψ) and f = h + mψ = h 1 + a 1 + mψ = g 1 + a 1 , which was to be proved. Conversely, if f = g 1 + a 1 with some g 1 ∈ L(D, m, ψ) and a 1 additive, then f − mψ = g 1 − mψ + a 1 is Wright-convex as a sum of a convex function and an additive function. This shows that f is (m, ψ)-lower Wright-convex. which, by Lemma 5, implies that A = a 1 + a 2 is affine. Denote a = a 1 and g = f − a. Then which implies that g ∈ L(D, m, ψ) because h 1 is convex. Also Mψ − g = Mψ − f + a = h 2 + a 2 + a = h 2 + A, which implies that g ∈ U(D, m, ψ) because h 2 + A is convex. Thus g ∈ B(D, m, ψ) and f = g + a, which finishes the proof.