On T-Schur Convex Maps

We introduce and examine the notion of T-Schur convexity which is naturally connected with Schur convexity. As a particular case, we consider T-Wright convex maps which generalize a well-known and intensively investigated class of t-Wright convex functions. We discuss several properties of this class of functions. In the last part of the paper we give a characterization of T-Wright affine maps i.e. maps satisfying the corresponding functional equation.


Introduction
The relations between the inequality of convexity assumed for all numbers t ∈ [0, 1] and between the same inequality assumed for one fixed number t are now well known and classical. We will write a few words concerning these relations later in the paper.
Similarly natural is the question about the inequality of Schur convexity assumed for one fixed doubly stochastic matrix which, up to our knowledge, has not been studied yet. We will study the properties of such maps (which we call T -Schur convex) in this paper. But before we do this we need to recall some basic facts and notions connected with the notion of Schur convexity.
The relation of majorization defined above turns out to be a pre-ordering relation i.e. it is reflexive and transitive. This notation and terminology was introduced by Hardy et al. [5].
Recall that an n × n matrix S = [s ij ] is said to be doubly stochastic, if The well-known Hardy, Littlewood and Pólya theorem says that x ≺ y, if and only if, x = Sy for some doubly stochastic matrix S (in general, the matrix S is not unique).
The functions that preserve the order of majorization (in Schur's honor who first considered them in 1923 [16]) are said to be convex in the sense of Schur. Thus we say that a function f : W → R, where W ⊆ R n is Schur convex, if for all x, y ∈ W the implication holds. In the case, where W = I n with some interval I ⊆ R the above condition is equivalent to the following one f (Sx) ≤ f (x) for x ∈ I n and for all doubly stochastic matrices S ∈ R n×n .
The Schur convex functions have many important applications in analytic inequalities, elementary quantum mechanics and quantum information theory. A survey of results concerning a majorization and Schur convex functions may be found in an extensive monograph by Arnold et al. [1]. Now, assume that D is a convex subset of a real linear space. Recall that a function f : If the above inequality is satisfied for all x, y ∈ D and fixed number t ∈ (0, 1), then we say that f is a t-convex. If t = 1 2 then f is said to be convex in the sense of Jensen.
Obviously, each convex function is t-convex for all t ∈ (0, 1), in particular, convex in the sense of Jensen. The converse implication does not hold in general. Indeed, fix t ∈ (0, 1). Any discontinuous additive function a : R → R i.e. a solution of the Cauchy's functional equation satisfying additionally the condition is an example of t-convex and Jensen-convex function which is not convex. (proof of the existence of such function can be found, for example, in [7], Theorem 5.4.2). On the other hand, every t-convex function has to be convex in the sense of Jensen. This result was proved by Kuhn in [8]. An easy proof of this fact was done by Daróczy and Páles in [4].
In 1954 Wright [18] introduced a new convexity property. A function Clearly, each convex and additive function is Wright convex, and each Wrightconvex functions is convex in the sense of Jensen.
The following theorem shows the connection between the classes of Schur convex and Wright convex functions.
Theorem 1 [12]. Let D ⊆ R m be a nonempty open and convex set, f : where w : D → R is a convex function, and a : R m → R is an additive function.
If the inequality (1) is satisfied for all x, y ∈ D and a fixed number t ∈ (0, 1), then we say that f is a t-Wright convex function. The definition of t-Wright convex functions was introduced by Matkowski in [11]. The connection between t-Wright convexity and Jensen convexity was investigated in [10,15]. In [15] the necessary and sufficient topological conditions under which every t-Wright convex function has to be Jensen convex are given. In [10] the authors solved an algebraic problem posed by Matkowski in [11], who asked whether a t-Wright convex function with a t ∈ (0, 1) has to be Jensen convex? In [10] Maksa et al. gave the positive answer to the problem of Matkowski for all rational t ∈ (0, 1) and certain algebraic values of t. However, they proved that if t is either transcendental or the distance of some of the algebraic (maybe complex) conjugate of t from 1 2 is at least 1 2 , then there exists a function which is t-Wright convex but not Jensen convex.
In the class of continuous functions the notions of: convexity, Jensen convexity, Wright convexity and t-Wright convexity coincide.
The organization of the paper is as follows: in Sect. 2 we introduce the notion of T -Schur convexity and we give some background results which are the starting point for our further considerations. In Sect. 3 we introduce a notion of T -Wright convexity as a natural generalization of usual t-Wright convexity. We prove that the local boundedness at a point of T -Wright convex maps implies its local boundedness at every point, moreover, we show that any semi-continuous T -Wright convex map has to be convex. Section 4 is devoted to the separation theorem for T -Wright convex maps. We show that if f and −g are T -Wright convex functions satisfying g ≤ f then there exists a T -Wright affine function h such that g ≤ h ≤ f . In the last section we give a characterization of T -Wright affine maps. This result generalizes a theorem proved by Lajko [9] who gave a characterization of t-Wright affine function.

T -Schur Convex Maps
Through this paper (unless explicitly stated otherwise) D stands for a convex subset of a real linear space, n ∈ N, n ≥ 2 is a fixed number and T ∈ R n×n is a fixed doubly stochastic matrix. Motivated by the concept of Schur-convexity we introduce the notion of T -Schur-convexity in the following way.
If f : D n → R is a function such that −f is T -Schur convex then f is called T -Schur concave. If f is at the same time T -Schur convex and T -Schur concave then we say that it is a T -Schur affine. In this case f satisfies the following functional equation This definition together with some results was presented by the first author at the XII International Symposium on Generalized Convexity and Monotonicity held in Hajduszoboszlo, Hungary from August 27 to September 2, 2017, whereas a very particular case (n = 2) the class of T -Schur convex functions was examined by Burai and Makó in [3]. Now, given a function f : D n → R we consider the set Obviously, for any function f the above set is nonempty, because the identity matrix I is a member of W f .

Proposition 1.
For every function f : D → R the following implication holds: in particular, Proof. Take arbitrarily T, S ∈ W f . Then, directly from the definition, we get For a fixed matrix T define the set where I is the identity matrix. Let us define the relation T on D n in the following manner It is easy to observe that if T ∈ R n×n is a doubly stochastic matrix which is not a permutation matrix then the relation T defines a partial order on D n , moreover, a function f : D n → R is a T -Schur convex if and only if Now, we prove a separation type theorem for T -Schur convex maps.
Proof. First we define two sequences of functions f k , g k : D n → R by the formulas By the assumption for k ∈ N, x ∈ D n . Then (g k ) k∈N is an increasing and bounded above sequence of functions (similarly (f k ) k∈N is a decreasing and bounded below sequence of functions). Putting we clearly have which means that h is a T -Schur affine map and this finishes the proof. To quote the next result which we will use in the sequel we need the following definition.

Definition 2.
A doubly stochastic matrix T ∈ R n×n is called semi-positive if all entries of some power T m are positive.
Theorem 3 [2]. If T ∈ R n×n is semi-positive doubly stochastic matrix, then As the following two simple examples show, the above theorem without the assumption of semi-positivity of T is not true.
has the property I = I 2 = I 3 = . . . = I m , m ∈ N and therefore lim m→∞ I m = I.

Theorem 4. Let D be a convex subset of a real linear topological space, let
T ∈ R n×n be a semi-positive doubly stochastic matrix, and let f : Proof. Let x = (x 1 , . . . , x n ) ∈ D n and denote by x = x1+···+xn n , . . . , x1+···+xn n . Using the semi-continuity of f and Theorem 3 we get Vol. 75 (2020) On T -Schur Convex Maps Page 7 of 22 30

T -Wright Convex Maps
In this section, motivated by [12], we will consider a T -Schur convex sums i.e. T -Schur convex maps g : D n → R of the form: where f : D → R is a given function defined on a convex subset D of a real linear space. Let T = (t ij ), i, j = 1, 2, . . . , n be a fixed doubly stochastic matrix. The function g of the form (2) is T -Schur convex if and only if it satisfies the following functional inequality: Observe that the class of functions satisfying the inequality (3) generalizes the class of t-Wright convex functions. Indeed, an arbitrary doubly stochastic matrix T ∈ R 2×2 is of the form thus a function f : D → R satisfies the inequality (3), with n = 2 if and only if it is a t-Wright-convex function. This allows us to formulate the following definition. In the proof of our next result (and in the sequel) we will use the following theorem which is a particular case of Lemma 3.7 from [6]. As an immediate consequence of the above result and Theorem 4 we obtain the following theorem. Putting x 1 := x, x 2 = . . . = x n = y we get so f is a 1 n -convex function and due to the Kuhn's result [8] (see also [4]) convex in the sense of Jensen.
Let D be a subset of a topological space. Recall that a function f : D → R is called locally bounded (locally bounded above, locally bounded below) at a point x 0 ∈ D if there exists a neighbourhood U of x 0 such that the function f is bounded (bounded above, bounded below) on U ∩ D. The next theorem refers to the local boundedness below of T -Wright convex maps and generalized corresponding theorem for t-Wright convex functions obtained in [13] (Theorem 2, p. 404).

Theorem 7. Let D be an open and convex subset of a locally convex real linear topological space, let T ∈ R n×n be a semi-positive doubly stochastic matrix and let f : D → R be a T -Wright convex function. If f is locally bounded below at a point x 0 ∈ D then it is locally bounded below at every point x ∈ D.
Proof. By the assumption there is a neighbourhood U x0 of x 0 and a real number m such that Without loss of generality we may assume that U x0 is a convex set. For an arbitrary number k ∈ N 0 = N ∪ {0} we put Note that V k is a convex neighbourhood of x 0 , k ∈ N 0 . We will prove by induction that for all k ∈ N 0 If k = 0 then the above inequality coincides with (4). Assume (5) for some k ∈ N. Fix an arbitrary point y ∈ V k+1 . There exists a z ∈ U x0 such that

Vol. 75 (2020)
On T -Schur Convex Maps Page 9 of 22 30 From the convexity of the set V k we get Now, using the fact that the function φ : R → X given by the formula Since by Theorem 3 lim k→∞ T k = 1 n E, then where T k = (t k ij ) i,j=1,...,n , k ∈ N. In view of the fact that T k ∈ W f , on account of Proposition 1 and from the formula (3) for which together with the induction assumption implies that or, equivalently, Thus m f is a function defined in D and with values in ∞). Note that definition (6) implies that Observe that, if for some In the proof of our next result we will use the Theorem 5 and the following theorem which is a particular cases of Theorem 4.1 from [6].
We will show that m f is a T -Wright convex function. To do it, fix x 1 , . . . , x n ∈ D and ε > 0 arbitrarily. Put By the definition of m f there exists a convex neighbourhood of zero U such that Vol. 75 (2020) On On the other hand, there exist the points r i ∈ U xi such that By the convexity of U, we get By (7) and (8) and T -Wright convexity of f we have Passing here with ε to zero, we obtain the T -Wright convexity of m f . From Theorem 8 we infer that the function m f is lower semi-continuous in D. Consequently, m f is convex in D on account of Theorem 6.

Separation Theorem for T -Wright Convex Function
In this section we deal with the separation problem for T -Wright convex maps. Let n ∈ N, n ≥ 2 and let k ∈ {1, . . . , n}. Throughout this section A k and B k will denote two disjoint sets such that A k ∪ B k = {1, . . . , n}\{k}.
In the proof of the main result of this section we will need the following technical lemma (we use here the convention that the sum over an empty set of indexes is equal to zero). .
Proof. To prove the first assertion suppose that h is a T -Wright-convex function i.e.
h(x i ), x 1 , . . . , x n ∈ D. Subtracting the expression i∈A k h(x i ) + j∈B k g(x j ) from both sides of the above inequality, we get Now, taking the infimum with respect to all x j ∈ D, j ∈ B k , we obtain In order to prove the second statement suppose that g is a T -Wright concave map. We will show that the function h k is T -Wright convex. To this end fix the points y 1 , . . . , y n ∈ D arbitrarily. Fix arbitrary real numbers s 1 , . . . , s n such that h k (y j ) < s j , j = 1, . . . , n.
By definition of h k there exist points u p j ∈ D for p = 1, . . . , n, j ∈ B k such that Therefore, using the T -Wright convexity of h and −g we obtain n p=1 Vol. 75 (2020) On Tending in the above inequalities with s p to h k (y p ), p = 1, . . . , n we get the T -Wright convexity of h k . Now, we prove the separation type theorem for T -Wright convex map. The corresponding theorem for t-Wright convex functions was proved in [14].
then there exists a T -Wright affine function h : D → R such that Proof. Observe that without loss of generality we may assume that Note that H = ∅, since f ∈ H. The pair (H, ) yields on partially ordered set, where an order relation is defined as follows We will show that any chain in H has a lower bound in H. Let L ⊂ H be an arbitrary chain. Define the function k : D → [−∞, ∞) by the formula Clearly, which, in particular, implies that k has finite values. Now, we will show that k is a T -Wright convex map. To prove it fix x 1 , . . . , x n ∈ D and a number ε > 0 arbitrarily. There exist k 1 , . . . , k n ∈ H such that Therefore, by the definition of k and T -Wright convexity of k 1 , . . . , k n we obtain where p = min {k 1 , . . . , k n }. By Kuratowski-Zorn lemma there exists a minimal element h in H. Since h ∈ H, the inequalities By Lemma 1 the function belongs to the family H, moreover, By the minimality of h the inequality holds for every k ∈ {1, . . . , n} and all x 1 , . . . , x k−1 , x k , x k+1 , . . . , x n ∈ D. In particular, putting k = 1, we get or equivalently, Vol. 75 (2020) On belongs to the family H, moreover, Therefore by the minimality of h we have or equivalently, for all x 1 , . . . , x n ∈ D.
Repeating this procedure n-times in the last step we obtain For arbitrarily chosen x 1 , . . . , x n−1 ∈ D we define the function h n : D → R (as a function of one variable) by the formula It follows from Lemma 1 (applied for k = n, A k = {1, . . . , n − 1}, B k = ∅) that h n ∈ H, moreover, Using again the minimality of h, we obtain which ends the proof.

The Corresponding Functional Equation
Let X be a real linear space and let D be a convex subset of X. In this part of the paper we will deal with the corresponding functional equation: However without any additional assumption on f nothing interesting can be said about the solutions. For example every function f : D n → R of the form f (x 1 , .., x n ) := Φ(x 1 + · · · + x n ), with arbitrary Φ : X → R is a solution to the above equation. Therefore we will give a representation of T -Wright affine functions i.e. we solve the functional equation To deal with this equation some definitions and results from the paper of Székelyhidi [17] will be needed. In this part of the paper G will be a topological, Abelian group and H will be an Abelian group.
holds for i = 1, . . . , n whenever x 1 , . . . , x i−1 , x i ,x i , x i+1 , . . . , x n , x i +x i ∈ U We will use also the following notation, if n is a positive integer and A : U n → H is a given function then the diagonalization of A, diagA : U → H is defined by diagA(x) := A(x, . . . , x), x ∈ U.
In the next definition we introduce the notion of local polynomials. holds whenever x − x 0 ∈ U (here U 0 = U and a 0-additive function is a constant). Now we may quote the main result from [17].