Characterization of Inner Product Spaces by Strongly Schur-Convex Functions

Involving the notion of strongly Schur-convex functions we give a new characterization of inner product spaces among norm spaces. We also present a representation theorem for functions which generate strongly Schur-convex sums.


Introduction
A rich collection of characterizations of inner product spaces among norm spaces can be found in Amir's book [8] (see also [1,Chpt. 11], [5][6][7]9,21]). The best known characterization was given in the classic paper of Jordan and von Neumann [12]. Namely, a norm space (X, · ) is an inner product space if and only if the following equality holds true for all x, y ∈ X. We call this equality the Jordan-von Neumann identity or the parallelogram law. In the paper [20] the authors show a characterization of functions which generate strongly Schur-convex sums, obtaining equivalently a representation of such functions. In particular, they prove that the following three sentences are equivalent.
(3) There exists a convex function g and an additive function a such that For details, see Theorem 7, p. 180 [20]; it should be mentioned, though, that the domains of investigated functions are subsets of an inner product space; i.e. the unitarity of a space is a sufficient condition for such a result. We will prove that it is also a necessary condition, which, together with the previously mentioned theorem from [20] will give a new characterization of inner product spaces among norm spaces. Moreover, we will obtain a representation theorem for functions generating strongly Schur-convex sums, and it will be a counterpart of the classical Hardy-Littlewood-Pólya majorization theorem (see [11]), works of Schur [22], Karamata [13], Ng [18] and Kominek [14]. The construction of presented theorems in this work was inspired by the works [3,19]. It is worth noting that beside theoretical applications Schur-convex functions have also practical applications in data transmission in cellular networks (see [15]). Thus it seems that research on these functions can be very useful.

Main Result
In this paper X will be a space and X n , where n ≥ 2 is a natural number, will be a Cartesian product of n-copies of the space X (i.e. X n := X × · · · × X n-times ).
We will start with recalling some definitions presented, for example, in the literature given in brackets and also discussed there, respectively. Definition 2.1. [16,20,22] Let X be a real vector space. For x, y ∈ X n we say that x is majorized by y, written x y, if x = y · P for a doubly stochastic matrix P (i.e. a matrix of degree n containing nonnegative elements with all rows and columns summing up to 1).

Definition 2.2.
[20] Let (X, · ) be a real normed space and D be a convex subset of X. We say that a function F : D n → R is strongly Schur-convex with modulus c > 0 if Definition 2.4. Let (X, · ) be a real normed space and D be a convex subset of X. We say that a function f : D → R generates strongly Schur-convex sums with modulus c > 0 if for all natural numbers n ≥ 2 the function F : The following theorem gives a new characterization of inner product spaces. To be clear, the phrase "For all functions f : D → R" is understood as "For all subsets D of X and all functions f defined on D".  [20] (Theorem 7, p. 180). Assume that (1) holds true. Immediately from the definition of functions f generating strongly Schur-convex sums with modulus c for each x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ D n (n ≥ 2) such that x y, we have the following two equivalent inequalities The function f (·) := c|| · || 2 satisfies the above inequality, and it means that it generates strongly Schur-convex sums with modulus c. Thus, from the adopted assumption, the function c|| · || 2 : X → R is strongly Wright-convex with modulus c and, consequently, it satisfies the following inequality for all x, y ∈ X and t ∈ (0, 1). Dividing this inequality by c and taking t = 1 2 we obtain the inequality 2 x + y 2 Multiplying the above inequality by 2 and using the homogeneous axiom of norm, we can write the result in the following form Now, putting the standard substitution (i.e. u = x + y and v = x − y) in the last inequality, we get the reverse inequality and it means that the norm || · || satisfies the parallelogram law, which, together with the paper of Jordan-von Neumann [12], implies that (X, || · ||) is an inner product space. The proof is finished. A representation theorem for functions generating strongly Schur-convex sums looks as follows.
Theorem 2.7. Let (X, · ) be a real normed space. The following conditions are equivalent: (1) For a c > 0 and a function f : D → R, f generates strongly Schur-convex sums with modulus c; (2) There exists a convex function g : D → R and an additive function a : X → R such that Proof. Assuming (1) and using the definition of functions f generating strongly Schur-convex sums with modulus c (Definition 2.4), we conclude that for each x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ D n (n ≥ 2) such that x y we receive the following inequality which is equivalent to the inequality It means that the function h(·) := f (·) − c|| · || 2 generates Schur-convex sums. Thus, from the theorem of Ng characterizing functions generating Schurconvex sums (see [14,18]), there exists a convex function g : D → R and an additive function a : X → R such that and consequently, f is of the form To prove (2) ⇒ (1), observe that the function h defined as before (i.e. h(·) := f (·) − c|| · || 2 ) is a sum of convex and additive functions. And using once more the aforementioned Ng's theorem, it generates Schur-convex sums, i.e.
Combining Theorems 2.5 and 2.7 we immediately obtain the following theorem. Notice that the implication (2) ⇒ (1) in Theorem 2.8 is also proved in [17].
Remark 2.9. We can replace the first condition in Theorem 2.8 with the following: (1') For all c > 0, a function f : X → R is strongly Wright-convex with modulus c if and only if there exists a convex function g : X → R and an additive function a : X → R such that Remark 2. 10. In virtue of Remarks 2.6 and 2.9, taking (1 ) instead of (1) in Theorems 2.5 and 2.8, respectively, we get stronger implications (1 ) ⇒ (2) than (1) ⇒ (2), but the implications (2) ⇒ (1 ) become weaker than (2) ⇒ (1).
We end this work with two examples which show that in Theorem 2.5 neither the assumption that f generates strongly Schur-convex sums implies that f is strongly Wright-convex, nor conversely. The examples' construction is based on the ideas of the examples from [10,19].
Observe the following equivalent inequalities.