Characterization of Inner Product Spaces by Strongly Schur-Convex Functions

Abstract

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.

1 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 $(X,\left\| \cdot \right\| )$ is an inner product space if and only if the following equality

$$\begin{aligned} \left\| x+y\right\| ^{2}+\left\| x-y\right\| ^{2}=2\left\| x\right\| ^{2}+2\left\| y\right\| ^{2}, \end{aligned}$$

holds true for all $x,y\in X$. 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.

• (1) Function f generates strongly Schur-convex sums.

• (2) Function f is strongly Wright-convex.

• (3) There exists a convex function g and an additive function a such that $f=g+a+c\left\| \cdot \right\| ^2$.

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.

2 Main Result

In this paper X will be a space and $X^{n}$, where $n\ge 2$ is a natural number, will be a Cartesian product of n-copies of the space X (i.e. $X^{n}:=\underbrace{X\times \cdots \times X}_{n\text {-}times}$).

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 $x,y\in X^{n}$ we say that xis majorized byy, written $x\preccurlyeq y$, if

$$\begin{aligned} x=y\cdot P \end{aligned}$$

for a doubly stochastic matrix P (i.e. a matrix of degree n containing non-negative elements with all rows and columns summing up to 1).

Definition 2.2

[20] Let $(X,\left\| \cdot \right\| )$ be a real normed space and D be a convex subset of X. We say that a function $F:D^{n}\rightarrow {\mathbb {R}}$ is strongly Schur-convex with modulus $c>0$ if

$$\begin{aligned} F(x)\le F(y)-c\left( |||y|||^{2}-|||x|||^{2}\right) \end{aligned}$$

for all $x,y \in D$ such that $x\preccurlyeq y$ and $|||\cdot |||$ is the Cartesian norm in the space $X^{n}$ (i.e. $|||x|||=\sqrt{||x_{1}||^{2}+\cdots +||x_{n}||^{2}}$ for n-tuples $x=(x_{1},\ldots ,x_{n})\in X^{n}$).

Definition 2.3

[17] Let $(X,\left\| \cdot \right\| )$ be a real normed space and D be a convex subset of X. We say that a function $f:D\rightarrow \mathbb R$ is strongly Wright-convex with modulus $c>0$ if

$$\begin{aligned} f\left( tx+(1-t)y\right) +f\left( (1-t)x+ty\right) \le f(x)+f(y)-2ct(1-t)||x-y||^{2} \end{aligned}$$

for all $x,y\in D$ and $t\in [0,1]$.

Definition 2.4

Let $(X,\left\| \cdot \right\| )$ be a real normed space and D be a convex subset of X. We say that a function $f:D\rightarrow \mathbb R$ generates strongly Schur-convex sums with modulus $c>0$ if for all natural numbers $n\ge 2$ the function $F:D^{n}\rightarrow {\mathbb {R}}$ defined by formula $F(x_{1},...,x_{n}):=f(x_{1})+\cdots +f(x_{n})$, is strongly Schur-convex with modulus c.

The following theorem gives a new characterization of inner product spaces. To be clear, the phrase “For all functions $f:D\rightarrow \mathbb R$” is understood as “For all subsets D of X and all functions f defined on D”.

Theorem 2.5

Let $(X,\left\| \cdot \right\| )$ be a real normed space. The following conditions are equivalent:

• (1) For all $c>0$ and all functions $f:D\rightarrow {\mathbb {R}}$, f generates strongly Schur-convex sums with modulus c if and only if f is strongly Wright-convex with modulus c;

• (2) $(X,\left\| \cdot \right\| )$ is an inner product space.

Proof

The implication $(2)\Rightarrow (1)$ is proved in [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 $x=(x_{1},\ldots ,x_{n})$, $y=(y_{1},\ldots ,y_{n})\in D^{n}$ ($n\ge 2$) such that $x\preccurlyeq y$, we have the following two equivalent inequalities

$$\begin{aligned}&\displaystyle F(x)\le F(y)-c\left( |||y|||^{2}-|||x|||^{2}\right) , \\&\displaystyle f(x_{1})-c||x_{1}||^{2}+\cdots +f(x_{n})-c||x_{n}||^{2}\le f(y_{1})-c||y_{1}||^{2}+\cdots +f(y_{n})-c||y_{n}||^{2}. \end{aligned}$$

The function $f(\cdot ):=c||\cdot ||^{2}$ satisfies the above inequality, and it means that it generates strongly Schur-convex sums with modulus c. Thus, from the adopted assumption, the function $c||\cdot ||^{2}:X\rightarrow {\mathbb {R}}$ is strongly Wright-convex with modulus c and, consequently, it satisfies the following inequality

$$\begin{aligned} c||tx+(1-t)y||^{2}+c||(1-t)x+ty||^{2}\le c||x||^{2}+c||y||^{2}-2ct(1-t)||x-y||^{2}, \end{aligned}$$

for all $x,y\in X$ and $t\in (0,1)$. Dividing this inequality by c and taking $t=\frac{1}{2}$ we obtain the inequality

$$\begin{aligned} 2\left\| \frac{x+y}{2}\right\| ^{2}\le ||x||^{2}+||y||^{2}-\frac{1}{2}||x-y||^{2}, \ \ \ x,y\in X. \end{aligned}$$

Multiplying the above inequality by 2 and using the homogeneous axiom of norm, we can write the result in the following form

$$\begin{aligned} \left\| x+y\right\| ^{2}+||x-y||^{2}\le 2||x||^{2}+2||y||^{2}, \ \ \ x,y\in X. \end{aligned}$$

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 $||\cdot ||$ satisfies the parallelogram law, which, together with the paper of Jordan-von Neumann [12], implies that $(X, ||\cdot ||)$ is an inner product space. The proof is finished. $\square$

Remark 2.6

We can replace the first condition in Theorem 2.5 with the following:

• (1’) For all $c>0$, a function $f:X\rightarrow {\mathbb {R}}$ generates strongly Schur-convex sums with modulus c if and only if f is strongly Wright-convex with modulus c.

A representation theorem for functions generating strongly Schur-convex sums looks as follows.

Theorem 2.7

Let $(X,\left\| \cdot \right\| )$ be a real normed space. The following conditions are equivalent:

• (1) For a $c>0$ and a function $f:D\rightarrow {\mathbb {R}}$, f generates strongly Schur-convex sums with modulus c;

• (2) There exists a convex function $g:D\rightarrow {\mathbb {R}}$ and an additive function $a:X\rightarrow {\mathbb {R}}$ such that

  $$\begin{aligned} f(x)=g(x)+a(x)+c\left\| x\right\| ^{2}, \ \ x\in D. \end{aligned}$$

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 $x=(x_{1},\ldots ,x_{n})$, $y=(y_{1},\ldots ,y_{n})\in D^{n}$ ($n\ge 2$) such that $x\preccurlyeq y$ we receive the following inequality

$$\begin{aligned} F(x)\le F(y)-c\left( |||y|||^{2}-|||x|||^{2}\right) , \end{aligned}$$

which is equivalent to the inequality

$$\begin{aligned} f(x_{1})-c||x_{1}||^{2}+\cdots +f(x_{n})-c||x_{n}||^{2}\le f(y_{1})-c||y_{1}||^{2}+\cdots +f(y_{n})-c||y_{n}||^{2}. \end{aligned}$$

It means that the function $h(\cdot ):=f(\cdot )-c||\cdot ||^{2}$ generates Schur-convex sums. Thus, from the theorem of Ng characterizing functions generating Schur-convex sums (see [14, 18]), there exists a convex function $g:D\rightarrow {\mathbb {R}}$ and an additive function $a:X\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} h(x)=g(x)+a(x), \ \ x\in D, \end{aligned}$$

and consequently, f is of the form

$$\begin{aligned} f(x)=g(x)+a(x)+c\left\| x\right\| ^{2}, \ \ x\in D. \end{aligned}$$

To prove $(2)\Rightarrow (1)$, observe that the function h defined as before (i.e. $h(\cdot ):=f(\cdot )-c||\cdot ||^{2}$) is a sum of convex and additive functions. And using once more the aforementioned Ng’s theorem, it generates Schur-convex sums, i.e.

$$\begin{aligned} h(x_{1})+\cdots +h(x_{n})\le h(y_{1})+\cdots +h(y_{n}), \ \ x\preccurlyeq y. \end{aligned}$$

Which is equivalent to the following inequalities

$$\begin{aligned}&f(x_{1})-c||x_{1}||^{2}+\cdots +f(x_{n})-c||x_{n}||^{2}\le f(y_{1})-c||y_{1}||^{2}+\cdots +f(y_{n})-c||y_{n}||^{2}, \\&f(x_{1})+\cdots +f(x_{n})\le f(y_{1})+\cdots +f(y_{n})\\&\quad -c\left( ||y_{1}||^{2}+\cdots +||y_{n}||^{2}-||x_{1}||^{2}-\cdots -||x_{n}||^{2}\right) , \end{aligned}$$

for all $x\preccurlyeq y$. Finally, taking the function

$$\begin{aligned} F(x_{1},\ldots , x_{n}):=f(x_{1})+\cdots +f(x_{n}) \end{aligned}$$

and the Cartesian norm $|||\cdot |||$ in the space $X^{n}$, we get the inequality

$$\begin{aligned} F(x)\le F(y)-c\left( |||y|||^{2}-|||x|||^{2}\right) , \ \ x\preccurlyeq y. \end{aligned}$$

It proves that f generates strongly Schur-convex sums with modulus c. This finishes the proof. $\square$

Combining Theorems 2.5 and 2.7 we immediately obtain the following theorem. Notice that the implication $(2)\Rightarrow (1)$ in Theorem 2.8 is also proved in [17].

Theorem 2.8

Let $(X,\left\| \cdot \right\| )$ be a real normed space. The following conditions are equivalent:

• (1) For all $c>0$ and all functions $f:D\rightarrow {\mathbb {R}}$, f is strongly Wright-convex with modulus c if and only if there exists a convex function $g:D\rightarrow {\mathbb {R}}$ and an additive function $a:X\rightarrow {\mathbb {R}}$ such that

  $$\begin{aligned} f(x)=g(x)+a(x)+c\left\| x\right\| ^{2}, \ \ x\in D; \end{aligned}$$

• (2) $(X,\left\| \cdot \right\| )$ is an inner product space.

Remark 2.9

We can replace the first condition in Theorem 2.8 with the following:

• (1’) For all $c>0$, a function $f:X\rightarrow {\mathbb {R}}$ is strongly Wright-convex with modulus c if and only if there exists a convex function $g:X\rightarrow {\mathbb {R}}$ and an additive function $a:X\rightarrow {\mathbb {R}}$ such that

  $$\begin{aligned} f(x)=g(x)+a(x)+c\left\| x\right\| ^{2}, \ \ x\in X. \end{aligned}$$

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 $(1')\Rightarrow (2)$ than $(1)\Rightarrow (2)$, but the implications $(2)\Rightarrow (1')$ become weaker than $(2)\Rightarrow (1)$.

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].

Example 2.11

Let $X={\mathbb {R}}^{2}$, $\left\| x\right\| =\left| x_{1}\right| +\left| x_{2}\right|$, for $x=(x_{1},x_{2})$ and $f(x)=\left\| x\right\| ^{2}$. Observe the following equivalent inequalities.

$$\begin{aligned}&f(x_{1})+\ldots +f(x_{n})\le f(y_{1})+\ldots +f(y_{n})-\left( |||y|||^{2}-|||x|||^{2}\right) , \\&f(x_{1})+\ldots +f(x_{n})\le f(y_{1})+\ldots +f(y_{n})\\&\quad -\left( \left\| y_{1}\right\| ^{2}+\ldots +\left\| y_{n}\right\| ^{2}-\left\| x_{1}\right\| ^{2}-\ldots -\left\| x_{n}\right\| ^{2}\right) , \\&\left\| x_{1}\right\| ^{2}+\ldots +\left\| x_{n}\right\| ^{2}\le \left\| y_{1}\right\| ^{2}+\ldots +\left\| y_{n}\right\| ^{2}\\&\quad -\left( \left\| y_{1}\right\| ^{2}+\ldots +\left\| y_{n}\right\| ^{2}-\left\| x_{1}\right\| ^{2}-\ldots -\left\| x_{n}\right\| ^{2}\right) , \end{aligned}$$

which means that f generates Schur-convex sums with modulus $c=1$. However, f is not strongly Wright-convex with modulus $c=1$. Indeed, for $x=(1,0)$, $y=(0,1)$ and $t=\frac{1}{2}$ we have

$$\begin{aligned} f\left( \frac{x+y}{2}\right) =1>0=\frac{f(x)+f(y)}{2}-\frac{1}{4}\left\| x-y\right\| ^{2}. \end{aligned}$$

Example 2.12

Let $X={\mathbb {R}}^{2}$, $\left\| x\right\| =\left| x_{1}\right| +\left| x_{2}\right|$, for $x=(x_{1},x_{2})$. We show that the function $f(x)=x_{1}^{2}+x_{2}^{2}$ is strongly Wright-convex with modulus $c=\frac{1}{2}$. First of all, observe that f is strongly midconvex with modulus $c=\frac{1}{2}$ (i.e. $f\left( \frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2}-\frac{c}{4}\left\| x-y\right\| ^{2}$, with $c=\frac{1}{2}$). Indeed, for arbitrary $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$ we have

$$\begin{aligned} f\left( \frac{x+y}{2}\right) =\frac{1}{4}\left( x_{1}^{2}+2x_{1}y_{1}+y_{1}^{2}+x_{2}^{2}+2x_{2}y_{2}+y_{2}^{2}\right) \end{aligned}$$

and

$$\begin{aligned} \frac{f(x)+f(y)}{2}-\frac{1}{8}\left\| x-y\right\| ^{2}= & {} \frac{3}{8}\left( x_{1}^{2}+y_{1}^{2}+x_{2}^{2}+y_{2}^{2}\right) \\&+\frac{1}{4}\left( x_{1}y_{1}+x_{2}y_{2}-\left| x_{1}-y_{1}\right| \left| x_{2}-y_{2}\right| \right) . \end{aligned}$$

Thus

$$\begin{aligned} \frac{f(x)+f(y)}{2}-\frac{1}{8}\left\| x-y\right\| ^{2}-f\left( \frac{x+y}{2}\right) =\frac{1}{8}\left( \left| x_{1}-y_{1}\right| -\left| x_{2}-y_{2}\right| \right) ^{2}\ge 0, \end{aligned}$$

which means that f is strongly midconvex with modulus $c=\frac{1}{2}$. And consequently, f satisfies the inequality

$$\begin{aligned} f\left( tx+(1-t)y\right) \le tf(x)+(1-t)f(y)-\frac{1}{2}t(1-t)\left\| x-y\right\| ^{2}, \end{aligned}$$

(2.1)

for all dyadic numbers $t\in (0,1)$ (see [2] and also a generalization in [4]). Moreover, f is continuous, thus it is a strongly convex function with modulus $c=\frac{1}{2}$ (i.e. f satisfies (2.1) for all $t\in (0,1)$). Now, interchanging t with $1-t$ in (2.1) and adding the obtained inequality side by side to the inequality (2.1), we conclude that f is strongly Wright-convex with modulus $c=\frac{1}{2}$.

Suppose that f generates strongly Schur-convex sums with modulus $c=\frac{1}{2}$. Then, from Theorem 2.7, there exists a convex function g and an additive function a such that

$$\begin{aligned} f(x)=g(x)+a(x)+ \frac{1}{2}\Vert x\Vert ^2. \end{aligned}$$

Hence the function

$$\begin{aligned} h(x):=f(x) - \frac{1}{2}\Vert x\Vert ^2 = g(x)+a(x) \end{aligned}$$

would be midconvex, but it is not. Indeed, for $x=(-1,1)$ and $y=(1,1)$ we get

$$\begin{aligned} h\left( \frac{x+y}{2}\right) =\frac{1}{2}>0=\frac{h(x)+h(y)}{2}. \end{aligned}$$