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# Superstability of the functional equation related to distance measures

*Journal of Inequalities and Applications*
**volumeÂ 2015**, ArticleÂ number:Â 352 (2015)

## Abstract

The functional equation related to a distance measure

can be generalized as follows:

where *f* is an information measure, *P* and *Q* are in the set of *n*-ary discrete complete probability, and \(\sigma_{i}\) is a permutation for each \(i=0, 1, \ldots, n-1\).

In this paper, we investigate the superstability of the above functional equation and also four generalized functional equations:

## 1 Introduction

Baker *et al.* in [1] introduced that if *f* satisfies the stability inequality \(|E_{1}(f)-E_{2}(f)|\leq \varepsilon\), then either *f* is bounded or \(E_{1}(f)=E_{2}(f)\). This is now frequently referred to as *superstability*. Baker [2] also proved the superstability of the cosine functional equation (also called the dâ€™Alembert functional equation).

In this paper, let \((G, \cdot)\) be a commutative group and *I* denote the open unit interval \((0, 1)\). Also let \(\mathbb{R}\) denote the set of real numbers and \({\mathbb{R}_{+}} = \{ x \in\mathbb{R} \mid x > 0 \}\) be a set of positive real numbers. Further, let

denote the set of all *n*-ary discrete complete probability distributions (without zero probabilities), that is, \(\Gamma_{n}^{0}\) is the class of discrete distributions on a finite set Î© of cardinality *n* with \(n \geq2\). Almost all similarity, affinity or distance measures \(\mu_{n} : \Gamma_{n}^{0} \times \Gamma_{n}^{0} \to\mathbb{R}_{+}\) that have been proposed between two discrete probability distributions can be represented in the *sum*-*form*

where \(\phi: I \times I \to\mathbb{R}\) is a real-valued function on unit square, or a monotonic transformation of the right-hand side of (1.1), that is,

where \(\psi: \mathbb{R} \to\mathbb{R}_{+}\) is an increasing function on \(\mathbb{R}\). The function *Ï•* is called a *generating function*. It is also referred to as the *kernel* of \(\mu_{n} (P,Q)\).

In information theory, for *P* and *Q* in \(\Gamma_{n}^{0}\), the symmetric divergence of degree *Î±* is defined as

For all \(P, Q \in\Gamma_{n}^{0}\), we define the product

In [3], Chung *et al.* characterized all symmetrically compositive sum-form distance measures with a measurable generating function. The following functional equation

holding for all \(p,q, r, s \in I\) was instrumental in their characterization of symmetrically compositive sum-form distance measures. They proved the following theorem giving the general solution of this functional equation (DM):

Suppose\(f : I^{2} \to\mathbb{R}\)satisfies(DM)for all\(p,q, r, s \in I\).Then$$f(p, q) = M_{1} (p) M_{2} (q) + M_{1} (q) M_{2} (p), $$

where\(M_{1} , M_{2} : \mathbb{R} \to\mathbb{C}\)are multiplicative functions.Further,either\(M_{1}\)and\(M_{2}\)are both real or\(M_{2}\)is the complex conjugate of\(M_{1}\).The converse is also true.

In [4] and [5], Kim (second author) and Sahoo obtained the superstability results of the equation (DM), its stability and four generalizations of (DM), namely

for all \(p, q, r, s \in G\).

The above equation (DM) characterized by distance measures can be considered by characterization of a symmetrically compositive sum-form information measurable functional equation.

The functional equation (DM) can be generalized as follows. Let \(f:\Gamma_{n}^{0} \rightarrow R\) be a function and

for all \(P=(p_{1} , p_{2} , \ldots, p_{n} ), Q =(q_{1} , q_{2} , \ldots, q_{n} ) \in\Gamma_{n}^{0} \), where \(\sigma_{i}:I^{n} \rightarrow I^{n}\) is a permutation defined by

for each \(i \in N \), and define \(P\cdot Q :=(p_{1} q_{1}, p_{2} q_{2} , \ldots, p_{n} q_{n} )\).

For other functional equations with the information measure, the interested reader should refer to [6â€“9] and [10â€“12].

This paper aims to investigate the superstability of (IM) and also four generalized functional equations of (IM) as well as that of the following type functional equations:

for all \(P, Q \in G\).

## 2 Results

In this section, we investigate the superstability of the pexiderized equation related to (IM).

### Theorem 1

*Let*
\(f,g,h: G^{n} \rightarrow\mathbb{R}\)
*and*
\(\phi: G^{n} \rightarrow \mathbb{R}_{+}\)
*be functions satisfying*

*and*
\(|f(X)-g(X)| \leq M \)
*for all*
\(X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}\)
*and some constant*
*M*. *Then either*
*g*
*is bounded or*
*h*
*is a solution of* (IM).

### Proof

Let *g* be an unbounded solution of inequality (2.1). Then there exists a sequence \(\{(Z_{m})=(z_{1m} , z_{2m} , \ldots, z_{nm}) \mid m \in N \}\) in \(G^{n}\) such that \(0 \neq|g(Z_{m})|\rightarrow\infty\) as \(m\rightarrow\infty\).

Letting \(X=Z_{m} \), *i.e.*, \(x_{i} =z_{im} \) in (2.1) for each *i* and dividing \(|g(Z_{m} )|\), we have

Passing to the limit as \(m\rightarrow\infty\), we obtain that

By (2.1), we have

as \(m\rightarrow\infty\). Also, for each *j*,

as \(m\rightarrow\infty\). Note that \(\sigma_{i}(X\cdot Y)=\sigma_{i}(X)\cdot\sigma _{i}(Y)\), \(\sigma_{i}(\sigma_{j}(Y))=\sigma_{i+j}(Y)\), \(\sigma _{n+j}(Y)=\sigma_{j}(Y) \) and \(\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{1} (X) \cdot\sigma_{i+1}(Y) )=\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{1} (X)\cdot\sigma_{i}(Y) )\). Thus, from (2.2), (2.3) and (2.4), we obtain

â€ƒâ–¡

### Theorem 2

*Let*
\(f,g,h: G^{n} \rightarrow\mathbb{R}\)
*and*
\(\phi: G^{n} \rightarrow \mathbb{R}_{+}\)
*be functions satisfying*

*and*
\(|f(X)-h(X)| \leq M \)
*for all*
\(X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}\)
*and some constant*
*M*. *Then either*
*h*
*is bounded or*
*g*
*is a solution of* (IM).

### Proof

Assume that there exists a sequence \(\{(Z_{m})=(z_{1m} , z_{2m} , \ldots, z_{nm}) \mid m \in N \}\) in \(G^{n}\) such that \(\lim_{m to \infty} |h(Z_{m})|=\infty\) with \(|h(Z_{m})|\neq0\) for each *m*.

Letting \(Y=Z_{m} \), *i.e.*, \(x_{i} =z_{im} \) in (2.5) for each *i* and dividing \(|h(Z_{m} )|\), we have

Passing to the limit as \(m\rightarrow\infty\), we obtain that

By (2.5), we have

as \(m\rightarrow\infty\). Also, for each *j*,

as \(m\rightarrow\infty\).

By using (2.5), (2.6) and (2.7), let us go through the same procedure as in TheoremÂ 1, then we arrive at the required result.â€ƒâ–¡

### Corollary 1

*Let*
\(f: G^{n} \rightarrow\mathbb{R}\)
*and*
\(\phi: G^{n} \rightarrow \mathbb{R}_{+}\)
*be functions satisfying*

*for all*
\(X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}\). *Then either*
*f*
*is bounded or*
*f*
*is a solution of* (IM).

### Proof

By Theorems 1 and 2, it is trivial.â€ƒâ–¡

### Corollary 2

*Let*
\(f: G^{n} \rightarrow\mathbb{R}\)
*and*
\(\phi: G^{n} \rightarrow \mathbb{R}_{+}\)
*be functions satisfying*

*for all*
\(X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}\). *Then either*
*f* (*or*
*g*) *is bounded or*
*g*
*satisfies* (IM). *And also*
\(\{f,g\}\)
*satisfies* (GIMfg).

### Proof

By TheoremÂ 1, we have that either *f* is bounded or *g* satisfies (IM). Also, it follows from (2.10) that

Thus if *f* is bounded, then *g* is bounded. Hence, by TheoremÂ 1, in the case *g* is unbounded, *g* also is a solution of (IM).

Let *g* be unbounded. By a similar method as the calculation in TheoremÂ 2 with the unboundedness of *g*, we have

for all \(X, Z_{m} \in G^{n} \) and \(0 \neq|g(Z_{m} )|\rightarrow\infty\) as \(m\rightarrow\infty\).

From a similar calculation as that in TheoremÂ 1 and TheoremÂ 2, we obtain the required result.â€ƒâ–¡

### Corollary 3

*Let*
\(f, g: G^{n} \rightarrow\mathbb{R}\)
*and*
\(\phi: G^{n} \rightarrow \mathbb{R}_{+}\)
*be functions satisfying*

*for all*
\(X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}\). *Then either*
*f* (*or*
*g*) *is bounded or*
*g*
*satisfies* (IM). *And also*
\(\{f,g\}\)
*satisfies* (GIMgf).

### Proof

By TheoremÂ 2, we have that either *f* is bounded or *g* is a solution of (IM). Suppose that *g* be unbounded, then *f* is unbounded. Hence, by TheoremÂ 2, *g* also is a solution of (IM). By a similar method as the calculation in TheoremÂ 1 with the unboundedness of *g*, we have

for all \(X, Z_{m} \in G^{n} \) and \(0 \neq|g(Z_{m} )|\rightarrow\infty\) as \(m\rightarrow\infty\).

From a similar calculation as that in CorollaryÂ 2 we obtain the required result.â€ƒâ–¡

### Corollary 4

*Let*
\(f, g, h: G^{n} \rightarrow\mathbb{R}\)
*and*
\(\phi: G^{n} \rightarrow\mathbb{R}_{+}\)
*be functions satisfying*

*and*
\(\max\{ |f(X)-g(X)|, |f(X)-h(X)|\} \leq M \)
*for all*
\(X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}\)
*and for some*
*M*. *Then either*
*f* (*or*
*g*, *or*
*h*) *is bounded or*
*g*
*and*
*h*
*are solutions of* (IM).

### Corollary 5

*Let*
\(f, g: G^{n} \rightarrow\mathbb{R}\)
*and*
\(\phi: G^{n} \rightarrow \mathbb{R}_{+}\)
*be functions satisfying*

*and*
\(\{ |f(X)-g(X)|\} \leq M \)
*for all*
\(X=(x_{1}, \ldots, x_{n}), Y=(y_{1} , \ldots, y_{n} ) \in G^{n}\)
*and for some*
*M*. *Then either*
*f* (*or*
*g*) *is bounded or*
*g*
*satisfies* (IM).

## 3 Discussion

We consider the functional equation

for all \(X, Y \in G^{n}\), where \(f: G^{n} \rightarrow\mathbb {R}\) is the unknown function to be determined, and \(\sigma_{i} (X)=(x_{i+1},x_{i+2}, \ldots, x_{n}, x_{1} , x_{2} , \ldots, x_{i} )\). If \(n=2\), the solution of the above functional equation is known on the semigroup \(S=(0,1)\) when the semigroup operation is multiplicationÂ [3]. It is not known when \(n\geq3\), but there is a special solution of it.

For example, let \(X=(x _{1} ,x _{2} , \ldots,x _{n} )\) and \(Y=(y _{1}, y _{2} , \ldots,y _{n} )\). And define \(f(X)=f(x _{1}, x _{2}, \ldots,x _{n} ):= \sum_{i=1} ^{n} \frac{1}{ x _{i}}\). Then *f* is a solution of the above equation. Thus our results are not limited. We expect to know the general solution of it.

## 4 Conclusions

In the present paper we considered generalized functional equations related to distance measures and investigated the stability of them. We extended for two-variables in (DM) to *n*-variables in (IM). That is, the following functional equation satisfies the property of superstability

where *f* is an information measure, *P* and *Q* are in the set of *n*-ary discrete complete probability, and \(\sigma_{i}\) is a permutation for each \(i=0, 1, \ldots, n-1\).

Also the pexiderized functional equation of the above equation satisfies the property of superstability.

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## Acknowledgements

This research was supported by the Daejeon University Fund (2014).

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Lee, Y.W., Kim, G.H. Superstability of the functional equation related to distance measures.
*J Inequal Appl* **2015**, 352 (2015). https://doi.org/10.1186/s13660-015-0880-4

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DOI: https://doi.org/10.1186/s13660-015-0880-4