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 d’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

$$ \Gamma_{n}^{0} = \Biggl\{ P = ( p_{1} , p_{2} , \ldots, p_{n} ) \Bigm| 0 < p_{k} < 1 , \sum_{k=1}^{n} p_{k} = 1 \Biggr\} $$

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 Ω 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

$$ \mu_{n} (P, Q ) = \sum_{k=1}^{n} \phi( p_{k} , q_{k} ) , $$
(1.1)

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,

$$ \mu_{n} (P, Q ) = \psi \Biggl( \sum_{k=1}^{n} \phi( p_{k} , q_{k} ) \Biggr) , $$
(1.2)

where \(\psi: \mathbb{R} \to\mathbb{R}_{+}\) is an increasing function on \(\mathbb{R}\). The function ϕ 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 α is defined as

$$ J_{n, {\alpha}} (P, Q) = {1 \over {{2^{\alpha-1} - 1}}} \Biggl[ \sum _{k=1}^{n} \bigl( p_{k}^{\alpha} q_{k}^{1-\alpha} + p_{k}^{1- \alpha} q_{k}^{\alpha} \bigr) - 2 \Biggr] . $$

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

$$ P\cdot R = ( p_{1} r_{1} , p_{1} r_{2} , \ldots, p_{1}r_{m}, p_{2} r_{1} , \ldots, p_{2} r_{m} , \ldots, p_{n} r_{m} ). $$

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

$$ f(pr, qs) + f(ps, qr) = f(p, q) f(r, s) $$
(DM)

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

$$\begin{aligned}& f(pr,qs)+f(ps,qr)=f(p,q) g(r,s), \end{aligned}$$
(DMfg)
$$\begin{aligned}& f(pr,qs)+f(ps,qr)=g(p,q) f(r,s), \end{aligned}$$
(DMgf)
$$\begin{aligned}& f(pr,qs)+f(ps,qr)=g(p,q) g(r,s), \end{aligned}$$
(DMgg)
$$\begin{aligned}& f(pr,qs)+f(ps,qr)=g(p,q) h(r,s) \end{aligned}$$
(DMgh)

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

$$ \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)f(Q) $$
(IM)

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

$$\sigma_{i}(x_{1} , x_{2} , \ldots, x_{n} ):=(x_{i+1}, x_{i+2}, \ldots, x_{n} , x_{1} , x_{2} , \ldots, x_{i} ) $$

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 [6–9] and [10–12].

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:

$$\begin{aligned}& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)g(Q), \end{aligned}$$
(GIMfg)
$$\begin{aligned}& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=g(P)f(Q), \end{aligned}$$
(GIMgf)
$$\begin{aligned}& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=g(P)g(Q), \end{aligned}$$
(GIMgg)
$$\begin{aligned}& \sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=g(P)h(Q) \end{aligned}$$
(GIMgh)

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

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)h(Y) \Biggr\vert \leq\phi(Y) $$
(2.1)

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

$$ \biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{i} (Y) )}{g(Z_{m} )}-h(Y)\biggr\vert \leq\frac{\phi(Y)}{|g(Z_{m} )|}. $$

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

$$ h(Y)=\lim_{m\to\infty}\frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{i} (Y))}{g(Z_{m} )}. $$
(2.2)

By (2.1), we have

$$\begin{aligned}& \biggl\vert \frac{\sum_{i=0}^{n-1} f((Z_{m} \cdot X) \cdot\sigma_{i} (Y))-g(Z_{m} \cdot X )h(Y)}{g(Z_{m} )}\biggr\vert \\& \quad \leq\frac{\phi(Y)}{|g(Z_{m} )|}\rightarrow0 \end{aligned}$$
(2.3)

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

$$\begin{aligned}& \biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{j}(X) \cdot \sigma_{i} (Y))-g(Z_{m} \cdot\sigma_{j}(X) )h(Y)}{g(Z_{m} )}\biggr\vert \\& \quad \leq\frac{\phi(Y)}{|g(Z_{m} )|}\rightarrow0 \end{aligned}$$
(2.4)

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

$$\begin{aligned}& \Biggl\vert \sum_{i=0}^{n-1} h\bigl(X\cdot \sigma_{i} (Y) \bigr)-h(X)h(Y) \Biggr\vert \\& \quad =\lim_{m\to\infty}\Biggl\vert \sum _{i=0}^{n-1} \frac{\sum_{j=0}^{n-1} f(Z_{m} \cdot\sigma_{j}(X \cdot\sigma_{i} (Y)))}{g(Z_{m} )} -h(X)h(Y) \Biggr\vert \\& \quad =\lim_{m\to\infty}\biggl\vert \frac{\sum_{j=0}^{n-1} f(Z_{m} \cdot\sigma _{j}(X\cdot \sigma_{0}(Y)))+\sum_{j=0}^{n-1} f(Z_{m} \cdot\sigma _{j}(X\cdot\sigma_{1} (Y)))}{g(Z_{m} )} \\& \qquad {} + \cdots+\frac{\sum_{j=0}^{n-1} f(Z_{m} \cdot\sigma_{j}(X\cdot \sigma_{n-1} (Y)))}{g(Z_{m} )} -h(X)h(Y) \biggr\vert \\& \quad =\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{0}(X\cdot\sigma_{i}(Y)))+\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{1}(X\cdot\sigma_{i} (Y)))}{g(Z_{m} )} \\& \qquad {} + \cdots+\frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{n-1}(X\cdot \sigma_{i} (Y)))}{g(Z_{m} )} -h(X)h(Y) \biggr\vert \\& \quad =\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{0}(X)\cdot\sigma_{i}(Y))+\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{1}(X)\cdot\sigma_{i} (Y))}{g(Z_{m} )} \\& \qquad {} + \cdots+\frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{n-1}(X)\cdot \sigma_{i} (Y))}{g(Z_{m} )} -h(X)h(Y) \biggr\vert \\& \quad \leq\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot \sigma_{0}(X)\cdot\sigma_{i} (Y))-g(Z_{m} \cdot\sigma _{0}(X))h(Y)}{g(Z_{m} )}\biggr\vert \\& \qquad {} +\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot \sigma_{1}(X)\cdot\sigma_{i} (Y))-g(Z_{m} \cdot\sigma _{1}(X))h(Y)}{g(Z_{m} )}\biggr\vert \\& \qquad {} + \cdots+\lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma_{n-1}(X)\cdot\sigma_{i} (Y))-g(Z_{m} \cdot\sigma _{n-1}(X))h(Y)}{g(Z_{m} )} \biggr\vert \\& \qquad {} + \lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot \sigma_{i}(X))\cdot h(Y)}{g(Z_{m} )}-h(X)h(Y) \biggr\vert \\& \qquad {} + \lim_{m\to\infty}\biggl\vert \frac{\sum_{i=0}^{n-1} (g-f)(Z_{m} \cdot \sigma_{i}(X))\cdot h(Y)}{g(Z_{m} )}\biggr\vert \\& \quad = \bigl\vert h(X)h(Y)-h(X)h(Y)\bigr\vert =0. \end{aligned}$$

 □

Theorem 2

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

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)h(Y) \Biggr\vert \leq\phi(X) $$
(2.5)

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

$$ \biggl\vert \frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i} (Z_{m} ) )}{h(Z_{m} )}-g(X )\biggr\vert \leq\frac{\phi(X)}{|h(Z_{m} )|}. $$

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

$$ g(X)=\lim_{m\to\infty}\frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i} (Z_{m} ))}{h(Z_{m} )}. $$
(2.6)

By (2.5), we have

$$\begin{aligned}& \biggl\vert \frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i} (Y \cdot Z_{m} ))-g( X )h(Y\cdot Z_{m})}{h( Z_{m} )}\biggr\vert \\& \quad \leq\frac{\phi(X)}{|h(Z_{m} )|}\rightarrow0 \end{aligned}$$
(2.7)

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

$$\begin{aligned}& \biggl\vert \frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i}(Y \cdot\sigma _{j}(Z_{m} )))-g(X) h(Y \cdot\sigma_{j}( Z_{m}))}{h(Z_{m} )}\biggr\vert \\& \quad \leq\frac{\phi(X)}{|h(Z_{m} )|}\rightarrow0 \end{aligned}$$
(2.8)

as \(m\rightarrow\infty\).

By using (2.5), (2.6) and (2.7), let us go through the same procedure as in Theorem 1, then we arrive at the required result. □

Corollary 1

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

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-f(X)f(Y) \Biggr\vert \leq\max \bigl\{ \phi(X),\phi(Y)\bigr\} $$
(2.9)

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. □

Corollary 2

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

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-f(X)g(Y) \Biggr\vert \leq\min \bigl\{ \phi(X),\phi(Y)\bigr\} $$
(2.10)

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

$$\bigl\vert g(Y)\bigr\vert \leq\frac{\phi(X)+\sum_{i=0}^{n-1}|f(\sigma_{i}(Y) )|}{|f(X)|}. $$

Thus if f is bounded, then g is bounded. Hence, by Theorem 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 Theorem 2 with the unboundedness of g, we have

$$ f(X)=\lim_{m\to\infty}\frac{\sum_{i=0}^{n-1} f(X \cdot\sigma_{i} (Z_{m} ))}{g(Z_{m} )} $$
(2.11)

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 Theorem 1 and Theorem 2, we obtain the required result. □

Corollary 3

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

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)f(Y) \Biggr\vert \leq\min \bigl\{ \phi(X),\phi(Y)\bigr\} $$
(2.12)

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 Theorem 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 Theorem 2, g also is a solution of (IM). By a similar method as the calculation in Theorem 1 with the unboundedness of g, we have

$$ f(X)=\lim_{m\to\infty}\frac{\sum_{i=0}^{n-1} f(Z_{m} \cdot\sigma _{i} (X ))}{g(Z_{m} )} $$
(2.13)

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 Corollary 2 we obtain the required result. □

Corollary 4

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

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)h(Y) \Biggr\vert \leq\min \bigl\{ \phi(X),\phi(Y)\bigr\} $$
(2.14)

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

$$ \Biggl\vert \sum_{i=0}^{n-1} f\bigl(X\cdot\sigma_{i} (Y) \bigr)-g(X)g(Y) \Biggr\vert \leq\min \bigl\{ \phi(X),\phi(Y)\bigr\} $$
(2.15)

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

$$ \sum_{i=0}^{n-1} f\bigl(X\cdot \sigma_{i} (Y) \bigr)=f(X)f(Y) $$

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 multiplication [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

$$\sum_{i=0}^{n-1} f\bigl(P\cdot \sigma_{i} (Q) \bigr)=f(P)f(Q), $$

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.