# Comment on "Functional inequalities associated with Jordan-von Neumann type additive functional equations"

## Abstract

Park et al. proved the Hyers-Ulam stability of some additive functional inequalities. There is a fatal error in the proof of Theorem 3.1. We revise the statements of the main theorems and prove the revised theorems.

2010 Mathematics Subject Classification: Primary 39B62; 39B72; 39B52.

## 1 Introduction and preliminaries

Ulam  gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

We are given a group G and a metric group G' with metric ρ(·,·). Given ϵ > 0, does there exist a δ > 0 such that if f : GG' satisfies ρ(f (xy), f(x) f(y)) < δ for all x,y G, then a homomorphism h : GG' exists with ρ(f(x), h(x)) < ϵ for all x G?

Hyers  considered the case of approximately additive mappings f: EE', where E and E' are Banach spaces and f satisfies Hyers' inequality

$∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥\le \epsilon$

for all x,y E. It was shown that the limit

$L\left(x\right)=\underset{n\to \infty }{\text{lim}}\frac{f\left({2}^{n}x\right)}{{2}^{n}}$

exists for all x E and that L : EE' is the unique additive mapping satisfying

$∥f\left(x\right)-L\left(x\right)∥\le \epsilon .$

Rassias  provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded.

Theorem 1.1. (Rassias). Let f : EE' be a mapping from a normed vector space E into a Banach space E' subject to the inequality

$∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥\le \epsilon \left({∥x∥}^{p}+{∥y∥}^{p}\right)$
(1.1)

for all x,y E, where ϵ and p are constants with ϵ > 0 and p < 1. Then the limit

$L\left(x\right)=\underset{n\to \infty }{\text{lim}}\frac{f\left({2}^{n}x\right)}{{2}^{n}}$

exists for all x E and L : EE' is the unique additive mapping which satisfies

$∥f\left(x\right)-L\left(x\right)∥\le \frac{2\epsilon }{2-{2}^{p}}{∥x∥}^{p}$
(1.2)

for all x E. If p < 0 then inequality (1.1) holds for x,y ≠ 0 and (1.2) for x ≠ 0.

Rassias  during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. Gajda  following the same approach as in Rassias  gave an affirmative solution to this question for p > 1. It was shown by Gajda , as well as by Rassias and Šemrl  that one cannot prove a Rassias' type theorem when p = 1 (cf. the books of Czerwik  and Hyers et al. ).

Rassias  followed the innovative approach of Rassias' theorem  in which he replaced the factor xp+ ypby xp· yqfor p,q with p + q ≠ 1. Găvruta  provided a further generalization of Rassias' theorem. During the last two decades a number of papers and research monographs have been published on various generalizations and applications of the Hyers-Ulam stability to a number of functional equations and mappings (see ).

Throughout this article, let G be a 2-divisible abelian group. Assume that X is a normed space with norm || · || X and that Y is a Banach space with norm || · || Y .

Gilányi  showed that if f satisfies the functional inequality

$∥2f\left(x\right)+2f\left(y\right)-f\left(x{y}^{-1}\right)∥\le ∥f\left(xy\right)∥$
(1.3)

then f satisfies the Jordan-von Neumann functional equation

$2f\left(x\right)+2f\left(y\right)=f\left(xy\right)+f\left(x{y}^{-1}\right).$

See also . Gilányi  and Fechner  proved the Hyers-Ulam stability of the functional inequality (1.3).

Park et al.  proved the Hyers-Ulam stability of the following functional inequalities

$∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥\le ∥2f\left(\frac{x+y+z}{2}\right)∥,$
(1.4)
$∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥\le ∥f\left(x+y+z\right)∥,$
(1.5)
$∥f\left(x\right)+f\left(y\right)+2f\left(z\right)∥\le ∥2f\left(\frac{x+y}{2}+z\right)∥.$
(1.6)

But there is an error in the 8th line on the 6th page in the proof of [18, Theorem 3.1]. We revise the statements of the main theorems and prove the revised theorems.

In Section 2, we prove the Hyers-Ulam stability of the functional inequality (1.4).

In Section 3, we prove the Hyers-Ulam stability of the functional inequality (1.5).

In Section 4, we prove the Hyers-Ulam stability of the functional inequality (1.6).

## 2 Stability of a functional inequality associated with a 3-variable Jensen additive functional equation

Proposition 2.1. [18, Proposition 2.1] Let f : GY be a mapping such that

${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_{Y}\le {∥2f\left(\frac{x+y+z}{2}\right)∥}_{Y}$

for all x, y, z G. Then f is Cauchy additive.

We prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type 3-variable Jensen additive functional equation.

Theorem 2.2. Let r > 1 and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_{Y}\le {∥2f\left(\frac{x+y+z}{2}\right)∥}_{Y}+\theta \left({∥x∥}_{X}^{r}+{∥y∥}_{X}^{r}+{∥z∥}_{X}^{r}\right)$
(2.1)

for all x,y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{{2}^{r}+2}{{2}^{r}-2}\theta {∥x∥}_{X}^{r}$
(2.2)

for all x X.

Proof. Letting y = x and z = -2x in (2.1), we get

${∥2f\left(x\right)-f\left(2x\right)∥}_{Y}={∥2f\left(x\right)+f\left(-2x\right)∥}_{Y}\le \left(2+{2}^{r}\right)\theta {∥x∥}_{X}^{r}$
(2.3)

for all x X. So

${∥f\left(x\right)-2f\left(\frac{x}{2}\right)∥}_{Y}\le \frac{2+{2}^{r}}{{2}^{r}}\theta {∥x∥}_{X}^{r}$

for all x X. Hence

$\begin{array}{ll}\hfill {∥{2}^{l}f\left(\frac{x}{{2}^{l}}\right)-{2}^{m}f\left(\frac{x}{{2}^{m}}\right)∥}_{Y}& \le \sum _{j=l}^{m-1}{∥{2}^{j}f\left(\frac{x}{{2}^{j}}\right)-{2}^{j+1}f\left(\frac{x}{{2}^{j+1}}\right)∥}_{Y}\phantom{\rule{2em}{0ex}}\\ \le \frac{2+{2}^{r}}{{2}^{r}}\sum _{j=l}^{m-1}\frac{{2}^{j}}{{2}^{rj}}\theta {∥x∥}_{X}^{r}\phantom{\rule{2em}{0ex}}\end{array}$
(2.4)

for all nonnegative integers m and l with m > l and all x X.

It follows from (2.4) that the sequence $\left\{{2}^{n}f\left(\frac{x}{{2}^{n}}\right)\right\}$ is a Cauchy sequence for all x X. Since Y is complete, the sequence $\left\{{2}^{n}f\left(\frac{x}{{2}^{n}}\right)\right\}$ converges. So one can define the mapping h : XY by

$h\left(x\right):=\underset{n\to \infty }{\text{lim}}{2}^{n}f\left(\frac{x}{{2}^{n}}\right)$

for all x X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.4), we get (2.2).

It follows from (2.1) that

$\begin{array}{c}{∥h\left(x\right)+h\left(y\right)+h\left(z\right)∥}_{Y}=\underset{n\to \infty }{\text{lim}}{2}^{n}{∥f\left(\frac{x}{{2}^{n}}\right)+f\left(\frac{y}{{2}^{n}}\right)+f\left(\frac{z}{{2}^{n}}\right)∥}_{Y}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\le \underset{n\to \infty }{\text{lim}}{2}^{n}{∥2f\left(\frac{x+y+z}{{2}^{n+1}}\right)∥}_{Y}+\underset{n\to \infty }{\text{lim}}\frac{{2}^{n}\theta }{{2}^{nr}}\left({∥x∥}_{X}^{r}+{∥y∥}_{X}^{r}+{∥z∥}_{X}^{r}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}={∥2h\left(\frac{x+y+z}{2}\right)∥}_{Y}\end{array}$

for all x, y, z X. So

${∥h\left(x\right)+h\left(y\right)+h\left(z\right)∥}_{Y}\le {∥2h\left(\frac{x+y+z}{2}\right)∥}_{Y}$

for all x, y, z X. By Proposition 2.1, the mapping h : XY is Cauchy additive.

Now, let T : XY be another Cauchy additive mapping satisfying (2.2). Then we have

$\begin{array}{ll}\hfill {∥h\left(x\right)-T\left(x\right)∥}_{Y}& ={2}^{n}{∥h\left(\frac{x}{{2}^{n}}\right)-T\left(\frac{x}{{2}^{n}}\right)∥}_{Y}\phantom{\rule{2em}{0ex}}\\ \le {2}^{n}\left({∥h\left(\frac{x}{{2}^{n}}\right)-f\left(\frac{x}{{2}^{n}}\right)∥}_{Y}+{∥T\left(\frac{x}{{2}^{n}}\right)-f\left(\frac{x}{{2}^{n}}\right)∥}_{Y}\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{2\left({2}^{r}+2\right){2}^{n}}{\left({2}^{r}-2\right){2}^{nr}}\theta {∥x∥}_{X}^{r},\phantom{\rule{2em}{0ex}}\end{array}$

which tends to zero as n → ∞ for all x X. So we can conclude that h(x) = T(x) for all x X. This proves the uniqueness of h. Thus the mapping h : XY is a unique Cauchy additive mapping satisfying (2.2).

Theorem 2.3. Let r < 1 and θ be positive real numbers, and let f : XY be an odd mapping satisfying (2.1). Then there exists a unique Cauchy additive mapping h : XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{2+{2}^{r}}{2-{2}^{r}}\theta {∥x∥}_{X}^{r}$
(2.5)

for all x X.

Proof. It follows from (2.3) that

${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_{Y}\le \frac{2+{2}^{r}}{2}\theta {∥x∥}_{X}^{r}$

for all x X. Hence

$\begin{array}{ll}\hfill {∥\frac{1}{{2}^{l}}f\left({2}^{l}x\right)-\frac{1}{{2}^{m}}f\left({2}^{m}x\right)∥}_{Y}& \le \sum _{j=l}^{m-1}{∥\frac{1}{{2}^{j}}f\left({2}^{j}x\right)-\frac{1}{{2}^{j+1}}f\left({2}^{j+1}x\right)∥}_{Y}\phantom{\rule{2em}{0ex}}\\ \le \frac{2+{2}^{r}}{2}\sum _{j=l}^{m-1}\frac{{2}^{rj}}{{2}^{j}}\theta {∥x∥}_{X}^{r}\phantom{\rule{2em}{0ex}}\end{array}$
(2.6)

for all nonnegative integers m and l with m > l and all x X.

It follows from (2.6) that the sequence $\left\{\frac{1}{{2}^{n}}f\left({2}^{n}x\right)\right\}$ is a Cauchy sequence for all x X. Since Y is complete, the sequence $\left\{\frac{1}{{2}^{n}}f\left({2}^{n}x\right)\right\}$ converges. So one can define the mapping h : XY by

$h\left(x\right):=\underset{n\to \infty }{\text{lim}}\frac{1}{{2}^{n}}f\left({2}^{n}x\right)$

for all x X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.5).

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 2.4. Let $r>\frac{1}{3}$ and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_{Y}\le {∥2f\left(\frac{x+y+z}{2}\right)∥}_{Y}+\theta \cdot {∥x∥}_{X}^{r}\cdot {∥y∥}_{X}^{r}\cdot {∥z∥}_{X}^{r}$
(2.7)

for all x, y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{{2}^{r}\theta }{{8}^{r}-2}{∥x∥}_{X}^{3r}$
(2.8)

for all x X.

Proof. Letting y = x and z = - 2x in (2.7), we get

${∥2f\left(x\right)-f\left(2x\right)∥}_{Y}={∥2f\left(x\right)+f\left(-2x\right)∥}_{Y}\le {2}^{r}\theta {∥x∥}_{X}^{3r}$
(2.9)

for all x X. So

${∥f\left(x\right)-2f\left(\frac{x}{2}\right)∥}_{Y}\le \frac{{2}^{r}}{{8}^{r}}\theta {∥x∥}_{X}^{3r}$

for all x X. Hence

$\begin{array}{ll}\hfill {∥{2}^{l}f\left(\frac{x}{{2}^{l}}\right)-{2}^{m}f\left(\frac{x}{{2}^{m}}\right)∥}_{Y}& \le \sum _{j=l}^{m-1}{∥{2}^{j}f\left(\frac{x}{{2}^{j}}\right)-{2}^{j+1}f\left(\frac{x}{{2}^{j+1}}\right)∥}_{Y}\phantom{\rule{2em}{0ex}}\\ \le \frac{{2}^{r}}{{8}^{r}}\sum _{j=l}^{m-1}\frac{{2}^{j}}{{8}^{rj}}\theta {∥x∥}_{X}^{3r}\phantom{\rule{2em}{0ex}}\end{array}$
(2.10)

for all nonnegative integers m and l with m > l and all x X.

It follows from (2.10) that the sequence $\left\{{2}^{n}f\left(\frac{x}{{2}^{n}}\right)\right\}$ is a Cauchy sequence for all x X. Since Y is complete, the sequence $\left\{{2}^{n}f\left(\frac{x}{{2}^{n}}\right)\right\}$ converges. So one can define the mapping h : XY by

$h\left(x\right):=\underset{n\to \infty }{\text{lim}}{2}^{n}f\left(\frac{x}{{2}^{n}}\right)$

for all x X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.8).

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 2.5. Let $r<\frac{1}{3}$ and θ be positive real numbers, and let f : XY be an odd mapping satisfying (2.7). Then there exists a unique Cauchy additive mapping h:XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{{2}^{r}\theta }{2-{8}^{r}}{∥x∥}_{X}^{3r}$
(2.11)

for all x X.

Proof. It follows from (2.9) that

${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_{Y}\le \frac{{2}^{r}}{2}\theta {∥x∥}_{X}^{3r}$

for all x X. Hence

$\begin{array}{ll}\hfill {∥\frac{1}{{2}^{l}}f\left({2}^{l}x\right)-\frac{1}{{2}^{m}}f\left({2}^{m}x\right)∥}_{Y}& \le \sum _{j=l}^{m-1}{∥\frac{1}{{2}^{j}}f\left({2}^{j}x\right)-\frac{1}{{2}^{j+1}}f\left({2}^{j+1}x\right)∥}_{Y}\phantom{\rule{2em}{0ex}}\\ \le \frac{{2}^{r}}{2}\sum _{j=l}^{m-1}\frac{{8}^{rj}}{{2}^{j}}\theta {∥x∥}_{X}^{r}\phantom{\rule{2em}{0ex}}\end{array}$
(2.12)

for all nonnegative integers m and l with m > l and all x X.

It follows from (2.12) that the sequence $\left\{\frac{1}{{2}^{n}}f\left({2}^{n}x\right)\right\}$ is a Cauchy sequence for all x X. Since Y is complete, the sequence $\left\{\frac{1}{{2}^{n}}f\left({2}^{n}x\right)\right\}$ converges. So one can define the mapping h : XY by

$h\left(x\right):=\underset{n\to \infty }{\text{lim}}\frac{1}{{2}^{n}}f\left({2}^{n}x\right)$

for all x X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.12), we get (2.11).

The rest of the proof is similar to the proof of Theorem 2.2.

## 3 Stability of a functional inequality associated with a 3-variable Cauchy additive functional equation

Proposition 3.1. [18, Proposition 2.2] Let f : GY be a mapping such that

${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_{Y}\le {∥f\left(x+y+z\right)∥}_{Y}$

for all x, y, z G. Then f is Cauchy additive.

We prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type 3-variable Cauchy additive functional equation.

Theorem 3.2. Let r > 1 and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_{Y}\le {∥f\left(x+y+z\right)∥}_{Y}+\theta \left({∥x∥}_{X}^{r}+{∥y∥}_{X}^{r}+{∥z∥}_{X}^{r}\right)$
(3.1)

for all x, y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{{2}^{r}+2}{{2}^{r}-2}\theta {∥x∥}_{X}^{r}$

for all x X.

Proof. Letting y = x and z = - 2x in (3.1), we get

${∥2f\left(x\right)-f\left(2x\right)∥}_{Y}={∥2f\left(x\right)+f\left(-2x\right)∥}_{Y}\le \left(2+{2}^{r}\right)\theta {∥x∥}_{X}^{r}$
(3.2)

for all x X.

The rest of the proof is the same as in the proof of Theorem 2.2.

Theorem 3.3. Let r < 1 and θ be positive real numbers, and let f : XY be an odd mapping satisfying (3.1). Then there exists a unique Cauchy additive mapping h:XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{2+{2}^{r}}{2-{2}^{r}}\theta {∥x∥}_{X}^{r}$

for all x X.

Proof. It follows from (3.2) that

${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_{Y}\le \frac{2+{2}^{r}}{2}\theta {∥x∥}_{X}^{r}$

for all x X.

The rest of the proof is the same as in the proofs of Theorems 2.2 and 2.3.

Theorem 3.4. Let $r>\frac{1}{3}$ and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

${∥f\left(x\right)+f\left(y\right)+f\left(z\right)∥}_{Y}\le {∥f\left(x+y+z\right)∥}_{Y}+\theta \cdot {∥x∥}_{X}^{r}\cdot {∥y∥}_{X}^{r}\cdot {∥z∥}_{X}^{r}$
(3.3)

for all x, y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{{2}^{r}\theta }{{8}^{r}-2}{∥x∥}_{X}^{3r}$

for all x X.

Proof Letting y = x and z = - 2x in (3.3), we get

${∥2f\left(x\right)-f\left(2x\right)∥}_{Y}={∥2f\left(x\right)+f\left(-2x\right)∥}_{Y}\le {2}^{r}\theta {∥x∥}_{X}^{3r}$
(3.4)

for all x X.

The rest of the proof is the same as in the proofs of Theorems 2.2 and 2.4.

Theorem 3.5. Let $r<\frac{1}{3}$ and θ be positive real numbers, and let f : XY be an odd mapping satisfying (3.3). Then there exists a unique Cauchy additive mapping h : XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{{2}^{r}\theta }{2-{8}^{r}}{∥x∥}_{X}^{3r}$

for all x X.

Proof. It follows from (3.4) that

${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_{Y}\le \frac{{2}^{r}}{2}\theta {∥x∥}_{X}^{3r}$

for all x X.

The rest of the proof is the same as in the proofs of Theorems 2.2 and 2.5.

## 4 Stability of a functional inequality associated with the Cauchy-Jensen functional equation

Proposition 4.1. [18, Proposition 2.3] Let f :GY be a mapping such that

${∥f\left(x\right)+f\left(y\right)+2f\left(z\right)∥}_{Y}\le {∥2f\left(\frac{x+y}{2}+z\right)∥}_{Y}$

for all x, y, z G. Then f is Cauchy additive.

We prove the Hyers-Ulam stability of a functional inequality associated with a Jordan-von Neumann type Cauchy-Jensen functional equation.

Theorem 4.2. Let r > 1 and θ be nonnegative real numbers, and let f : XY be an odd mapping such that

${∥f\left(x\right)+f\left(y\right)+2f\left(z\right)∥}_{Y}\le {∥2f\left(\frac{x+y}{2}+z\right)∥}_{Y}+\theta \left({∥x∥}_{X}^{r}+{∥y∥}_{X}^{r}+{∥z∥}_{X}^{r}\right)$
(4.1)

for all x, y, z X. Then there exists a unique Cauchy additive mapping h : XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{{2}^{r}+1}{{2}^{r}-2}\theta {∥x∥}_{X}^{r}$

for all x X.

Proof. Replacing x by 2x and letting y = 0 and z = -x in (4.1), we get

${∥f\left(2x\right)-2f\left(x\right)∥}_{Y}={∥f\left(2x\right)+2f\left(-x\right)∥}_{Y}\le \left(1+{2}^{r}\right)\theta {∥x∥}_{X}^{r}$
(4.2)

for all x X. So

${∥f\left(x\right)-2f\left(\frac{x}{2}\right)∥}_{Y}\le \frac{1+{2}^{r}}{{2}^{r}}\theta {∥x∥}_{X}^{r}$

for all x X.

The rest of the proof is similar to the proof of Theorem 2.2.

Theorem 4.3. Let r < 1 and θ be positive real numbers, and let f : XY be an odd mapping satisfying (4.1). Then there exists a unique Cauchy additive mapping h : XY such that

${∥f\left(x\right)-h\left(x\right)∥}_{Y}\le \frac{1+{2}^{r}}{2-{2}^{r}}\theta {∥x∥}_{X}^{r}$

for all x X.

Proof. It follows from (4.2) that

${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_{Y}\le \frac{1+{2}^{r}}{2}\theta {∥x∥}_{X}^{r}$

for all x X.

The rest of the proof is similar to the proofs of Theorems 2.2 and 2.3.

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Correspondence to Jung Rye Lee.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Park, C., Lee, J.R. Comment on "Functional inequalities associated with Jordan-von Neumann type additive functional equations". J Inequal Appl 2012, 47 (2012). https://doi.org/10.1186/1029-242X-2012-47 