Ratio Mathematica Volume 41, 2021, pp. 7-18

On the stability of a multiplicative type sum
form functional equation

Surbhi Madan*
Shveta Grover†

Dhiraj Kumar Singh‡§

Abstract

In this paper we intend to discuss the stability of a sum form func-
tional equation

n∑
i=1

m∑
j=1

f (piqj) =
n∑
i=1

k (pi)
m∑
j=1

q
β
j

where f,k are real valued mappings each having the domain I; (p1,
. . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm; n ≥ 3, m ≥ 3 are fixed integers
and β is a fixed positive real power different from 1 satisfying the
conventions 0β := 0 and 1β := 1.
Keywords: Additive mapping; bounded mapping; functional equa-
tion; stability of a sum form functional equation.
2020 AMS subject classifications: 39B52, 39B82. 1

*Department of Mathematics, Shivaji College (University of Delhi), Raja Garden, Ring Road,
New Delhi-110027, India; surbhimadan@gmail.com, surbhi@shivaji.du.ac.in.

†Department of Mathematics, University of Delhi, Delhi 110007, India; srk-
grover9@gmail.com.

‡Department of Mathematics, Zakir Husain Delhi College (University of Delhi), Jawaharlal
Nehru Marg, Delhi 110002, India; dhiraj426@rediffmail.com, dksingh@zh.du.ac.in.

§Corresponding author
1Received on November 23, 2021. Accepted on December 27, 2021. Published on December 31,
2021. doi: 10.23755/rm.v41i0.690. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This
paper is published under the CC-BY licence agreement.

7



S. Madan, S. Grover, D.K. Singh

1 Introduction
Throughout this paper, N denotes the set of natural numbers; R denotes the set

of real numbers and I denotes the closed unit interval [0, 1]. For n ∈ N,

Γn =

{
(p1, . . . ,pn); pi ≥ 0, i = 1, . . . ,n;

n∑
i=1

pi = 1

}

denotes the set of all finite n-component complete discrete probability distribu-
tions.

A mapping a : I → R is said to be additive on I or on the unit triangle
∆ = {(x,y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ x + y ≤ 1} if it satisfies the equation
a(x + y) = a(x) + a(y) for all (x,y) ∈ ∆. Further, a mapping A : R → R is
said to be additive on R if it satisfies the equation A(x + y) = A(x) + A(y) for
all x ∈ R, y ∈ R. An interesting relation between these two additive mappings
was established by Daróczy and Losonczi (2). They proved that additive mapping
a : I → R can be uniquely extended to the set of real numbers.

The stability of functional equations has intrigued mathematicians for more
than eight decades now. The problem primarily aims to study the functional in-
equality corresponding to the given functional equation. Thereafter, it focuses on
examining the proximity of their solutions. It needs to be remarked that without
knowing the general solution of an equation we can not discuss its stability.

The seminal problem entered the corpus of sum form functional equations
with the paper of Maksa (9). One of the interesting aspect explored in this field is
to obtain general solution and discuss the stability of those sum form functional
equations which are useful in characterizing entropies. Working in this direction,
Nath and Singh (15; 16; 18); Singh and Grover (25) have recently addressed few
sum form functional equations. These equations were useful in characterizing the
Shannon entropies (19) defined as:

Hn (p1, . . . ,pn) = −
n∑
i=1

pilog2pi (with 0 log2 0 := 0) (1.1)

where Hn : Γn → R, n ∈ N and (p1, . . . ,pn) ∈ Γn.
With the aim to further delve in the area of entropies, Havrda and Charvát

(4) generalized the Shannon entropies (1.1) by introducing entropies of degree β
defined as:

Hβn (p1, . . . ,pn) = (1 − 2
1−β)−1

[
1 −

n∑
i=1

p
β
i

]
(1.2)

8



On the stability of a multiplicative . . .

where Hβn : Γn → R, n ∈ N; (p1, . . . ,pn) ∈ Γn and β is a fixed positive real
power different from 1, such that 0β := 0 and 1β := 1.

Losonczi and Maksa (8) were first to address the sum form functional equa-
tion that characterized the entropies (1.2) by considering the multiplicative type
functional equation

n∑
i=1

m∑
j=1

f (piqj) =
n∑
i=1

f (pi)
m∑
j=1

f (qj) (1.3)

where f : I → R; (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm and n ≥ 3,m ≥ 3
are fixed integers. Thereafter, Nath and Singh have analysed Pexiderized forms of
(1.3), containing two and three unknown mappings for n ≥ 3,m ≥ 3 being fixed
integers in (11) and (12).

It needs to be highlighted that many research papers in reference to the sum
form functional equations characterizing several entropies have been written. In
brief, these papers reflected upon: some generalizations; Pexiderized forms; im-
portance and applications. Some significant contributions are: Nath and Singh
(10; 17); Singh and Dass (20); Singh and Grover (21; 22; 23). The primary focus
of these authors had been to obtain the general solution (or solutions) of the sum
form functional equations for fixed integers n ≥ 3, m ≥ 3 or n ≥ 3, m ≥ 2.
The stability problem for some of these sum form functional equations has been
discussed by the authors but for most of them it remains unaddressed.

One of the interesting equation which motivated us is the functional equation

n∑
i=1

m∑
j=1

f (piqj) =
n∑
i=1

f (pi)
m∑
j=1

q
β
j (1.4)

where f : I → R; (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm and β is a fixed positive
real power different from 1 satisfying 0β := 0 and 1β := 1. In the recent past, Nath
and Singh (13) have studied the equation (1.4) and obtained its general solutions
for n ≥ 3,m ≥ 3 being fixed integers. The authors have further explained the
relation of these solutions with entropies (1.2). The stability of (1.4) is established
by Singh and Grover (24) for n ≥ 3,m ≥ 3 being fixed integers. Indeed, with the
goal of getting a deeper insight of the equation (1.4), Nath and Singh (14); Garg,
Grover and Singh (3) have recently studied a Pexiderized form (1.4), that is

n∑
i=1

m∑
j=1

f (piqj) =
n∑
i=1

k (pi)
m∑
j=1

q
β
j (1.5)

where f : I → R, k : I → R; (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm and β is a
fixed positive real power different from 1 satisfying 0β := 0 and 1β := 1. In (14)

9



S. Madan, S. Grover, D.K. Singh

and (3), the authors have obtained the general solutions of (1.5) for n ≥ 3,m ≥ 3
and n ≥ 3,m ≥ 2. The authors have further reflected upon the significance of
general solutions in the light of entropies of degree β and diversity index. So
far the stability problem remains unaddressed. The objective of this paper is to
discuss the stability of functional equation (1.5).

For the problem of stability concerning functional equations, we refer to the
survey paper of Hyers and Rassias (5) and Hyers, Isac and Rassias (6). The prob-
lem of stability of the functional equation (1.5) is given along the following lines:
Let n ≥ 3, m ≥ 3 be fixed integers; 0 ≤ ε ∈ R be fixed. Find all the mappings
f : I → R, k : I → R satisfying the functional inequality∣∣∣∣∣

n∑
i=1

m∑
j=1

f(piqj) −
n∑
i=1

k(pi)
m∑
j=1

q
β
j

∣∣∣∣∣ ≤ ε (1.6)
for all (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm.

Below we will provide some known results.

Lemma 1.1 ((8)). Suppose a mapping φ : I → R satisfies the functional equation
n∑
i=1

φ(pi) = c1 for all (p1, . . . ,pn) ∈ Γn, n ≥ 3 a fixed integer and c1 a real

constant. Then there exists an additive mapping a : R → R such that φ(p) =
a(p) − 1

n
a(1) + c1

n
for all p ∈ I.

Lemma 1.2 ((9)). Let 0 ≤ ε ∈ R, n ≥ 3 be fixed integer and ψ : I → R

be a mapping which satisfies the functional inequality
∣∣∣∣ n∑
i=1

ψ(pi)

∣∣∣∣ ≤ ε for all
(p1, . . . ,pn) ∈ Γn. Then there exist an additive mapping A1 : R → R and a
mapping B1 : R → R such that |B1(p)| ≤ 18ε for all p ∈ R, B(0) = 0 and
ψ(p) −ψ(0) = A1(p) + B1(p) for all p ∈ I.

Lemma 1.3 ((7)). Let A2 : R → R be an additive mapping, M : I → R a
multiplicative mapping, B2 : R → R a bounded mapping and c2 ∈ R. If A2(p) =
M(p)+c2 for all p ∈ I, then A2(p) = dp, p ∈ R for some d ∈ R and M(p) = 0 or
M(p) = p, p ∈ I. Also if A2(p) = M(p) + B2(p) for all p ∈ I, then A2(p) = dp,
p ∈ R for some d ∈ R and M(p) = 0 or M(p) = pα, p ∈ I for some 0 ≤ α ∈ R.

Lemma 1.4 ((26)). If f is a solution of the functional equation f(x+y) = f(x) +
f(y) which is bounded over an interval [a,b], then it is of the form f(x) = c3x for
some real number c3.

Lemma 1.5 ((3)). Let n ≥ 3, m ≥ 2 be fixed integers; β be fixed positive real
power different from 1 satisfying the conventions 0β := 0, 1β := 1 and f : I → R,

10



On the stability of a multiplicative . . .

k : I → R. The pair (f,k) satisfies (1.5) if and only if there exist the additive
mappings a1,a2 : R → R and c ∈ R such that

(i) f(p) = cpβ + a1(p) − 1nma1(1),

(ii) k(p) = cpβ + a2(p) − 1na2(1).

}

2 The stability of the functional equation (1.5)
In this section our primary aim is to find the solutions of inequality (1.6).

Thereafter, we need to observe: What is the difference between these solutions
and the solutions (given by Lemma 1.5) of equation (1.5)? In the sense of Hyers,
Isac and Rassias (6), if the difference is only a bounded mapping, we would say
that functional equation (1.5) is stable. Following this we establish the the main
result as follows:

Theorem 2.1. Let n ≥ 3, m ≥ 3 be fixed integers; β be fixed positive real power
different from 1 satisfying the conventions 0β := 0 and 1β := 1; ε be a nonnegative
real constant and let f : I → R, k : I → R be real valued mappings. Suppose the
pair (f,k) satisfies (1.6), then there exist the additive mappings a1,a2 : R → R,
the bounded mappings b1,b2 : R → R and 0 6= c,c ∈ R such that

(i) f(p) −f(0) = cpβ + a1(p) + b1(p),

(ii) k(p) −k(0) = cpβ + a2(p) + b2(p)

}
(2.1)

with

(i) |b1(p)| ≤ 1296εc (2m + 1), b1(0) = 0,

(ii) |b2(p)| ≤ 1296εc (2m + 1) + 18ε, b2(0) = 0.

}
(2.2)

Proof. Let us put q1 = 1, q2 = . . . = qm = 0 in (1.6). We get∣∣∣∣∣
n∑
i=1

[
f(pi) + (m− 1)f(0) −k(pi)

]∣∣∣∣∣ ≤ ε
for all (p1, . . . ,pn) ∈ Γn. By Lemma 1.2, there exists an additive mapping A1 :
R → R and a mapping B∗1 : R → R with |B∗1 (p)| ≤ 18ε, B∗1 (0) = 0, such that

f(p) −k(p) −f(0) + k(0) = A1(p) + B∗1 (p)

for all p ∈ I. From this, we obtain the expression

k(p) = f(p) −A1(p) −B1(p) (2.3)

11



S. Madan, S. Grover, D.K. Singh

where B1 : R → R, defined as B1(x) = B∗1 (x) + f(0) − k(0) is a bounded
mapping. With the aid of (2.3), inequality (1.6) can be written as∣∣∣∣∣

n∑
i=1

[
m∑
j=1

f(piqj) − [f(pi) −A1(1)pi −B1(pi)]
m∑
j=1

q
β
j

]∣∣∣∣∣ ≤ ε
for all (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm; n ≥ 3, m ≥ 3 being fixed integers.
By Lemma 1.2, there exists a mapping A2 : R × Γm → R, additive in the first
variable and a mapping B2 : R × Γm → R, bounded in the first variable by 18ε
with B2(0; q1, . . . ,qm) = 0, such that

m∑
j=1

f(pqj) − [f(p) −A1(1)p−B1(p) −f(0) + B1(0)]
m∑
j=1

q
β
j −mf(0)

= A2(p; q1, . . . ,qm) + B2(p; q1, . . . ,qm). (2.4)

Let x ∈ I and (r1, . . . ,rm) ∈ Γm be an arbitrary probability distribution. Now,
replacing p successively by xrt, t = 1, . . . ,m in (2.4); summing the resulting m
equations so obtained and using the additivity of the mapping A2 : R × Γm → R
in the first variable, we have
m∑
t=1

m∑
j=1

f(xrtqj)−

[
m∑
t=1

f(xrt)−A1(1)x−
m∑
t=1

B1(xrt)−mf(0)+mB1(0)

]
m∑
j=1

q
β
j

−m2f(0) = A2(x; q1, . . . ,qm) +
m∑
t=1

B2(xrt; q1, . . . ,qm) (2.5)

for all x ∈ I, (q1, . . . ,qm) ∈ Γm and (r1, . . . ,rm) ∈ Γm. Now for p = x and
q1 = r1, . . . ,qm = rm, functional equation (2.4) gives

m∑
t=1

f(xrt) = [f(x) −A1(1)x−B1(x) −f(0) + B1(0)]
m∑
t=1

r
β
t + mf(0)

+ A2(x; r1, . . . ,rm) + B2(x; r1, . . . ,rm). (2.6)

From (2.5) and (2.6), we get
m∑
t=1

m∑
j=1

f(xrtqj) − [f(x) −A1(1)x−B1(x) −f(0) + B1(0)]
m∑
t=1

r
β
t

m∑
j=1

q
β
j

−m2f(0) =
[
A2(x; r1, . . . ,rm)+B2(x; r1, . . . ,rm)−A1(1)x−

m∑
t=1

B1(xrt)

+ mB1(0)

] m∑
j=1

q
β
j + A2(x; q1, . . . ,qm) +

m∑
t=1

B2(xrt; q1, . . . ,qm). (2.7)

12



On the stability of a multiplicative . . .

We see that, the left hand side of (2.7) is commutative in rt and qj, t = 1, . . . ,m;
j = 1, . . . ,m (Acźel (1)). So, the commutativity on the right hand side implies

A2(x; r1, . . . ,rm)

[
1 −

m∑
j=1

q
β
j

]
−A2(x; q1, . . . ,qm)

[
1 −

m∑
t=1

r
β
t

]

=
m∑
t=1

B2(xrt; q1, . . . ,qm) −
m∑
j=1

B2(xqj; r1, . . . ,rm)

+

[
B2(x; r1, . . . ,rm) −A1(1)x−

m∑
t=1

B1(xrt) + mB1(0)

]
m∑
j=1

q
β
j

−

[
B2(x; q1, . . . ,qm) −A1(1)x−

m∑
j=1

B1(xqj) + mB1(0)

]
m∑
t=1

r
β
t .

For fixed (q1, . . . ,qm) ∈ Γm and (r1, . . . ,rm) ∈ Γm, the right hand side of the
above equation is bounded on I while the left hand side is additive in x ∈ I. So,
by Lemma 1.4, it follows that

[A2(x; r1, . . . ,rm) −xA2(1; r1, . . . ,rm)]

[
1 −

m∑
j=1

q
β
j

]

= [A2(x; q1, . . . ,qm) −xA2(1; q1, . . . ,qm)]

[
1 −

m∑
t=1

r
β
t

]
. (2.8)

Now, we assert that for m ≥ 3 and fixed positive real power β 6= 1, 1−
m∑
t=1

r
β
t

does not vanish identically on Γm. To the contrary, suppose 1−
m∑
t=1

r
β
t vanishes

identically on Γm. Then, 1 =
m∑
t=1

r
β
t for all (r1, . . . ,rm) ∈ Γm. By Lemma 1.1,

there exists an additive mapping A : R → R such that rβ = A(r) with A(1) =
1. By Lemma 1.3, rβ = 0 or rβ = r for all r ∈ I. This gives a contradiction
as for the former case, our supposition 1 =

m∑
t=1

r
β
t is contradicted while for the

latter case, our assumption β 6= 1 is contradicted. This proves our assertion and so
there exists a probability distribution (r∗1, . . . ,r

∗
m) ∈ Γm such that 1−

m∑
t=1

r
∗β
t 6= 0.

Therefore with the substitution rt = r∗t , t = 1, . . . ,m, functional equation (2.8)
reduces to

A2(x; q1, . . . ,qm) = A0(x)

[
1 −

m∑
j=1

q
β
j

]
+ x A2(1; q1, . . . ,qm) (2.9)

13



S. Madan, S. Grover, D.K. Singh

where A0 : R → R defined as

A0(x) =

[
1 −

m∑
t=1

r∗
β

t

]−1
[A2(x; r

∗
1, . . . ,r

∗
m) −x A2(1; r

∗
1, . . . ,r

∗
m)]

is an additive mapping with A0(1) = 0. Further from (2.4), we have

A2 (1; q1, . . . ,qm) =
m∑
j=1

f(qj)−[f(1)−A1(1)−B1(1)−f(0)+B1(0)]
m∑
j=1

q
β
j

−mf(0) −B2 (1; q1, . . . ,qm) . (2.10)

From (2.4), (2.9), (2.10) with A0(1) = 0, we gather that

m∑
j=1

[f(pqj)−f(0)−A0(pqj)− (f(1)−f(0))pqj]−[f(p)−f(0)−A0(p)

−p(f(1)−f(0))]
m∑
j=1

q
β
j −p

m∑
j=1

[f(qj)−f(0)−A0(qj)−(f(1)−f(0))qj]

= [B1(0)−B1(p)+pB1(1)−pB1(0)]
m∑
j=1

q
β
j −pB2 (1; q1, . . . ,qm) (2.11)

for all p ∈ I and (q1, . . . ,qm) ∈ Γm.
Now, define a mapping F : I → R as

F(x) = f(x) −f(0) −A0(x) − (f(1) −f(0))x (2.12)

for all x ∈ I. With the aid of (2.12), equation (2.11) can be written as
m∑
j=1

F(pqj) −F(p)
m∑
j=1

q
β
j −p

m∑
j=1

F(qj)

= [B1(0)−B1(p)+pB1(1)−pB1(0)]
m∑
j=1

q
β
j −pB2 (1; q1, . . . ,qm) . (2.13)

It clearly follows from (2.12), that F(0) = 0. Also we observe, the right hand side
of (2.13) is bounded by 18ε(2m + 1), consequently by Lemma 1.2, along with
F(0) = 0, there exists a mapping A3 : I×R → R, additive in the second variable
and a mapping B3 : I×R → R, bounded in the second variable by 324ε(2m + 1)
with B3(p; 0) = 0, such that

F(pq) − qβ F(p) −p F(q) = A3(p; q) + B3(p; q). (2.14)

14



On the stability of a multiplicative . . .

Define a mapping K : I × I → R as

K(p; q) = F(pq) − qβ F(p) −p F(q) (2.15)

for all p ∈ I, q ∈ I. With the help of (2.15), it can be verified easily that

F(pqr) −pq F(r) − qβrβ F(p) −prβ F(q) = K (pq; r) + rβK (p; q)
= K (p; qr) + pK(q; r) (2.16)

for all p ∈ I, q ∈ I and r ∈ I. From (2.14), (2.15) and (2.16), it follows that

A3 (p; qr) + pA3 (q; r) −A3 (pq; r)
= B3 (pq; r) + r

βA3 (p; q) + r
βB3 (p; q) −B3 (p; qr) −pB3 (q; r) . (2.17)

Apparently, the left hand side of (2.17) is additive in r ∈ I, while its right hand
side is bounded on I. Consequently by Lemma 1.4, its left hand side must be
linear therefore, we get

A3 (p; qr)+pA3 (q; r)−A3 (pq; r) =r [A3 (p; q) + pA3 (q; 1)−A3 (pq; 1)] . (2.18)

Now, for the substitution r = 1, equation (2.17) gives

pA3 (q; 1) −A3 (pq; 1) = B3 (pq; 1) −pB3(q; 1) . (2.19)

From (2.17), (2.18) and (2.19), we obtain(
rβ − r

)
A3 (p; q) = rB3 (pq; 1) − rpB3(q; 1) −B3 (pq; r)

− rβB3 (p; q) + B3 (p; qr) + pB3 (q; r) . (2.20)

Since for fixed positive real number β 6= 1, we have ‘rβ − r’ does not vanish
identically on I, there exists some r∗ ∈ I, such that

r∗β − r∗ 6≡ 0. (2.21)

Using this in (2.20), it follows that

A3 (p; q) =
(
r∗β − r∗

)−1{
r∗B3 (pq; 1) − r∗pB3(q; 1) −B3 (pq; r∗)

− r∗βB3 (p; q) + B3 (p; qr∗) + pB3 (q; r∗)
}

for all p ∈ I, q ∈ I. This yield that additive mapping A3(p; q) is bounded in the
second variable also. Hence by Lemma 1.4, it must be linear therein. So,

A3 (p; q) = qA3(p; 1) (2.22)

15



S. Madan, S. Grover, D.K. Singh

for all p ∈ I, q ∈ I. From (2.19), with the substitution q = 1 it follows that

A3 (p; 1) = pA3 (1; 1) −B3 (p; 1) + pB3(1; 1) (2.23)

for all p ∈ I. From (2.22) and (2.23), it can be concluded that mapping A3(p; q) is
bounded. Moreover, from (2.14) and (2.23) with F(1) = 0 we obtain, A3(p; 1) =
−B3(p; 1). Consequently we get |A3(p; q)| ≤ 324ε(2m + 1). Hence, using this in
(2.14), it follows that

|F(pq) − qβ F(p) −p F(q)| ≤ 648ε(2m + 1) (2.24)

for all p ∈ I, q ∈ I. Now on interchanging the places of p and q in the functional
inequality (2.24), we have

|F(pq) −pβ F(q) − q F(p)| ≤ 648ε(2m + 1). (2.25)

Applying triangle inequality to functional inequalities (2.24) and (2.25), we obtain

|(qβ − q) F(p) − (pβ −p) F(q)| ≤ 1296ε(2m + 1) (2.26)

where p ∈ I, q ∈ I and β is a fixed positive real power different from 1. With the
aid of (2.21) we get, q∗β − q∗ 6≡ 0 for some q∗ ∈ I. On taking c := (q∗β − q∗)−1 ∈
R; c := F(q∗)(q∗β − q∗)−1 ∈ R in (2.26), it follows that there exist a mapping
b1 : R → R such that (2.2)(i) holds for all p ∈ R and

F(p) = cpβ − cp + b1(p) (2.27)

for all p ∈ I. Thus, the solution (2.1)(i) follows from (2.12) and (2.27) by defining
additive mapping a1 : R → R as a1(x) = A0(x)+(f(1)−f(0))x−cx. Further, the
solution (2.1)(ii) with (2.2)(ii) follows from (2.1)(i), (2.2)(i) and (2.3) by defining
additive mapping a2 : R → R as a2(x) = a1(x) − A1(x) and bounded mapping
b2 : R → R as b2(x) = b1(x) −B∗1 (x). This completes the proof.

3 Acknowledgements
The third author is grateful for the support from the SERB-MATRICS scheme

(MTR/2020/000508) of the Department of Science and Technology, Government
of India.

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