International Journal of Analysis and Applications

ISSN 2291-8639

Volume 1, Number 1 (2013), 1-17

http://www.etamaths.com

GENERALIZED ULAM-HYERS STABILITY OF THE

HARMONIC MEAN FUNCTIONAL EQUATION IN TWO

VARIABLES

K. RAVI1,∗, J.M. RASSIAS2 AND B.V. SENTHIL KUMAR3

Abstract. In this paper, we find the solution and prove the generalized Ulam-

Hyers stability of the harmonic mean functional equation in two variables. We

also provide counterexamples for singular cases.

1. INTRODUCTION

In 1940, S.M. Ulam [47] raised the following question concerning the stability

of group homomorphisms:

“Let G be a group and H be a metric group with metric d(., .). Given � > 0,

does there exist a δ > 0 such that if a function f : G → H satisfies

d(f(xy),f(x)f(y)) < δ

for all x,y ∈ G, then there exists a homomorphism a : G → H with d(f(x),a(x)) <

� for all x ∈ G?”

In 1941, D.H. Hyers [20] gave an answer to the Ulam’s stability problem. He

proved the following celebrated theorem.

Theorem 1.1. [D.H. Hyers] Let X,Y be Banach spaces and let f : X → Y be a

mapping satisfying

(1.1) ‖f(x + y) −f(x) −f(y)‖≤ �

2010 Mathematics Subject Classification. 39B22, 39B52, 39B72.

Key words and phrases. Harmonic mean; additive functional equation; reciprocal functional

equation; Generalized Hyers-Ulam stability.

c©2013 Authors retain the copyrights of their papers, and all

open access articles are distributed under the terms of the Creative Commons Attribution License.

1



2 K. RAVI, J.M. RASSIAS AND B.V. SENTHIL KUMAR

for all x,y ∈ X. Then the limit

(1.2) a(x) = lim
n→∞

f(2nx)

2n

exists for all x ∈ X and a : X → Y is the unique additive mapping satisfying

(1.3) ‖f(x) −a(x)‖≤ �

for all x ∈ X.

In 1950, Aoki [2] generalized the Hyers theorem for additive mappings. In

1978, Th.M. Rassias [43] provided a generalized version of the theorem of Hyers

which permitted the cauchy difference to become unbounded. Th.M. Rassias proved

the following theorem for sum of powers of norms.

Theorem 1.2. [Th.M. Rassias] Let X and Y be two Banach spaces. Let θ ∈

[0,∞) and let p ∈ [0, 1). If a function f : X → Y satisfies the inequality

(1.4) ‖f(x + y) −f(x) −f(y)‖≤ θ(‖x‖p + ‖y‖p)

for all x,y ∈ X. Then there exists a unique linear mapping T : X → Y such that

(1.5) ‖f(x) −T(x)‖≤
2θ

2 − 2p
‖x‖p

for all x ∈ X. Moreover, if f(tx) is continuous in t for each fixed x ∈ X, then the

function T is linear.

The theorem of Th.M. Rassias was later extended for all p 6= 1. The stability

phenomenon that was presented by Th.M. Rassias is called the generalized Hyers-

Ulam stability. In 1982, J.M. Rassias [33] gave a further generalization of the result

of D.H. Hyers and proved the following theorem using weaker conditions controlled

by a product of powers of norms.

Theorem 1.3. [J.M. Rassias] Let f : E → E′ be a mapping from a normed

vector space E into a Banach space E′ subject to the inequality

(1.6) ‖f(x + y) −f(x) −f(y)‖≤ �‖x‖p‖y‖p



GENERALIZED ULAM-HYERS STABILITY 3

for all x,∈ E, where � and p are constants with � > 0 and 0 ≤ p < 1
2

. Then the

limit

(1.7) L(x) = lim
n→∞

f(2nx)

2n

exists for all x ∈ E and L : E → E′ is the unique additive mapping which satisfies

(1.8) ‖f(x) −L(x)‖≤
�

2 − 22p
‖x‖2p

for all x ∈ E. Moreover, if f(tx) is continuous in t for each fixed x of X, then the

function L is linear.

The above mentioned stability involving a product of powers of norms is called

Ulam-Gavruta-Rassias stability by various authors ([4], [5], [6], [9], [11], [18], [19],

[30], [31], [37], [38], [41], [45], [46]). Very recently, J.M. Rassias introduced mixed

type product sum of powers of norms in [38]. The investigation of stability of

functional equations involving with the mixed type product sum of powers of norms

is known as J.M. Rassias stability. Several functional equations and its J.M. Rassias

stability were investigated by many mathematicians ([5], [6], [18], [19], [39], [41],

[45]).

Theorem 1.4. [42] Let (E,⊥) be an orthogonality normed space with norm ‖.‖E
and (F,‖.‖F ) be a Banach space. Let f : E → F be a mapping which satisfies the

inequality∥∥f(mx + y) + f(mx−y) − 2f(x + y) − 2f(x−y) − 2(m2 − 2)f(x) + 2f(y)∥∥
F

≤ �
{
‖x‖pE ‖y‖

p
E +

(
‖x‖2pE + ‖y‖

2p
E

)}
for all x,y ∈ E with x⊥y, where � and p are constants with �,p > 0 and either

m > 1; p < 1 or m < 1; p > 1 with m 6= 0;m 6= ±1; m 6= ±
√

2 and −1 6= |m|p−1 < 1.

Then the limit

Q(x) = lim
n→∞

f(mnx)

m2n

exists for all x ∈ E and Q : E → F is the unique orthogonally Euler-Lagrange

quadratic mapping such that

‖f(x) −Q(x)‖F ≤
�

2 |m2 −m2p|
‖x‖2pE

for all x ∈ E.



4 K. RAVI, J.M. RASSIAS AND B.V. SENTHIL KUMAR

In the last two decades, several form of quadratic, cubic and quartic functional

equations and its Hyers-Ulam-Rassias stability were discussed by various authors

([3], [7], [12], [21], [24], [27], [29], [32], [34], [35]). Recently, the mixed type of

functional equations that is having additive and quadratic, quadratic and quartic,

additive and cubic, additive, quadratic, cubic and quartic property were investi-

gated in the literature ([8], [13], [14], [15], [16], [17], [18], [22], [26], [28], [36], [41],

[42], [44]). Very recently, we find that the functional equations are dealt in vari-

ous spaces like fuzzy normed spaces, random normed spaces, quasi-Banach spaces,

quasi-normed linear spaces by various authors, one can refer to ([10], [15], [16],

[17], [18], [23], [25], [26], [42], [44]). But, not many papers are available in the

reciprocal functional equation (RFE) and Reciprocal difference functional equation

(RDFE). K. Ravi and B.V. Senthil Kumar [39] introduced and investigated a new

2-dimensional reciprocal functional equation (RFE)

(1.9) f(x + y) =
f(x)f(y)

f(x) + f(y)

and obtained some interesting results on the Ulam-Gavruta-Rassias stability for

the equation (1.9). In 2009, J.M. Rassias introduced the Reciprocal Difference

Functional equation (RDF equation)

(1.10) r

(
x + y

2

)
−r(x + y) =

r(x)r(y)

r(x) + r(y)

and the Reciprocal Adjoint Functional equation (RAF equation)

(1.11) r

(
x + y

2

)
+ r(x + y) =

3r(x)r(y)

r(x) + r(y)
.

and discussed Ulam stability problem as well as the extended Ulam stability prob-

lem for the equation (1.10) and (1.11) in [40].

In this paper, we obtain the solution and generalized Ulam-Hyers stability of

the harmonic mean functional equation in two variables of the form

(1.12) H
( xu
x + u

,
yv

y + v

)
=

H(x,y)H(u,v)

H(x,y) + H(u,v)

which is originating from the harmonic mean of two positive real numbers x and

y. The function H(x,y) = 2xy
x+y

is a solution of the functional equation (1.12). The

above function H(x,y) represents the harmonic mean of x and y. We also provide

counterexamples for singular cases.



GENERALIZED ULAM-HYERS STABILITY 5

Before we proceed to the main theorem, we present some definitions, which

will be useful to prove our theorems.

Definition 1.5. The harmonic mean of two positive real numbers x and y is the

reciprocal of arithmetic mean of the reciprocals of x and y. Thus, harmonic mean

of x and y is

Harmonic mean =
2

1
x

+ 1
y

=
2xy

x + y
.

Definition 1.6. A mapping a : X → Y between Banach spaces is called additive

if a satisfies the functional equation

(1.13) a(x + y) = a(x) + a(y).

The function a(x) = cx is a solution of the functional equation (1.13).

Definition 1.7. A function r : X → Y between sets of non-zero real numbers is

called reciprocal if r satisfies the functional equation (1.9). The function r(x) = c
x

is a solution of the functional equation (1.9).

Definition 1.8. Let X be set of positive real numbers. A function H : X×X → R

be such that H(x,y) = 2xy
x+y

is called harmonic mean function if it satisfies the

functional equation (1.12). The functional equation (1.12) is called harmonic mean

functional equation.

Throughout this paper, let X be set of positive real numbers.

2. RELATION BETWEEN (1.9) AND (1.12)

Theorem 2.1. Let r : X → R be a mapping satisfying r(x) = c
x

where c ∈ R−{0}.
If H : X ×X → R is a mapping given by

(2.1) H(x,y) =
2

r(x) + r(y)

for all x,y ∈ X, then H satisfies (1.12).



6 K. RAVI, J.M. RASSIAS AND B.V. SENTHIL KUMAR

Proof. From (2.1), we have

H
( xu
x + u

,
yv

y + v

)
=

2

r
(
xu
x+u

)
+ r
(
yv
y+v

)
=

2

c
[
x+u
xu

]
+ c
[
y+v
yv

]
=

2/c
1
x

+ 1
u

+ 1
y

+ 1
v

(2.2)

and
H(x,y)H(u,v)

H(x,y) + H(u,v)
=

2
r(x)+r(y)

2
r(u)+r(v)

2
r(x)+r(y)

+ 2
r(u)+r(v)

=
2

r(x) + r(u) + r(y) + r(v)

=
2/c

1
x

+ 1
u

+ 1
y

+ 1
v

.(2.3)

From (2.2) and (2.3), we see that H satisfies (1.12). �

3. SOLUTION OF FUNCTIONAL EQUATION (1.12)

Theorem 3.1. A mapping H : X × X → R satisfies (1.12) if and only if there
exists an additive mapping A : X → R such that

H(x,y) =
2cA(x)A(y)

A(x) + A(y)

for all x,y ∈ X, where c ∈ R−{0}.

Proof. Let H(x,y) be a solution of (1.12). Then by Theorem 2.1, H will be of

the form (2.1). That is, H(x,y) = 2
r(x)+r(y)

, for all x,y ∈ X. In particular,
H(x,x) = 2

r(x)+r(x)
= x

c
, for all x ∈ X. Now, we define a mapping A : X → R

be such that A(x) = H(x,x) = x
c
, for all x ∈ X. Then, clearly A is an additive

mapping. Further,

2cA(x)A(y)

A(x) + A(y)
=

2cx
c
y
c

x
c

+ y
c

=
2xy

x + y

= H(x,y), for all x,y ∈ X.



GENERALIZED ULAM-HYERS STABILITY 7

Conversely, if there exists an additive mapping A : X → R such that H(x,y) =
2cA(x)A(y)
A(x)+A(y)

, for all x,y ∈ X, then

H
( xu
x + u

,
yv

y + v

)
=

2cA
(
xu
x+u

)
A
(
yv
y+v

)
A
(
xu
x+u

)
+ A

(
yv
y+v

)

=
2 xy
x+y

uv
u+v

xy
x+y

+ uv
u+v

=
2
(
c

x
c
y
c

x
c
+
y
c

)(
c

u
c
v
c

u
c
+v
c

)
(
c

x
c
y
c

x
c
+
y
c

)
+
(
c

u
c
v
c

u
c
+v
c

)
=

[
2cA(x)A(y)
A(x)+A(y)

][
2cA(u)A(v)
A(u)+A(v)

]
[
2cA(x)A(y)
A(x)+A(y)

]
+
[
2cA(u)A(v)
A(u)+A(v)

]
=

H(x,y)H(u,v)

H(x,y) + H(u,v)

for all x,u,y,v ∈ X. �

4. GENERALIZED ULAM-HYERS STABILITY OF FUNCTIONAL

EQUATION (1.12)

Theorem 4.1. Let H : X×X → R be a mapping for which there exists a function
ϕ : X ×X ×X ×X → R with the condition

(4.1) lim
n→∞

2nϕ(2−nx, 2−nx, 2−ny, 2−ny) = 0

such that the functional inequality

(4.2)
∣∣∣H( xu

x + u
,
yv

y + v

)
−

H(x,y)H(u,v)

H(x,y) + H(u,v)

∣∣∣ ≤ ϕ(x,u,y,v)
holds for all x,u,y,v ∈ X. Then there exists a unique harmonic mean mapping
M : X ×X → R satisfying the functional equation (1.12) and

(4.3) |M(x,y) −H(x,y)| ≤
∞∑
i=0

2i+1ϕ(2−ix, 2−ix, 2−iy, 2−iy)

for all x,y ∈ X. The mapping M(x,y) is defined by

M(x,y) = lim
n→∞

2nH(2−nx, 2−ny)

for all x,y ∈ X.



8 K. RAVI, J.M. RASSIAS AND B.V. SENTHIL KUMAR

Proof. Replacing (x,u,y,v) by (x,x,y,y) in (4.2) and multiplying by 2, we obtain

(4.4)
∣∣∣2H(x

2
,
y

2

)
−H(x,y)

∣∣∣ ≤ 2ϕ(x,x,y,y).
Now, replacing (x,y) by (x

2
, y
2
) in (4.4), multiplying by 2 and summing the resulting

inequality with (4.4), we arrive

|22H
(
2−2x, 2−2y

)
−H(x,y)| ≤ 2

1∑
i=0

2iϕ(2−ix, 2−ix, 2−iy, 2−iy).

Using induction on a positive integer n, we get

|2nH
(
2−nx, 2−ny

)
−H(x,y)| ≤ 2

n−1∑
i=0

2iϕ(2−ix, 2−ix, 2−iy, 2−iy)

≤ 2
∞∑
i=0

2iϕ(2−ix, 2−ix, 2−iy, 2−iy)(4.5)

for all x,y ∈ X. In order to prove the convergence of the sequence {2nH(2−nx, 2−ny)},
replacing (x,y) by (2−px, 2−py) in (4.5) and multiplying by 2p, we find that for

n > p > 0

|2n+pH(2−n−px, 2−n−py) − 2pH(2−px, 2−py)|

= 2p|2nH(2−n−px, 2−n−py) −H(2−px, 2−py)|

≤ 2
∞∑
i=0

2p+iϕ(2−p−ix, 2−p−ix, 2−p−iy, 2−p−iy)

→ 0 as p →∞.

This shows that the sequence {2nH(2−nx, 2−ny)} is a Cauchy sequence in (R, |.|).
Since (R, |.|) is complete, the sequence {2nH(2−nx, 2−ny)} converges for all x,y ∈
X. Thus, we can define a function M : X×X → R by M(x,y) = lim

n→∞
2nH(2−nx, 2−ny).

Allow n → ∞ in (4.5), we arrive (4.3). To show that M satisfies (1.12), replacing
(x,u,y,v) by (2−nx, 2−nu, 2−ny, 2−nv) in (4.2) and multiplying by 2n, we obtain

2n
∣∣∣H( 2−nx2−nu

2−nx + 2−nu
,

2−ny2−nv

2−ny + 2−nv

)
−

H(2−nx, 2−ny)H(2−nu, 2−nv)

H(2−nx, 2−ny) + H(2−nu, 2−nv)

∣∣∣
= 2n

∣∣∣H(2−n( xu
x + u

)
, 2−n

( yv
y + v

))
−

H(2−nx, 2−ny)H(2−nu, 2−nv)

H(2−nx, 2−ny) + H(2−nu, 2−nv)

∣∣∣
≤ 2nϕ(2−nx, 2−nu, 2−ny, 2−nv).

(4.6)



GENERALIZED ULAM-HYERS STABILITY 9

Allow n →∞ in (4.6), we see that M satisfies (1.12) for all x,u,y,v ∈ X. To prove
M is unique mapping satisfying (1.12), let N : X × X → R be another mapping
which satisfies (1.12) and the inequality (4.3). Clearly N and M satisfy (1.12) and

using (4.3), we arrive

|N(x,y) −M(x,y)|

= 2n|N(2−nx, 2−ny) −M(2−nx, 2−ny)|

≤ 2n
(
|N(2−nx, 2−ny) −H(2−nx, 2−ny)| + |H(2−nx, 2−ny) −M(2−nx, 2−ny)|

)

≤ 4
∞∑
i=0

2n+iϕ(2−n−ix, 2−n−ix, 2−n−iy, 2−n−iy)

(4.7)

for all x,y ∈ X. Allow n → ∞ in (4.7) and using (4.1), we find that M is unique.
This completes the proof of Theorem 4.1. �

Theorem 4.2. Let H : X×X → R be a mapping satisfying (4.2) for all x,u,y,v ∈
X. Then there exists a unique harmonic mapping M : X ×X → R satisfying the
functional equation (1.12) and

(4.8) |H(x,y) −M(x,y)| ≤
∞∑
i=0

2−iϕ(2i+1x, 2i+1x, 2i+1y, 2i+1y)

for all x,y ∈ X. The mapping M(x,y) is defined by

M(x,y) = lim
n→∞

2−nH(2nx, 2ny)

for all x,y ∈ X, where ϕ : X ×X ×X ×X → R is a function.

Proof. The proof is obtained by replacing (x,u,y,v) by (x,x,y,y) in (4.2) and

proceeding further by similar arguments as in Theorem 4.1. �

Corollary 4.3. Let k1 ≥ 0 be fixed and a > 1 or a < 1. If a mapping H : X×X →
R satisfies the inequality∣∣∣H( xu

x + u
,
yv

y + v

)
−

H(x,y)H(u,v)

H(x,y) + H(u,v)

∣∣∣ ≤ k1(xa + ua + ya + va)
for all x,u,y,v ∈ X, then there exists a unique harmonic mean mapping M :
X ×X → R satisfying the functional equation (1.12) and

|M(x,y) −H(x,y)| ≤




4k1

(
2a

2a−2

)(
xa + ya

)
for a > 1

4k1

(
2a

2−2a

)(
xa + ya

)
for a < 1



10 K. RAVI, J.M. RASSIAS AND B.V. SENTHIL KUMAR

for all x,y ∈ X.

Proof. Choosing ϕ(x,u,y,v) = k1

(
xa + ua + ya + va

)
, for all x,u,y,v ∈ X in

Theorem 4.1, we arrive

|M(x,y) −H(x,y)| ≤ 4k1
( 2a

2a − 2

)(
xa + ya

)
, for all x,y ∈ X and a > 1

and using Theorem 4.2, we arrive

|M(x,y) −H(x,y)| ≤ 4k1
( 2a

2 − 2a
)(
xa + ya

)
, for all x,y ∈ X and a < 1.

�

Corollary 4.4. Let H : X × X → R be a mapping and there exist real numbers
p,q : σ = p + q > 1

2
or p,q : σ = p + q < 1

2
. If there exists k2 such that∣∣∣H( xu

x + u
,
yv

y + v

)
−

H(x,y)H(u,v)

H(x,y) + H(u,v)

∣∣∣ ≤ k2xpupyqvq
for all x,u,y,v ∈ X, then there exists a unique harmonic mean mapping M :
X ×X → R satisfying the functional equation (1.12) and

|M(x,y) −H(x,y)| ≤




2k2

(
22σ

22σ−2

)
x2py2q for σ > 1

2

2k2

(
22σ

2−22σ

)
x2py2q for σ < 1

2

for all x,y ∈ X.

Proof. Taking ϕ(x,u,y,v) = k2x
pupyqvq, for all x,u,y,v ∈ X in Theorem 4.1, we

arrive

|M(x,y) −H(x,y)| ≤ 2k2
( 22σ

22σ − 2

)
x2py2q, for all x,y ∈ X and σ >

1

2

and using Theorem 4.2, we arrive

|M(x,y) −H(x,y)| ≤ 2k2
( 22σ

2 − 22σ
)
x2py2q, for all x,y ∈ X and σ <

1

2
.

�

Corollary 4.5. Let k3 > 0 and r >
1
4

or r < 1
4

be real numbers, and H : X×X → R
be a mapping satisfying the functional inequality∣∣∣H( xu

x + u
,
yv

y + v

)
−

H(x,y)H(u,v)

H(x,y) + H(u,v)

∣∣∣
≤ k3

(
xruryrvr +

(
x4r + u4r + y4r + v4r

))



GENERALIZED ULAM-HYERS STABILITY 11

for all x,u,y,v ∈ X. Then there exists a unique harmonic mean mapping M :
X ×X → R satisfying the functional equation (1.12) and

|M(x,y) −H(x,y)| ≤




2k3

(
24r

24r−2

)[
x2ry2r + 2

(
x4r + y4r

)]
for r > 1

4

2k3

(
24r

2−24r

)[
x2ry2r + 2

(
x4r + y4r

)]
for r < 1

4

for all x,y ∈ X.

Proof. Choosing ϕ(x,u,y,v) = k3

(
xruryrvr +

(
x4r + u4r + y4r + v4r

))
, for all

x,u,y,v ∈ X in Theorem 4.1, we arrive

|M(x,y) −H(x,y)|

≤ 2k3
( 24r

24r − 2

)[
x2ry2r + 2

(
x4r + y4r

)]
for all x,y ∈ X and r >

1

4

and using Theorem 4.2, we arrive

|M(x,y) −H(x,y)|

≤ 2k3
( 24r

2 − 24r
)[
x2ry2r + 2

(
x4r + y4r

)]
for all x,y ∈ X and r <

1

4
.

�

Corollary 4.6. Let k4 > 0 and p >
1
2

or p < 1
2

be real numbers, and H : X×X → R
be a mapping satisfying the functional inequality∣∣∣∣H

(
xu

x + u
,
yv

y + v

)
−

H(x,y)H(u,v)

H(x,y) + H(u,v)

∣∣∣∣ ≤ k4
(

xpupypvp

xpup + ypvp

)
for all x,u,y,v ∈ X. Then there exists a unique harmonic mean mapping M :
X ×X → R satisfying the functional equation (1.12) and

|M(x,y) −H(x,y)| ≤




2k4

(
22p

22p−2

)(
x2py2p

x2p+y2p

)
for p > 1

2

2k4

(
22p

2−22p

)(
x2py2p

x|2p+y2p

)
for p < 1

2

for all x,y ∈ X.

Proof. Choosing ϕ(x,u,y,v) = k4

(
xpupypvp

xpup+ypvp

)
, for all x,u,y,v ∈ X in Theorem

4.1, we arrive

|M(x,y) −H(x,y)| ≤ 2k4
(

22p

22p − 2

)(
x2py2p

x2p + y2p

)
for all x,y ∈ X and p >

1

2

and using Theorem 4.2, we arrive

|M(x,y) −H(x,y)| ≤ 2k4
(

22p

2 − 22p

)(
x2py2p

x2p + y2p

)
for all x,y ∈ X and p <

1

2
.



12 K. RAVI, J.M. RASSIAS AND B.V. SENTHIL KUMAR

�

5. COUNTER-EXAMPLES

Now, we give an example to show that the functional equation (1.12) is not stable

for p = 1
2

in Corollary 4.6.

Let us define a function H : X ×X → R be

H(x,y) =

∞∑
n=0

φ(2nx, 2ny)

2n
,

for all x,y ∈ X where the function φ : X ×X → R is given by

φ(x,y) =



αxy
x+y

for x,y ∈ (0, 1)

α otherwise

with α > 0 is a constant. Then the function H satisfies the inequality

(5.1)

∣∣∣∣H
(

xu

x + u
,
yv

y + v

)
−

H(x,y)H(u,v)

H(x,y) + H(u,v)

∣∣∣∣ ≤ 6α
(

x
1
2 u

1
2 y

1
2 v

1
2

x
1
2 u

1
2 + y

1
2 v

1
2

)
for all x,u,y,v ∈ X. Then there does not exist a harmonic mapping M : X×X → R

and a constant λ > 0 such that

(5.2) |H(x,y) −M(x,y)| ≤ λ
(

xy

x + y

)
for all x,y ∈ X.

Proof. |H(x,y)| ≤
∑∞
n=0

|φ(2nx,2ny)|
|2n| ≤

∑∞
n=0

α
2n

= α
(
1 − 1

2

)−1
= 2α. Hence H is

bounded by 2α. If x
1
2 u

1
2 y

1
2 v

1
2

x
1
2 u

1
2 +y

1
2 v

1
2
≥ 1, then the left hand side of the inequality (5.1)

is less than 6α. Now, consider the case: 0 < x
1
2 u

1
2 y

1
2 v

1
2

x
1
2 u

1
2 +y

1
2 v

1
2
< 1. Then there exists a

k ∈ N such that

(5.3)
1

2k+1
<

x
1
2 u

1
2 y

1
2 v

1
2

x
1
2 u

1
2 + y

1
2 v

1
2

<
1

2k
.

Hence x
1
2 u

1
2 y

1
2 v

1
2

x
1
2 u

1
2 +y

1
2 v

1
2
< 1

2k
implies

2kx
1
2 u

1
2 y

1
2 v

1
2

x
1
2 u

1
2 + y

1
2 v

1
2

< 1

or 2
k
2 x

1
2 2

k
2 u

1
2 2

k
2 y

1
2 2

k
2 v

1
2

2
k
2 x

1
2 2

k
2 u

1
2 +2

k
2 y

1
2 2

k
2 v

1
2
< 1

or 2
k
2 x

1
2 2

k
2 u

1
2 2

k
2 y

1
2 2

k
2 v

1
2 < 2

k
2 x

1
2 2

k
2 u

1
2 + 2

k
2 y

1
2 2

k
2 v

1
2



GENERALIZED ULAM-HYERS STABILITY 13

or 1 < 2
k
2 x

1
2 2

k
2 u

1
2 +2

k
2 y

1
2 2

k
2 v

1
2

2
k
2 x

1
2 2

k
2 u

1
2 2

k
2 y

1
2 2

k
2 v

1
2

or 1 < 1
2
k
2 x

1
2

1

2
k
2 u

1
2

+ 1
2
k
2 y

1
2

1

2
k
2 v

1
2
.

Therefore,
1

2
k
2 x

1
2

1

2
k
2 u

1
2

+
1

2
k
2 y

1
2

1

2
k
2 v

1
2

> 1.

Hence
1

2
k
2 x

1
2

> 1,
1

2
k
2 u

1
2

> 1,
1

2
k
2 y

1
2

> 1,
1

2
k
2 v

1
2

> 1

which implies

2
k
2 x

1
2 < 1, 2

k
2 u

1
2 < 1, 2

k
2 y

1
2 < 1, 2

k
2 v

1
2 < 1

so that

2
k−1
2 x

1
2 <

1
√

2
, 2

k−1
2 u

1
2 <

1
√

2
, 2

k−1
2 y

1
2 <

1
√

2
, 2

k−1
2 v

1
2 <

1
√

2
,

2
k−1
2

(
xu

x + u

)1
2

<
1
√

2
, 2

k−1
2

(
yv

y + v

)1
2

<
1
√

2

and consequently

2k−1(x), 2k−1(u), 2k−1(y), 2k−1(v), 2k−1
(

xu

x + u

)
, 2k−1

(
yv

y + v

)
∈ (0, 1).

Therefore, for each n = 0, 1, 2, . . . ,k − 1, we have

2n(x), 2n(u), 2n(y), 2n(v), 2n
(

xu

x + u

)
, 2n

(
yv

y + v

)
∈ (0, 1)

and

φ

(
2n
(

xu

x + u

)
, 2n

(
yv

y + v

))
−

φ (2n(x), 2n(y)) φ (2n(u), 2n(v))

φ (2n(x), 2n(y)) + φ (2n(u), 2n(v))
= 0

for n = 0, 1, 2, . . . ,k − 1. Using (5.3) and definition of H, we obtain∣∣∣∣H
(

xu

x + u
,
yv

y + v

)
−

H(x,y)H(u,v)

H(x,y) + H(u,v)

∣∣∣∣
≤
∞∑
n=0

1

2n

∣∣∣∣φ
(

2n
(

xu

x + u

)
, 2n

(
yv

y + v

))
−

φ (2n(x), 2n(y)) φ (2n(u), 2n(v))

φ (2n(x), 2n(y)) + φ (2n(u), 2n(v))

∣∣∣∣
≤
∞∑
n=k

1

2n

∣∣∣∣φ
(

2n
(

xu

x + u

)
, 2n

(
yv

y + v

))
−

φ (2n(x), 2n(y)) φ (2n(u), 2n(v))

φ (2n(x), 2n(y)) + φ (2n(u), 2n(v))

∣∣∣∣
≤
∞∑
n=k

1

2n
3

2
α

≤
6α

2k+1
≤ 6α

(
x

1
2 u

1
2 y

1
2 v

1
2

x
1
2 u

1
2 + y

1
2 v

1
2

)
,



14 K. RAVI, J.M. RASSIAS AND B.V. SENTHIL KUMAR

that is the inequality (5.1) holds true.

We claim that the harmonic mean functional equation (1.12) is not stable for

p = 1
2

in Corollary 4.6.

Assume that there exists a harmonic mean mapping M : X ×X → R satisfying
(5.2). Therefore, we have

(5.4) |H(x,y)| ≤ (λ + 2)
(

xy

x + y

)
.

But we can choose a positive integer m with mα > λ + 2. If x ∈
(
0, 21−m

)
, then

2nx ∈ (0, 1) for all n = 0, 1, 2, . . . ,m− 1. For this x, we get

H(x,y) =

∞∑
n=0

φ(2nx, 2ny)

2n

≥
m−1∑
n=0

α
(

2nxy
xy

)
2n

= mα

(
xy

x + y

)
> (λ + 2)

(
xy

x + y

)
which in comparison with (5.4) is a contradiction. Therefore, the harmonic mean

functional equation (1.12) is not stable if p = 1
2

in Corollary 4.6. �

References

[1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ.

Press, 1989.

[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math.Soc. Japan,

2(1950), 64-66.

[3] J.H. Bae and K.W. Jun, On the generalized Hyers-Ulam-Rassias stability of quadratic func-

tional equation, Bull. Koeran. Math. Soc. 38(2001), 325-336.

[4] B. Bouikhalene and E. Elqorachi, Ulam-Gavruta-Rassias stability of the Pexider functional

equation, Int. J. Math. Stat. 7(2007), 27-39.

[5] H.X. Cao, J.R. Lv and J.M. Rassias, Superstability for generalized module left derivations

and generalized module derivations on a Banach module(I), J. Ineq. and Appl. Vol.2009,

Article ID 718020, 1-10.

[6] H.X. Cao, J.R. Lv and J.M. Rassias, Superstability for generalized module left derivation-

s and generalized module derivations on a Banach module(II), J. Pure & Appl. Math.

10(2)(2009), 1-8.

[7] I.S. Chang and H.M. Kim, On the Hyers-Ulam stability of quadratic functional equations,

J. Ineq. Appl. Math. 33(2002), 1-12.

[8] I.S. Chang and Y.S. Jung, Stability of functional equations deriving from cubic and quadratic

functions, J. Math. Anal. Appl. 283(2003), 491-500.

[9] Ch.Park, J.S. An and J. Cui, Isomorphisms and Derivations in Lie C*-Algebras, Abstract

and Applied Analysis, Vol.2007, Article ID 85737, 14 pages.



GENERALIZED ULAM-HYERS STABILITY 15

[10] Ch.Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull.

Sci. Math. 132(2) (2008), 87-96.

[11] C.G. Park and J.M. Rassias, Hyers-Ulam stability of an Euler-Lagrange type additive map-

ping, Int. J. Math. Stat. 7(2007), 112-125.

[12] S. Czerwik, On the stability of the quadratic mappings in normed spaces, Abh. Math. Sem.

Univ. Hamburg. 62 (1992), 59-64.

[13] M. Eshaghi Gordji, A. Ebadian and S. Zolfaghari, Stability of a functional equation deriving

from cubic and quartic functions, Abstract and Applied Analysis, Volume 2008, Article ID

801904, 17 pages.

[14] M. Eshaghi Gordji, S. Kaboli and S. Zolfaghari, Stability of a mixed type quadratic, cubic

and quartic functional equations, arXiv:0812.2939v1 Math FA, 15 Dec 2008.

[15] M. Eshaghi Gordji and M. B. Savadkouhi, Stability of cubic and quartic functional equation

in non-Archimedean spaces, Acta Appl. Math. DOI 10.1007/s10440-009-9512-7, 2009.

[16] M. Eshaghi Gordji and H. Khodaei, Solution and stability of generalized mixed type cu-

bic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Analysis,

71(2009), 5629-5643.

[17] M. Eshaghi Gordji, H. Khodaei and Th.M. Rassias, On the Hyers-Ulam-Rassias stability

of a generalized mixed type quartic, cubic, quadratic and additive functional equations in

quasic-Banach spaces, arXiv:0903.0834v2 Math FA, 24 Apr 2009.

[18] M. Eshaghi Gordji, S. Zolfaghari, J.M. Rassias and M.B. Savadkouhi, Solution and stability

of a mixed type cubic and quartic functional equation in quasi-Banach spaces, Abstract and

Applied Analysis, Volume 2009, Article ID 417473, 1-14.

[19] M. Eshaghi Gordji and H. Khodaei, On the generalized Hyers-Ulam-Rassias stability of

quadratic functional equations, Abstract and Applied Analysis, Vol.2009, Article ID 923476,

1-11.

[20] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A.

27(1941), 222-224.

[21] K.W. Jun and H.M. Kim, On the Hyers-Ulam-Rassias stabillity of a general cubic functional

equation, Math. Ineq. Appl. Vol.6(2) (2003), 289-302.

[22] K.W. Jun and H.M. Kim, Ulam stabillity problem for a mixed type of cubic and additive

functional equation, Bull. Belg. Math. Soc. 13(2006), 271-285.

[23] K.W. Jun and H.M. Kim, On the stabillity of Euler-Lagrange type cubic mappings in quasi-

Banach spaces, J. Math. Anal. Appl. 332(2007), 1335-1350.

[24] S.M. Jung, On the Hyers-Ulam -Rassias stability of a quadratic functional equation, J.

Math. Anal. Appl. 232(1999), 384-393.

[25] D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in

random normed spaces, J. Math. Anal. Appl. 343(2008), 567-572.

[26] F. Moradlou, H. Vaezi and G.Z. Eskandani, Hyers-Ulam-Rassias stability of a quadratic and

additive functional equation in quasi-Banach spaces, Mediterr. J. Math. 6(2009), 233-248.

[27] A. Najati, Hyers-Ulam-Rassias stability of a cubic functional equation, Bull. Korean Math.

Soc. 44(4) (2007), 825-840.



16 K. RAVI, J.M. RASSIAS AND B.V. SENTHIL KUMAR

[28] A. Najati and M.B. Moghimi, On the stability of a quadratic and additive functional equa-

tion, J. Math. Anal. Appl. 337(2008), 399-415.

[29] A. Najati and Ch. Park, On the stability of a cubic functional equation, Acta Mathematica

Sinica, English series, Vol.24(12) (2008), 1953-1964.

[30] P. Nakmahachalasint, Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities of an Ad-

ditive Functional Equation in Several variables, Internat. J. Math. and Math. Sc. (IJMMS)

Article ID 13437, 6 pages, 2007.

[31] P. Nakmahachalasint, On the generalized Ulam-Gavruta-Rassias stability of mixed-type lin-

ear and Euler-Lagrange-Rassias functional equations, Internat. J. Math. and Math. Sc.

(IJMMS) Article ID 63239, 10 pages, 2007.

[32] M. Petapirak and P. Nakmahachalasint, A quartic functional equation and its generalized

Hyers-Ulam-Rassias stability, Thai Journal of Mathematics, Special issue(Annual Meeting

in Mathematics, 2008), 77-84.

[33] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J.

Funct. Anal. 46(1982), 126-130.

[34] J.M. Rassias, Hyers-Ulam stability of the quadratic functional equation in several variables,

J. Ind. Math. Soc. 68(1992), 65-73.

[35] J.M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnic Mathe-

maticki, Vol.34(54) (1999), 243-252.

[36] J.M. Rassias, K. Ravi, M. Arunkumar and B.V. Senthil Kumar, Solution and Ulam stability

of mixed type cubic and additive Functional Equation, Functional equations, Difference

Inequalities and Ulam stability Notions, Nova Science Publishers, Chapter 13, (2010), 149-

175.

[37] K. Ravi and M. Arunkumar, On the Ulam-Gavruta-Rassias stability of the orthogonally

Euler-Lagrange type functional equation, Internat. J. Appl. Math. Stat.7(2007), 143-156.

[38] K. Ravi, M. Arunkumar and J.M. Rassias, Ulam stability for the orthogonally general Euler-

Lagrange type functional equation, Int. J. Math. Stat. 3(2008), A08, 36-46.

[39] K. Ravi and B.V. Senthil Kumar, Ulam-Gavruta-Rassias stability of Rassias Reciprocal

functional equation, Global J. of Appl. Math. and Math. Sci. 3(No.1-2),(Jan-Dec 2010),

57-79.

[40] K. Ravi, J.M. Rassias and B.V. Senthil Kumar, Ulam stability of Reciprocal Difference and

Adjoint Funtional Equations, The Australian J. Math. Anal.Appl. 8(1), Art.13 (2011), 1-18.

[41] K. Ravi, J.M. Rassias M. Arunkumar and R. Kodandan, Stability of a generalized mixed

type additive, quadratic, cubic and quartic functional equation, J. of Ineq. Pure & Appl.

Math. 10(4) (2009), 1-29.

[42] K. Ravi and B.V. Senthil Kumar, Stability of mixed type cubic and additive functional

equation in fuzzy normed spaces, Int. J. Math. Sci. Engg. Appl.4(I) (March 2010), 159-172.

[43] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math.

Soc. 72(1978), 297-300.

[44] R. Saadati, S.M. Vaezpour and Y.J. Cho, A note to paper “On the stability of cubic mappings

and quartic mappings in random normed spaces”, J. of Ineq. & Appl. Volume 2009, Article

ID 214530, 6 pages.



GENERALIZED ULAM-HYERS STABILITY 17

[45] M.B. Savadkouhi, M. Eshaghi Gordji, J.M. Rassias and N. Ghobadipour, Approximate

ternary Jordan derivations on Banach ternary algebras, J. Math. Phys. 50042303 (2009),

1-9.

[46] M.A. Sibaha, B. Bouikhalene and E. Elqorachi, Ulam-Gavruta-Rassias stability of a linear

functional equation, Internat. J.Appl. Math. Stat. 7(2007), 157-166.

[47] S.M. Ulam, Problems in Modern Mathematics, Chapter VI, Wiley-Interscience, New York,

1964.

1PG & Research Department of Mathematics, Sacred Heart College, Tirupattur -

635 601, TamilNadu, India

2Pedagogical Department E.E., Section of Mathematics and Informatics, National

and Capodistrian University of Athens, 4, Agamemnonos Str., Aghia Paraskevi, Athens,

Attikis 15342, GREECE

3Department of Mathematics, C. Abdul Hakeem College of Engg. and Tech., elvisharam

- 632 509, TamilNadu, India

∗Corresponding author