International Journal of Analysis and Applications
ISSN 2291-8639
Volume 13, Number 1 (2017), 64-69
http://www.etamaths.com

ON THE GENERALIZED OSTROWSKI TYPE INTEGRAL INEQUALITY FOR

DOUBLE INTEGRALS

MUSTAFA KEMAL YILDIZ1,∗ AND MEHMET ZEKI SARIKAYA2

Abstract. In this paper, we establish a new generalized Ostrowski type inequality for double inte-

grals involving functions of two independent variables by using fairly elementary analysis.

1. Introduction

In 1938, the classical integral inequality was established by Ostrowski [5] as follows:

Theorem 1.1. Let f : [a, b]→ R be a differentiable mapping on (a, b) whose derivative f
′

: (a, b)→ R
is bounded on (a, b), i.e., ‖f′‖∞ = sup

t∈(a,b)
|f′(t)| < ∞. Then, the inequality holds:

∣∣∣∣∣∣f(x) − 1b−a
b∫

a

f(t)dt

∣∣∣∣∣∣ ≤
[

1

4
+

(
x− a+b

2

)2
(b−a)2

]
(b−a)‖f′‖∞ (1.1)

for all x ∈ [a, b]. The constant 1
4

is the best possible.

In a recent paper [3], Barnett and Dragomir proved the following Ostrowski type inequality for
double integrals:

Theorem 1.2. Let f : [a, b]×[c, d]→ R be continuous on [a, b]×[c, d], f′′x,y =
∂2f
∂x∂y

exists on (a, b)×(c, d)
and is bounded, i.e., ∥∥f′′x,y∥∥∞ = sup

(x,y)∈(a,b)×(c,d)

∣∣∣∣∂2f(x, y)∂x∂y
∣∣∣∣ < ∞.

Then, we have the inequality:∣∣∣∣∣∣
b∫

a

d∫
c

f(s, t)dtds− (d− c)(b−a)f(x, y)

−


(b−a) d∫

c

f(x, t)dt + (d− c)
b∫

a

f(s, y)ds



∣∣∣∣∣∣ (1.2)

≤
[

1

4
(b−a)2 + (x−

a + b

2
)2
][

1

4
(d− c)2 + (y −

d + c

2
)2
]∥∥f′′x,y∥∥∞

for all (x, y) ∈ [a, b] × [c, d].

In [3], the inequality (1.2) is established by the use of integral identity involving Peano kernels.
In [7], Pachpatte obtained an inequality in the view (1.2) by using elementary analysis. The interested
reader is also refered to ( [3], [4], [6]- [13]) for Ostrowski type inequalities in several independent
variables and for recent weighted version of these type inequalities see [1], [2], [9] and [11].

Received 22nd July, 2016; accepted 19th September, 2016; published 3rd January, 2017.

2010 Mathematics Subject Classification. 26D07, 26D15.

Key words and phrases. integral inequality; Ostrowski’s inequality.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

64



ON THE GENERALIZED OSTROWSKI TYPE INTEGRAL INEQUALITY 65

Meanwhile, in [11] Sarikaya and Ogunmez gave the following interesting identity and by using this
indentity they establised some interesting integral inequalities:

Lemma 1.1. Let f : [a, b] × [c, d]→ R be an absolutely continuous function such that the partial
derivative of order

∂2f(t,s)
∂t∂s

exists for all (t, s) ∈ [a, b]× [c, d] and the weight function w : [a, b] → [0,∞)
is integrable, nonnegative and

m(a, b) =

b∫
a

w(t)dt < ∞. (1.3)

Then, we have

f(x, y) =
1

m(a, b)

b∫
a

w(t)f(t, y)dt +
1

m(c, d)

d∫
c

w(s)f(x, s)ds

−
1

m(a, b)m(c, d)


 b∫

a

d∫
c

w(t)w(s)f(t, s)dsdt−
b∫

a

d∫
c

p(x, t)q(y, s)
∂2f(t, s)

∂t∂s
dsdt


 (1.4)

where

p(x, t) =




p1(a, t) =
t∫
a

w(u)du, a ≤ t < x

p2(b, t) =
t∫
b

w(u)du, x ≤ t ≤ b

and

q(y, s) =




q1(c, s) =
s∫
c

w(v)dv, c ≤ s < y

q2(d, s) =
s∫
d

w(v)dv, y ≤ s ≤ d.

The main aim of this paper is to establish a new generalized Ostrowski type inequality for double
integrals involving functions of two independent variables and their partial derivatives.

2. Main Result

We begin with the following important result:

Lemma 2.1. Let f : [a, b] × [c, d]→ R be an absolutely continuous function such that the partial
derivative of order

∂2f(t,s)
∂t∂s

exists for all (t, s) ∈ [a, b]× [c, d], and the function p : [a, b]× [c, d] → [0,∞)
is integrable. Then, we have

 b∫
a

d∫
c

p(u, v)dvdu


f(x, y) − b∫

a

d∫
c

p(t, v)f(t, y)dvdt (2.1)

−
b∫

a

d∫
c

p(u, s)f(x, s)dsdu +

b∫
a

d∫
c

p(t, s)f(t, s)dsdt

=

b∫
a

d∫
c

P (x, t; y, s)
∂2f(t, s)

∂t∂s
dsdt



66 YILDIZ AND SARIKAYA

where

P (x, t; y, s) =




t∫
a

s∫
c

p(u, v)dvdu, a ≤ t < x, c ≤ s < y
t∫
a

s∫
d

p(u, v)dvdu, a ≤ t < x, y ≤ s ≤ d
t∫
b

s∫
c

p(u, v)dvdu, x ≤ t ≤ b, c ≤ s < y
t∫
b

s∫
d

p(u, v)dvdu, x ≤ t ≤ b, y ≤ s ≤ d.

Proof. By definitions of P (x, t; y, s), we have

b∫
a

d∫
c

P (x, t; y, s)
∂2f(t, s)

∂t∂s
dsdt =

x∫
a

y∫
c

[
t∫
a

s∫
c

p(u, v)dvdu

]
∂2f(t, s)

∂t∂s
dsdt

+
x∫
a

d∫
y

[
t∫
a

s∫
d

p(u, v)dvdu

]
∂2f(t, s)

∂t∂s
dsdt +

b∫
x

y∫
c

[
t∫
b

s∫
c

p(u, v)dvdu

]
∂2f(t, s)

∂t∂s
dsdt

+
b∫
x

d∫
y

[
t∫
b

s∫
d

p(u, v)dvdu

]
∂2f(t, s)

∂t∂s
dsdt.

Integrating by parts, we can state:

x∫
a

y∫
c

[
t∫
a

s∫
c

p(u, v)dvdu

]
∂2f(t, s)

∂t∂s
dsdt

=
x∫
a

[(
t∫
a

y∫
c

p(u, v)dvdu

)
∂f(t, y)

∂t
−

y∫
c

(
t∫
a

p(u, s)du

)
∂f(t, s)

∂t
ds

]
dt

=

(
x∫
a

y∫
c

p(u, v)dvdu

)
f(x, y) −

x∫
a

(
y∫
c

p(t, v)dv

)
f(t, y)dt

−
y∫
c

(
x∫
a

p(u, s)du

)
f(x, s)ds +

x∫
a

y∫
c

p(t, s)f(t, s)dsdt,

(2.2)

x∫
a

d∫
y

[
t∫
a

s∫
d

p(u, v)dvdu

]
∂2f(t, s)

∂t∂s
dsdt

= −
x∫
a

[(
t∫
a

y∫
d

p(u, v)dvdu

)
∂f(t, y)

∂t
+

d∫
y

(
t∫
a

p(u, s)du

)
∂f(t, s)

∂t
ds

]
dt

=

(
x∫
a

d∫
y

p(u, v)dvdu

)
f(x, y) −

x∫
a

(
d∫
y

p(t, v)dv

)
f(t, y)dt

−
d∫
y

(
x∫
a

p(u, s)du

)
f(x, s)ds +

x∫
a

d∫
y

p(t, s)f(t, s)dsdt,

(2.3)

b∫
x

y∫
c

[
t∫
b

s∫
c

p(u, v)dvdu

]
∂2f(t, s)

∂t∂s
dsdt

= −
b∫
x

[(
b∫
t

y∫
c

p(u, v)dvdu

)
∂f(t, y)

∂t
−

y∫
c

(
b∫
t

p(u, s)du

)
∂f(t, s)

∂t
ds

]
dt

=

(
b∫
x

y∫
c

p(u, v)dvdu

)
f(x, y) −

b∫
x

(
y∫
c

p(t, v)dv

)
f(t, y)dt

−
y∫
c

(
b∫
x

p(u, s)du

)
f(x, s)ds +

b∫
x

y∫
c

p(t, s)f(t, s)dsdt,

(2.4)



ON THE GENERALIZED OSTROWSKI TYPE INTEGRAL INEQUALITY 67

b∫
x

d∫
y

[
t∫
b

s∫
d

p(u, v)dvdu

]
∂2f(t, s)

∂t∂s
dsdt

=
b∫
x

[(
b∫
t

d∫
y

p(u, v)dvdu

)
∂f(t, y)

∂t
+

d∫
y

(
b∫
t

p(u, s)du

)
∂f(t, s)

∂t
ds

]
dt

=

(
b∫
x

d∫
y

p(u, v)dvdu

)
f(x, y) −

b∫
x

(
d∫
y

p(t, v)dv

)
f(t, y)dt

−
d∫
y

(
b∫
x

p(u, s)du

)
f(x, s)ds +

b∫
x

d∫
y

p(t, s)f(t, s)dsdt.

(2.5)

Adding (2.2)-(2.5) and rewriting, we easily deduce required identity (2.1) which completes the proof.
�

Remark 2.1. If take p(., .) ≡ 1 in Lemma 2.1, we get

f(x, y) −
1

(b−a)

b∫
a

f(t, y)dt

−
1

(d− c)

d∫
c

f(x, s)dsdu +
1

(b−a) (d− c)

b∫
a

d∫
c

f(t, s)dsdt

=
1

(b−a) (d− c)

b∫
a

d∫
c

P (x, t; y, s)
∂2f(t, s)

∂t∂s
dsdt

where

P (x, t; y, s) =




(t−a) (s− c) , a ≤ t < x, c ≤ s < y
(t−a) (s−d) , a ≤ t < x, y ≤ s ≤ d
(t− b) (s− c) , x ≤ t ≤ b, c ≤ s < y
(t− b) (s−d) , x ≤ t ≤ b, y ≤ s ≤ d.

which is given by Barnett and Dragomir in [3].

Remark 2.2. If take p(u, v) = w(u)w(v) in Lemma 2.1, then the Lemma 2.1 reduces to the Lemma
1.1 which is proved by Sarikaya and Ogunmez in [11].

Theorem 2.1. Let f : [a, b] × [c, d]→ R be an absolutely continuous function such that the partial
derivative of order

∂2f(t,s)
∂t∂s

exists and is bounded, i.e.,∥∥∥∥∂2f(t, s)∂t∂s
∥∥∥∥
∞

= sup
(t,s)∈(a,b)×(c,d)

∣∣∣∣∂2f(t, s)∂t∂s
∣∣∣∣ < ∞

for all (t, s) ∈ [a, b] × [c, d], the function p : [a, b] × [c, d] → [0,∞) is integrable. Then, we have∣∣∣∣∣∣

 b∫

a

d∫
c

p(u, v)dvdu


f(x, y) − b∫

a

d∫
c

p(t, v)f(t, y)dvdt

−
b∫

a

d∫
c

p(u, s)f(x, s)dsdu +

b∫
a

d∫
c

p(t, s)f(t, s)dsdt

∣∣∣∣∣∣ (2.6)
≤

∥∥∥∥∂2f(t, s)∂t∂s
∥∥∥∥
∞

x∫
a

(x−u)A(u, y)du +
b∫

x

(u−x)A(u, y)du

where

A(u, y) =

y∫
c

(y −v) |p(u, v)|dv +
d∫

y

(v −y) |p(u, v)|dv.



68 YILDIZ AND SARIKAYA

Proof. From Lemma 2.1 and using the properties of modulus, we observe that∣∣∣∣∣∣

 b∫

a

d∫
c

p(u, v)dvdu


f(x, y) − b∫

a

d∫
c

p(t, v)f(t, y)dvdt

−
b∫

a

d∫
c

p(u, s)f(x, s)dsdu +

b∫
a

d∫
c

p(t, s)f(t, s)dsdt

∣∣∣∣∣∣
≤

b∫
a

d∫
c

|P (x, t; y, s)|
∣∣∣∣∂2f(t, s)∂t∂s

∣∣∣∣dsdt (2.7)
≤

∥∥∥∥∂2f(t, s)∂t∂s
∥∥∥∥
∞

b∫
a

d∫
c

|P (x, t; y, s)|dsdt

≤
∥∥∥∥∂2f(t, s)∂t∂s

∥∥∥∥
∞




x∫
a

y∫
c


 t∫

a

s∫
c

|p(u, v)|dvdu


dsdt

+

x∫
a

d∫
y


 t∫

a

d∫
s

|p(u, v)|dvdu


dsdt + b∫

x

y∫
c


 b∫

t

s∫
c

|p(u, v)|dvdu


dsdt

+

b∫
x

d∫
y


 b∫

t

d∫
s

|p(u, v)|dvdu


dsdt




≤
∥∥∥∥∂2f(t, s)∂t∂s

∥∥∥∥
∞
{J1 + J2 + J3 + J4} .

Now, using the change of order of integration we get

J1 =

x∫
a

y∫
c


 t∫

a

s∫
c

|p(u, v)|dvdu


dsdt

=

x∫
a

t∫
a


 y∫

c

s∫
c

|p(u, v)|dvds


dudt

=

x∫
a

t∫
a


 y∫

c

(y −v) |p(u, v)|dv


dudt

=

y∫
c


 x∫

a

t∫
a

(y −v) |p(u, v)|dudt


dv (2.8)

=

x∫
a

y∫
c

(x−u) (y −v) |p(u, v)|dvdu

and similarly,

J2 =

x∫
a

d∫
y

(x−u) (v −y) |p(u, v)|dvdu, (2.9)

J3 =

b∫
x

y∫
c

(u−x) (y −v) |p(u, v)|dvdu, (2.10)



ON THE GENERALIZED OSTROWSKI TYPE INTEGRAL INEQUALITY 69

J4 =

b∫
x

d∫
y

(u−x) (v −y) |p(u, v)|dvdu. (2.11)

Thus, using (2.8), (2.9), (2.10) and (2.11) in (2.7), we obtain the inequality (2.6) and the proof is
completed. �

Remark 2.3. If we choose p(., .) ≡ 1 in Theorem 2.1, then the inequality (2.6) reduces the inequality
(1.2) which is proved by Barnett and Dragomir in [3].

Remark 2.4. If take p(u, v) = w(u)w(v) in Theorem 2.1, then the inequality (2.6) reduces∣∣∣∣∣∣f(x, y) − 1m(a, b)
b∫

a

w(t)f(t, y)dt

−
1

m(c, d)

d∫
c

w(s)f(x, s)ds +
1

m(a, b)m(c, d)

b∫
a

d∫
c

w(s)w(t)f(t, s)dsdt

∣∣∣∣∣∣
≤

∥∥∥∥∂2f(t, s)∂t∂s
∥∥∥∥
∞

x∫
a

(x−u)A(u, y)du +
b∫

x

(u−x)A(u, y)du

where

A(u, y) =

y∫
c

(y −v)w(u)w(v)dv +
d∫

y

(v −y)w(u)w(v)dv.

which is proved by Sarikaya and Ogunmez in [11].

References

[1] F. Ahmad, N. S. Barnett and S. S. Dragomir, New Weighted Ostrowski and Cebysev Type Inequalities, Nonlinear
Analysis: Theory, Methods & Appl., 71 (12) (2009), 1408-1412.

[2] F. Ahmad, A. Rafiq, N. A. Mir, Weighted Ostrowski type inequality for twice differentiable mappings, Global Journal

of Research in Pure and Applied Math., 2 (2) (2006), 147-154.
[3] N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature

formulae, Soochow J. Math., 27(1) (2001), 109-114.
[4] S. S. Dragomir, N. S. Barnett and P. Cerone, An n-dimensional version of Ostrowski’s inequality for mappings of

Hölder type, RGMIA Res. Pep. Coll., 2(2) (1999), 169-180.

[5] A. M. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment.
Math. Helv. 10 (1938), 226-227.

[6] B. G. Pachpatte, On an inequality of Ostrowski type in three independent variables, J. Math.Anal. Appl., 249

(2000), 583-591.
[7] B. G. Pachpatte, On a new Ostrowski type inequality in two independent variables, Tamkang J. Math., 32 (1)

(2001), 45-49
[8] B. G. Pachpatte, A new Ostrowski type inequality for double integrals, Soochow J. Math., 32 (2) (2006), 317-322.
[9] A. Rafiq and F. Ahmad, Another weighted Ostrowski-Grüss type inequality for twice differentiable mappings, Kragu-

jevac Journal of Mathematics, 31 (2008), 43-51.
[10] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, 79 (1) (2010), 129-134.

[11] M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski type integral inequality for double integrals, The
Arabian Journal for Science and Engineering (AJSE)-Mathematics, 36 (2011), 1153-1160.

[12] M. Z. Sarikaya and H. Yildirim, New inequalities for local fractional integrals, Iranian Journal of Science and
Technology (Sciences), in press.

[13] N. Ujević, Some double integral inequalities and applications, Appl. Math. E-Notes, 7 (2007), 93-101.

1Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon-Turkey

2Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-Turkey

∗Corresponding author: myildiz@aku.edu.tr


	1. Introduction
	2. Main Result
	References