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volume 3, issue 4, article 58, 2002.

Received 02 April, 2002;

accepted 20 June, 2002.

Communicated by:G. Anastassiou

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Journal of Inequalities in Pure and Applied Mathematics

ON MULTIVARIATE OSTROWSKI TYPE INEQUALITIES

B.G. PACHPATTE

57, Shri Niketen Colony Aurangabad - 431 001, (Maharashtra) India.

EMail:bgpachpatte@hotmail.com

2000c Victoria University ISSN (electronic): 1443-5756 069-02

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On Multivariate Ostrowski Type Inequalities

B.G. Pachpatte

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J. Ineq. Pure and Appl. Math. 3(4) Art. 58, 2002

Abstract

In the present paper we establish new multivariate Ostrowski type inequalities by using fairly elementary analysis.

2000 Mathematics Subject Classification:26D15, 26D20.

Key words: Multivariate, Ostrowski type inequalities, Many independent variables, n−fold integral.

1. Introduction

The following inequality is well known in the literature as Ostrowski’s integral inequality (see [5, p. 469]).

Letf : [a, b]→Rbe continuous on[a, b]and differentiable on(a, b)whose derivative f⁰ : (a, b) → Ris bounded on(a, b),i.e., kf⁰k_∞ = sup

t∈(a,b)

|f⁰(t)| <

∞.Then

f(x)− 1 b−a

Z b

a

f(t)dt

≤

"

1

4+ x− ^a+b₂ 2

(b−a)²

#

(b−a)kf⁰k_∞,

for allx∈[a, b].

Many generalisations, extensions and variants of this inequality have ap- peared in the literature, see [1] – [7] and the references given therein.

The main aim of this paper is to establish new inequalities similar to that of Ostrowski’s inequality involving functions of many independent variables and their first order partial derivatives. The analysis used in the proof is elementary and our results provide new estimates on these types of inequalities.

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2. Statement of Results

In what follows, R denotes the set of real nummbers, Rⁿ the n−dimensional Euclidean space. LetD={(x1, . . . , xn) :ai < xi < bi (i= 1, . . . , n)}andD¯ be the closure of D.For a functionu(x) : Rⁿ → R, we denote the first order partial derivatives by ^∂u(x)_∂x

i (i= 1, . . . , n)and R

Du(x)dx the n−fold integral Rb1

a1 · · ·Rbn

an u(x₁, . . . , x_n)dx₁. . . dx_n.

Our main results are established in the following theorems.

Theorem 2.1. Let f, g : Rⁿ → Rbe continuous functions on D¯ and differen- tiable onDwhose derivatives _∂x^∂f

i, _∂x^∂g

i are bounded, i.e.,

∂f

∂xi

_∞

= sup

x∈D

∂f(x)

∂xi

<∞,

∂g

∂x_i _∞

= sup

x∈D

∂g(x)

∂x_i

<∞.

Let the functionw(x)be defined, nonnegative, integrable for everyx∈ Dand R

Dw(y)dy >0.Then for everyx∈D,¯ (2.1)

f(x)g(x)− 1 2Mg(x)

Z

D

f(y)dy− 1 2Mf(x)

Z

D

g(y)dy

≤ 1 2M

n

X

i=1

|g(x)|

∂f

∂x_i _∞

+|f(x)|

∂g

∂x_i _∞

E_i(x),

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(2.2)

f(x)g(x)−

g(x)R

Dw(y)f(y)dy+f(x)R

Dw(y)g(y)dy 2R

Dw(y)dy

≤ R

Dw(y)Pn i=1

h|g(x)|

∂f

∂xi

_∞+|f(x)|

∂g

∂xi

_∞

i|x_i−y_i|dy 2R

Dw(y)dy ,

where

M = mesD=

n

Y

i=1

(bi−ai), dy=dy1. . . dyn and Ei(x) = Z

D

|xi−yi|dy.

Remark 2.1. If we take g(x) = 1 and hence _∂x^∂g

i = 0 in Theorem 2.1, then the inequalities (2.1) and (2.2) reduces to the inequalities established by Milo- vanovi´c in [3, Theorems 2 and 3] which in turn are the further generalisations of the well known Ostrowski’s inequality.

Theorem 2.2. Letf, g, _∂x^∂f

i, _∂x^∂g

i be as in Theorem2.1. Then for everyx∈D,¯ (2.3)

f(x)g(x)−f(x) 1

M Z

D

g(y)dy

− g(x) 1

M Z

D

f(y)dy

+ 1 M

Z

D

f(y)g(y)dy

≤ 1 M

Z

D

" _n X

i=1

∂f

∂x_i _∞

|x_i−y_i|

! _n X

i=1

∂g

∂x_i _∞

|x_i−y_i|

!#

dy,

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(2.4)

f(x)g(x)−f(x) 1

M Z

D

g(y)dy

− g(x) 1

M Z

D

f(y)dy

+ 1 M²

Z

D

f(y)dy Z

D

g(y)dy

≤ 1 M²

n

X

i=1

∂f

∂x_i ∞

E_i(x)

! _n X

i=1

∂g

∂x_i ∞

E_i(x)

! ,

whereM, dy andE_i(x)are as defined in Theorem2.1.

Remark 2.2. We note that in [1] Anastassiou has used a slightly different tech- nique to establish multivariate Ostrowski type inequalities. However, the in- equalities established in (2.3) and (2.4) are different from those given in [1] and our proofs are extremely simple. For an n−dimensional version of Ostrowski’s inequality for mappings of Hölder type, see [2].

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3. Proof of Theorem 2.1

Let x = (x₁, . . . , x_n) and y = (y₁, . . . , y_n) x∈D, y¯ ∈D

. From the n−

dimensional version of the mean value theorem, we have (see [8, p. 174] or [4, p. 121])

f(x)−f(y) =

n

X

i=1

∂f(c)

∂x_i (x_i −y_i), (3.1)

g(x)−g(y) =

n

X

i=1

∂g(c)

∂x_i (x_i−y_i), (3.2)

wherec= (y₁+α(x₁−y₁), . . . , y_n+α(x_n−y_n)) (0< α <1).

Multiplying both sides of (3.1) and (3.2) byg(x)andf(x)respectively and adding, we get

(3.3) 2f(x)g(x)−g(x)f(y)−f(x)g(y)

=g(x)

n

X

i=1

∂f(c)

∂x_i (x_i−y_i) +f(x)

n

X

i=1

∂g(c)

∂x_i (x_i−y_i). Integrating both sides of (3.3) with respect to y over D, using the fact that mesD >0and rewriting, we have

(3.4) f(x)g(x)− 1 2Mg(x)

Z

D

f(y)dy− 1 2Mf(x)

Z

D

g(y)dy

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= 1

2M

"

g(x) Z

D n

X

i=1

∂f(c)

∂xi

(x_i−y_i)dy

+f(x) Z

D n

X

i=1

∂g(c)

∂xi

(x_i−y_i)dy

# .

From (3.4) and using the properties of modulus we have

f(x)g(x)− 1 2Mg(x)

Z

D

f(y)dy− 1 2Mf(x)

Z

D

g(y)dy

≤ 1 2M

"

|g(x)|

Z

D n

X

i=1

∂f(c)

∂x_i

|x_i−y_i|dy

+|f(x)|

Z

D n

X

i=1

∂g(c)

∂x_i

|x_i−y_i|dy

#

≤ 1 2M

n

X

i=1

|g(x)|

∂f

∂x_i _∞

+|f(x)|

∂g

∂x_i _∞

E_i(x).

The proof of the inequality (2.1) is complete.

Multiplying both sides of (3.3) byw(y)and integrating the resulting identity with respect toy onD and following the proof of inequality (2.1), we get the desired inequality in (2.2).

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4. Proof of Theorem 2.2

From the hypotheses, as in the proof of Theorem 2.1, the identities (3.1) and (3.2) hold. Multiplying the left and right sides of (3.1) and (3.2) we get

(4.1) f(x)g(x)−f(x)g(y)−g(x)f(y) +f(y)g(y)

=

" _n X

i=1

∂f(c)

∂x_i (xi−yi)

# " _n X

i=1

∂g(c)

∂x_i (xi−yi)

# .

Integrating both sides of (4.1) with respect toyonDand rewriting, we have (4.2) f(x)g(x)−f(x)

1 M

Z

D

g(y)dy

−g(x) 1

M Z

D

f(y)dy

+ 1 M

Z

D

f(y)g(y)dy

= 1 M

Z

D

" _n X

i=1

∂f(c)

∂x_i (x_i−y_i)

# " _n X

i=1

∂g(c)

∂x_i (x_i−y_i)

# dy.

From (4.2) and using the properties of the modulus, we have

f(x)g(x)−f(x) 1

M Z

D

g(y)dy

−g(x) 1

M Z

D

f(y)dy

+ 1 M

Z

D

f(y)g(y)dy

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≤ 1 M

Z

D

" _n X

i=1

∂f(c)

∂xi

|x_i−y_i|

# " _n X

i=1

∂g(c)

∂xi

|x_i−y_i|

# dy

≤ 1 M

Z

D

" _n X

i=1

∂f

∂xi

∞

|x_i−y_i|

# " _n X

i=1

∂g

∂xi

∞

|x_i−y_i|

# dy, which is the required inequality in (2.3).

Integrating both sides of (3.1) and (3.2) with respect toyoverDand rewriting, we get

(4.3) f(x)− 1 M

Z

D

f(y)dy= 1 M

Z

D n

X

i=1

∂f(c)

∂x_i (x_i−y_i)dy, and

(4.4) g(x)− 1 M

Z

D

g(y)dy= 1 M

Z

D n

X

i=1

∂g(c)

∂x_i (xi−yi)dy, respectively. Multiplying the left and right sides of (4.3) and (4.4) we get (4.5) f(x)g(x)−f(x)

1 M

Z

D

g(y)dy

−g(x) 1

M Z

D

f(y)dy

+ 1 M²

Z

D

f(y)dy Z

D

g(y)dy

= 1 M²

Z

D n

X

i=1

∂f(c)

∂x_i (x_i−y_i)dy

! Z

D n

X

i=1

∂g(c)

∂x_i (x_i−y_i)dy

! .

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From (4.5) and using the properties of the modulus we have

f(x)g(x)−f(x) 1

M Z

D

g(y)dy

− g(x) 1

M Z

D

f(y)dy

+ 1 M²

Z

D

f(y)dy Z

D

g(y)dy

≤ 1 M²

Z

D n

X

i=1

∂f(c)

∂x_i

|xi−yi|dy

!Z

D

∂g(c)

∂x_i

|xi−yi|dy

≤ 1 M²

n

X

i=1

∂f

∂x_i _∞

Ei(x)

! _n X

i=1

∂g

∂x_i _∞

Ei(x)

! .

This is the desired inequality in (2.4) and the proof is complete.

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References

[1] G.A. ANASTASSIOU, Multivariate Ostrowski type inequalities, Acta Math. Hungar., 76 (1997), 267–278.

[2] N.S. BARNETT AND S.S DRAGOMIR, An Ostrowski type inequality for double integrals and applications for cubature for- mulae, RGMIA Res. Rep. Coll., 1(1) (1998), 13–22. [ONLINE]

http://rgmia.vu.edu.au/v1n1.html

[3] S.S DRAGOMIR, N.S. BARNETT ANDP. CERONE, Ann−dimensional version of Ostrowski’s inequality for mappings of the Hölder type, RGMIA Res. Rep. Coll., 2(2) (1999), 169–180. [ONLINE]

http://rgmia.vu.edu.au/v2n2.html

[4] G.V. MILOVANOVI ´C, On some integral inequalities, Univ. Beograd Publ.

Elek. Fak. Ser. Mat. Fiz., No. 496 – No. 541 (1975), 119-124.

[5] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities for func- tions and their integrals and derivatives, Kluwer Academic Publishers, Dordrecht, 1994.

[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 (2001), 45–49.

[8] W. RUDIN, Principles of Mathematical Analysis, McGraw-Hill Book Com- pany Inc., 1953.