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volume 3, issue 2, article 27, 2002.

Received 14 August, 2001.;

accepted 25 January, 2002.

Communicated by:R.N. Mohapatra

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Journal of Inequalities in Pure and Applied Mathematics

ON MULTIDIMENSIONAL GRÜSS TYPE INEQUALITIES

B.G. PACHPATTE

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

EMail:bgpachpatte@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 063-01

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On Multidimensional Grüss Type Inequalities

B.G. Pachpatte

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J. Ineq. Pure and Appl. Math. 3(2) Art. 27, 2002

http://jipam.vu.edu.au

Abstract

In this paper we establish some new multidimensional integral inequalities of the Grüss type by using a fairly elementary analysis.

2000 Mathematics Subject Classification:26D15, 26D20.

Key words: Multidimensional, Grüss type inequalities, Partial Derivatives, Continu- ous Functions, Bounded, Identities.

1. Introduction

The following inequality is well known in the literature as the Grüss inequality [2] (see also [4, p. 296]):

1 b−a

Z b

a

f(x)g(x)dx− 1

b−a Z b

a

f(x)dx 1 b−a

Z b

a

g(x)dx

≤(M−m) (N −n), provided thatf, g: [a, b]→Rare integrable on[a, b]and

m≤f(x)≤M, n≤g(x)≤N, for allx∈[a, b],wherem, M, n, N are given constants.

Since the appearance of the above inequality in 1935, it has evoked consider- able interest and many variants, generalizations and extensions have appeared, see [1, 4] and the references cited therein. The main purpose of the present paper is to establish some new integral inequalities of the Grüss type involving functions of several independent variables. The analysis used in the proofs is elementary and our results provide new estimates on inequalities of this type.

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2. Statement of Results

In what follows, R denotes the set of real numbers. Let ∆ = [a, b]× [c, d],

∆⁰ = (a, b)×(c, d),Ω = [a, k]×[b, m]×[c, n],Ω⁰ = (a, k)×(b, m)×(c, n) fora, b, c, d, k, m, ninR. For functionsαandβ defined respectively on∆and Ω,the partial derivatives^∂²_∂y∂x^α(x,y) and ^∂³_∂z∂y∂x^β(x,y,z) are denoted byD₂D₁α(x, y)and D₃D₂D₁β(x, y, z).Let

D={x= (x₁, . . . , x_n) :a_i < x_i < b_i (i= 1,2, . . . , n)}

andD¯ be the closure ofD.For a functionedefined onD¯ the partial derivatives

∂e(x)

∂xi (i= 1,2, . . . , n)are denoted byD_ie(x).

First, we give the following notations used to simplify the details of presen- tation.

A(D₂D₁f(x, y)) = A[a, c;x, y;b, d;D₂D₁f(s, t)]

= Z x

a

Z y

c

D₂D₁f(s, t)dtds− Z x

a

Z d

y

D₂D₁f(s, t)dtds

− Z b

x

Z y

c

D₂D₁f(s, t)dtds+ Z b

x

Z d

y

D₂D₁f(s, t)dtds, B(D₃D₂D₁f(r, s, t))

=B[a, b, c;r, s, t;k, m, n;D₃D₂D₁f(u, v, w)]

= Z r

a

Z s

b

Z t

c

D₃D₂D₁f(u, v, w)dwdvdu

− Z r

a

Z s

b

Z n

t

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− Z r

a

Z m

s

Z t

c

− Z k

r

Z s

b

Z t

c

D₃D₂D₁f(u, v, w)dwdvdu +

Z r

a

Z m

s

Z n

t

Z k

r

Z m

s

Z t

c

Z k

r

Z s

b

Z n

t

− Z k

r

Z m

s

Z n

t

D3D2D1f(u, v, w)dwdvdu,

E(f(x, y)) = E[a, c;x, y;b, d;f]

= 1

2[f(x, c) +f(x, d) +f(a, y) +f(b, y)]

− 1

4[f(a, c) +f(a, d) +f(b, c) +f(b, d)], L(f(r, s, t)) =L[a, b, c;r, s, t;k, m, n;f]

= 1

8[f(a, b, c) +f(k, m, n)]

− 1

4[f(r, b, c) +f(r, m, n) +f(r, m, c) +f(r, b, n)]

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− 1

4[f(a, s, c) +f(k, s, n) +f(a, s, n) +f(k, s, c)]

− 1

4[f(a, b, t) +f(k, m, t) +f(k, b, t) +f(a, m, t)]

+ 1

2[f(a, s, t) +f(k, s, t)] + 1

2[f(r, b, t) +f(r, m, t)]

+ 1

2[f(r, s, c) +f(r, s, n)]. Our main results are given in the following theorems.

Theorem 2.1. Letf, g : ∆ →Rbe continuous functions on∆;D₂D₁f(x, y), D2D1g(x, y)exist on∆⁰ and are bounded, i.e.

kD₂D₁fk_∞ = sup

(x,y)∈∆⁰

|D₂D₁f(x, y)|<∞, kD₂D₁gk_∞ = sup

(x,y)∈∆⁰

|D₂D₁g(x, y)|<∞.

Then (2.1)

Z b

a

Z d

c

f(x, y)g(x, y)dydx

− 1 2

Z b

a

Z d

c

[E(f(x, y))g(x, y) +E(g(x, y))f(x, y)]dydx

≤ 1

8(b−a) (d−c) Z b

a

Z d

c

(|g(x, y)| kD₂D₁fk_∞

+|f(x, y)| kD₂D₁gk_∞)dydx.

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Theorem 2.2. Letp, q : Ω→Rbe continuous functions onΩ;D₃D₂D₁p(r, s, t), D3D2D1q(r, s, t)exist onΩ⁰and are bounded, i.e.

kD₃D₂D₁pk_∞ = sup

(r,s,t)∈Ω⁰

|D₃D₂D₁p(r, s, t)|<∞, kD₃D₂D₁qk_∞ = sup

(r,s,t)∈Ω⁰

|D₃D₂D₁q(r, s, t)|<∞.

Then

Z k

a

Z m

b

Z n

c

p(r, s, t)q(r, s, t)dtdsdr (2.2)

− 1 2

Z k

a

Z m

b

Z n

c

[L(p(r, s, t))q(r, s, t) +L(q(r, s, t))p(r, s, t)]dtdsdr

≤ 1

16(k−a) (m−b) (n−c)

× Z k

a

Z m

b

Z n

c

(|q(r, s, t)| kD₃D₂D₁pk_∞ +|p(r, s, t)| kD3D2D1qk_∞)dtdsdr.

Theorem 2.3. Let f, g : Rⁿ → Rbe continuous functions on D¯ and differen- tiable onDwhose derivativesD_if(x),D_ig(x)are bounded, i.e.,

kD_ifk_∞ = sup

x∈D

|D_if(x)|<∞, kD_igk_∞ = sup

x∈D

|D_ig(x)|<∞.

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Then (2.3)

1 M

Z

D

f(x)g(x)dx− 1

M Z

D

f(x)dx 1 M

Z

D

g(x)dx

≤ 1 2M²

Z

D n

X

i=1

(|g(x)| kD_ifk_∞+|f(x)| kD_igk_∞)E_i(x)dx, where

M =

n

Y

i=1

(bi−ai), Ei(x) = Z

D

|xi−yi|dy, dx = dx1· · ·dxn, dy=dy1· · ·dyn.

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3. Proof of Theorem 2.1

From the hypotheses we have the following identities (see [6]):

f(x, y) = E(f(x, y)) + 1

4A(D₂D₁f(x, y)), (3.1)

g(x, y) = E(g(x, y)) + 1

4A(D₂D₁g(x, y)), (3.2)

for (x, y) ∈ ∆. Multiplying (3.1) by g(x, y)and (3.2) by f(x, y)and adding the resulting identities, then integrating on∆and rewriting we have

Z b

a

Z d

c

f(x, y)g(x, y)dydx (3.3)

= 1 2

Z b

a

Z d

c

(E(f(x, y))g(x, y) +E(g(x, y))f(x, y))dydx + 1

8 Z b

a

Z d

c

(A(D₂D₁f(x, y))g(x, y) + A(D₂D₁g(x, y))f(x, y))dydx.

From the properties of modulus and integrals, it is easy to see that (3.4) |A(D2D1f(x, y))| ≤

Z b

a

Z d

c

|D2D1f(s, t)|dtds,

(3.5) |A(D₂D₁g(x, y))| ≤ Z b

a

Z d

c

|D₂D₁g(s, t)|dtds.

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From (3.3) – (3.5) we observe that

Z b

a

Z d

c

f(x, y)g(x, y)dydx

− 1 2

Z b

a

Z d

c

(E(f(x, y))g(x, y) +E(g(x, y))f(x, y))dydx

≤ 1 8

Z b

a

Z d

c

(|g(x, y)| |A(D₂D₁f(x, y))|

+|f(x, y)| |A(D₂D₁g(x, y))|)dydx

≤ 1 8

Z b

a

Z d

c

|g(x, y)|

Z b

a

Z d

c

|D₂D₁f(s, t)|dtds +|f(x, y)|

Z b

a

Z d

c

|D₂D₁g(s, t)|dtds

dydx

≤ 1

8(b−a) (d−c) Z b

a

Z d

c

(|g(x, y)| kD₂D₁fk_∞ +|f(x, y)| kD2D1gk_∞)dydx,

which is the required inequality in (2.1). The proof is complete.

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4. Proof of Theorem 2.2

From the hypotheses we have the following identities (see [5]):

p(r, s, t) = L(p(r, s, t)) + 1

8B(D₃D₂D₁p(r, s, t)), (4.1)

q(r, s, t) = L(q(r, s, t)) + 1

8B(D₃D₂D₁q(r, s, t)), (4.2)

for (r, s, t) ∈ Ω. Multiplying (4.1) by q(r, s, t) and (4.2) by p(r, s, t) and adding the resulting identities, then integrating onΩand rewriting we have

Z k

a

Z m

b

Z n

c

p(r, s, t)q(r, s, t)dtdsdr (4.3)

= 1 2

Z k

a

Z m

b

Z n

c

[L(p(r, s, t))q(r, s, t) +L(q(r, s, t))p(r, s, t)]dtdsdr + 1

16 Z k

a

Z m

b

Z n

c

(B(D₃D₂D₁p(r, s, t))q(r, s, t) +B(D3D2D1q(r, s, t))p(r, s, t))dtdsdr.

From the properties of modulus and integrals, we observe that

|B(D₃D₂D₁p(r, s, t))| ≤ Z k

a

Z m

b

Z n

c

|D₃D₂D₁p(u, v, w)|dwdvdu, (4.4)

|B(D₃D₂D₁q(r, s, t))| ≤ Z k

a

Z m

b

Z n

c

|D₃D₂D₁q(u, v, w)|dwdvdu.

(4.5)

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Now, from (4.3) – (4.5) and following the same arguments as in the proof of Theorem2.1with suitable changes, we get the required inequality in (2.2).

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5. Proof of Theorem 2.3

Letx ∈ D, y¯ ∈ D. From then−dimensional version of the mean value theorem, we have (see [3]):

(5.1) f(x)−f(y) =

n

X

i=1

Dif(c) (xi−yi), wherec= (y₁+δ(x₁−y₁), . . . , y_n+δ(x_n−y_n)),0< δ <1.

Integrating (5.1) with respect toy,we obtain (5.2) f(x) mesD=

Z

D

f(y)dy+

n

X

i=1

Z

D

D_if(c) (x_i−y_i)dy, wheremesD=Qn

i=1(b_i−a_i) =M.Similarly, we obtain (5.3) g(x) mesD=

Z

D

g(y)dy+

n

X

i=1

Z

D

Dig(c) (xi−yi)dy.

Multiplying (5.2) byg(x)and (5.3) byf(x)and adding the resulting identities, then integrating onDand noting thatmes>0, we have

Z

D

f(x)g(x)dx (5.4)

= 1 M

Z

D

f(x)dx Z

D

g(x)dx

+ 1 2M

"

Z

D

g(x)

n

X

i=1

Z

D

D_if(c) (x_i−y_i)dy

! dx

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+ Z

D

f(x)

n

X

i=1

Z

D

D_ig(c) (x_i−y_i)dy

! dx

# .

From (5.4) we observe that

1 M

Z

D

f(x)g(x)dx− 1

M Z

D

f(x)dx 1 M

Z

D

g(x)dx

≤ 1 2M²

Z

D n

X

i=1

(|g(x)| kD_ifk_∞+|f(x)| kD_igk_∞)E_i(x)dx.

This is the required inequality in (2.3). The proof is complete.

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References

[1] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. Pure

& Appl. Math., 31(4) (2000), 379–415. ONLINE: RGMIA Research Report Collection, 1(2) (1998), 97–113.

[2] G. GRÜSS, Über das maximum des absoluten Betrages von

1 b−a

Rb

a f(x)g(x)dx − _(b−a)¹ 2

Rb

a f(x)dxRb

a g(x)dx, Math. Z., 39 (1935), 215–226.

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

Elek. Fak. Ser. Mat. Fiz., No498–No541 (1975), 112–124.

[4] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C AND A.M. FINK, Classical and new Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.

[5] B.G. PACHPATTE, On an inequality of Ostrowski type in three independent variables, J. Math. Anal. Appl., 249 (2000), 583–591.

[6] B.G. PACHPATTE, On a new Ostrowski type inequality in two independent variables, Tamkang J. Math., 32 (2001), 45–49.