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volume 4, issue 5, article 108, 2003.

Received 09 October, 2002;

accepted 15 December, 2003.

Communicated by:P. Cerone

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Journal of Inequalities in Pure and Applied Mathematics

NEW WEIGHTED MULTIVARIATE GRÜSS TYPE INEQUALITIES

B.G. PACHPATTE

57 Shri Niketan Colony, Near Abhinay Talkies, Aurangabad 431 001 (Maharashtra) India.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 104-02

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New Weighted Multivariate Grüss Type Inequalities

B.G. Pachpatte

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J. Ineq. Pure and Appl. Math. 4(5) Art. 108, 2003

http://jipam.vu.edu.au

Abstract

In this paper we establish some new weighted multidimensional Grüss type integral and discrete inequalities by using a fairly elementary analysis .

2000 Mathematics Subject Classification:26D15 , 26D20.

Key words: Multivariate Grüss type inequalities, Discrete inequalities, New esti- mates, Differentiable function, Partial derivatives, Forward difference operators, Mean value theorem .

1. Introduction

In 1935, G. Grüss [3] proved the following classical integral inequality (see, also [4, p. 296]):

1 b−a

Z b

a

f(x)g(x)− 1

b−a Z b

a

f(x)dx 1 b−a

Z b

a

g(x)dx

≤ 1

4(P −p) (Q−q), provided thatf andg are two integrable functions on[a, b]such that

p≤f(x)≤P, q ≤g(x)≤Q, for allx∈[a, b], wherep, P, q, Qare constants.

A large number of generalizations, extensions and variants of this inequality have been given by several authors since its discovery, see [1,2], [4] – [6] and the references given therein. The main purpose of this paper is to establish new weighted integral and discrete 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,RandNdenote the set of real and natural numbers respectively.

Let D_i[a, b] = {x_i :a_i < x_i < b_i} for i = 1, . . . , n, a_i, b_i ∈ R, D =

n

Q

i=1

D_i[a_i, b_i] andD¯ be the closure of D. For a differentiable function u(x) : D¯ →R, we denote the first order partial derivatives by ^∂u(x)_∂x

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

Du(x)dxthen-fold integral Z b1

a1

· · · Z bn

an

u(x₁, . . . , x_n)dx₁. . . dx_n.

If

∂u

∂xi

∞

= sup

x∈D

∂u(x)

∂xi

<∞, then we say that the partial derivatives ^∂u(x)_∂x

i are bounded. Let N_i[0, a_i] = {0,1,2, . . . , a_i}, a_i ∈ N, (i= 1, . . . , n) and B =

n

Q

i=1

N_i[0, a_i]. For a func- tionz(x) :B →Rwe define the first order forward difference operators as

∆₁z(x) =z(x₁+ 1, x₂, . . . , x_n)−z(x), . . . ,∆_nz(x)

=z(x₁, . . . , xn−1, x_n+ 1)−z(x)

and denote then-fold sum overBwith respect to the variabley= (y₁, . . . , y_n)∈ B by

X

y

z(y) =

a1−1

X

y1=0

· · ·

an−1

X

yn=0

z(y₁, . . . , y_n).

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Clearly,

X

y

z(y) =X

x

z(x) forx, y ∈B.

If k∆_izk_∞ = sup

x∈B

|∆_iz(x)| < ∞, then we say that the partial differences

∆iz(x)are bounded. The notation

xi−1

X

ti=yi

∆_iz(y₁, . . . , yi−1, t_i, x_i+1, . . . , x_n), x_i, y_i ∈N_i [0, a_i] (i= 1, . . . , n),

we mean fori= 1it isPx1−1

t1=y1∆₁z(t₁, x₂, . . . , x_n)and so on, and fori=nit is Pxn−1

tn=yn∆_n ×z(y₁, . . . , yn−1, t_n). We use the usual convention that the empty sum is taken to be zero. We use the following notations to simplify the details of presentation.

For continuous functionsp, q defined onD¯ and differentiable onD, w(x)a real-valued nonnegative and integrable function for everyx∈DwithR

Dw(x)dx

>0andxi, yi ∈Di[ai, bi], we set A[w, p, q] =

Z

D

w(x)p(x)q(x)dx

− 1

R

Dw(x)dx Z

D

w(x)p(x)dx Z

D

w(x)q(x)dx

,

H[p, x_i, y_i] =

n

X

i=1

∂p

∂x_i _∞

|x_i −y_i|.

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For the functionsp, q : B → Rwhose forward differences ∆_ip,∆_iqexist, w(x) a real-valued nonnegative function defined on B and P

x

w(x) > 0 and xi, yi ∈Ni[0, ai], we set

L[w, p, q]

=X

x

w(x)p(x)q(x)− 1 P

x

w(x) X

x

w(x)p(x)

! X

x

w(x)q(x)

! ,

M[p, xi, yi] =

n

X

i=1

k∆ipk_∞|xi−yi|.

Our main results on weighted Grüss type integral inequalities involving functions of many independent variables are embodied in the following theorem.

Theorem 2.1. Letf, g be real-valued continuous functions onD¯ and differen- tiable onDwhose derivatives _∂x^∂f

i,_∂x^∂g

i are bounded. Letw(x)be a real-valued, nonnegative and integrable function forx∈DandR

Dw(x)dx >0. Then (2.1) |A[w, f, g]|

≤ 1

2R

Dw(x)dx Z

D

w(x)

|g(x)|

Z

D

H[f, x_i, y_i]w(y)dy +|f(x)|

Z

D

H[g, x_i, y_i]w(y)dy

dx,

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(2.2) |A[w, f, g]|

≤ 1

R

Dw(x)dx2

Z

D

w(x) Z

D

H[f, xi, yi]w(y)dy

× Z

D

dx.

Remark 2.1. If we taken= 1andD=I ={a < x < b}in (2.1), then we get

Z b

a

w(t)f(t)g(t)dt− 1 Rb

aw(t)dt Z b

a

w(t)f(t)dt

Z b

a

w(t)g(t)dt

≤ 1

2Rb

aw(t)dt Z b

a

w(t)

|g(t)|

Z b

a

kf⁰k_∞|t−s|w(s)ds +|f(t)|

Z b

a

kg⁰k_∞|t−s|w(s)ds

dt.

Similarly, one can obtain the special version of (2.2). It is easy to see that the upper bound given on the right side in the above inequality (whenw(t) = 1) is different from those given by Grüss in [3].

The next theorem deals with the discrete versions of the inequalities in The- orem2.1.

Theorem 2.2. Letf, gbe real-valued functions defined onBand∆_if,∆_igare bounded. Let w(x) be a real-valued nonnegative function defined on B and

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P

x

w(x)>0. Then

(2.3) |L[w, f, g]| ≤ 1 2P

x

w(x) X

x

w(x)

"

|g(x)|X

y

M[f, x_i, y_i]w(y)

+|f(x)|X

y

M[g, x_i, y_i]w(y)

# ,

(2.4) |L(w, f, g)| ≤ 1

P

x

w(x) 2

X

x

w(x)

× X

y

M[f, xi, yi]w(y)

! X

y

M[g, xi, yi]w(y)

! .

Remark 2.2. In a recent paper [6] the author gave multidimensional Grüss type finite difference inequalities whose proofs were based on a certain finite difference identity. Here we note that the inequalities established in (2.3) and (2.4) are of more general type and can be considered as the weighted general- izations of the similar inequalities given in [6].

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3. Proof of Theorem 2.1

Let x = (x₁, . . . , x_n) ∈ D, y¯ = (y₁, . . . , y_n) ∈ D. From the n-dimensional version of the mean value theorem we have (see [7, p. 174])

(3.1) f(x)−f(y) =

n

X

i=1

∂f(c)

∂x_i (x_i−y_i) and

(3.2) g(x)−g(y) =

n

X

i=1

∂g(c)

∂x_i (x_i−y_i),

where c = (y₁+α(x₁−y₁), . . . , y_n+α(x_n−y_n)) (0< α <1). Multiply- ing 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). Multiplying both sides of (3.3) byw(y) and integrating the resulting identity with respect toyoverDwe have

(3.4) 2 Z

D

w(y)dy

f(x)g(x)

−g(x) Z

D

w(y)f(y)dy−f(x) Z

D

w(y)g(y)dy

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=g(x) Z

D n

X

i=1

∂f(c)

∂xi

(x_i −y_i)w(y)dy+f(x) Z

D n

X

i=1

∂g(c)

∂xi

(x_i−y_i)w(y)dy.

Next, multiplying both sides of (3.4) byw(x)and integrating the resulting identity with respect toxonDwe get

(3.5) 2 Z

D

w(y)dy Z

D

w(x)f(x)g(x)dx

− Z

D

w(x)g(x)dx Z

D

w(y)f(y)dy

− Z

D

w(x)f(x)dx Z

D

w(y)g(y)dy

= Z

D

w(x)g(x) Z

D n

X

i=1

∂f(c)

∂x_i (x_i−y_i)w(y)dy

! dx

+ Z

D

w(x)f(x) Z

D n

X

i=1

∂g(c)

! dx.

From (3.5) and using the properties of modulus we have

|A[w, f, g]|

≤ 1

2R

Dw(x)dx Z

D

w(x)|g(x)|

Z

D n

X

i=1

∂f(c)

∂x_i

|xi−yi|w(y)dy

! dx

+ Z

D

w(x)|f(x)|

Z

D n

X

i=1

∂g(c)

∂x_i

|x_i−y_i|w(y)dy

! dx

#

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≤ 1

2R

Dw(x)dx Z

D

w(x)

"

|g(x)|

Z

D n

X

i=1

∂f

∂xi

∞

|x_i−y_i|w(y)dy

+|f(x)|

Z

D n

X

i=1

∂g

∂xi

∞

|x_i−y_i|w(y)dy

# dx

= 1

2R

Dw(x)dx Z

D

w(x)

|g(x)|

Z

D

H[f, x_i, y_i]w(y)dy +|f(x)|

Z

D

dx.

This is the required inequality in (2.1).

Multiplying both sides of (3.1) and (3.2) byw(y)and integrating the resulting identities with respect toyonDwe get

(3.6)

Z

D

w(y)dy

f(x)− Z

D

w(y)f(y)dy

= Z

D n

X

i=1

∂f(c)

∂x_i (x_i−y_i)w(y)dy and

(3.7)

Z

D

w(y)dy

g(x)− Z

D

w(y)g(y)dy

= Z

D n

X

i=1

∂g(c)

∂x_i (x_i−y_i)w(y)dy.

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Multiplying the left sides and right sides of (3.6) and (3.7) we get (3.8)

Z

D

w(y)dy 2

f(x)g(x)− Z

D

w(y)dy

f(x) Z

D

w(y)g(y)dy

− Z

D

w(y)dy

g(x) Z

D

w(y)f(y)dy +

Z

D

w(y)f(y)dy Z

D

w(y)g(y)dy

= Z

D n

X

i=1

∂f(c)

! Z

D n

X

i=1

∂g(c)

! . Multiplying both sides of (3.8) by w(x)and integrating the resulting identity with respect toxonD, by simple calculations we obtain

(3.9) Z

D

w(x)f(x)g(x)dx

− 1

R

Dw(y)dy Z

D

w(x)f(x)dx Z

D

w(x)g(x)dx

= 1

R

Dw(y)dy2

Z

D

w(x) Z

D n

X

i=1

∂f(c)

∂x_i (x_i −y_i)w(y)dy

!

× Z

D n

X

i=1

∂g(c)

! dx.

From (3.9) and following the proof of the inequality (2.1) with suitable modifi- cations we get the required inequality in (2.2). The proof is complete.

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Remark 3.1. Multiplying the left sides and right sides of (3.1) and (3.2), then multiplying the resulting identity byw(y), integrating it with respect toyonD, again multiplying the resulting identity byw(x), integrating it with respect tox overD and following the similar arguments as in the proofs of (2.1), (2.2) we have

(3.10) |A[w, f, g]|

≤ 1

2R

Dw(x)dx Z

D

w(x) Z

D

H[f, x_i, y_i]H[g, x_i, y_i]w(y)dy

dx.

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4. Proof of Theorem 2.2

For x = (x₁, . . . , x_n), y = (y₁, . . . , y_n) in B, it is easy to observe that the following identities hold (see [6]):

(4.1) f(x)−f(y) =

n

X

i=1

(_x

i−1

X

ti=yi

∆_if(y₁, . . . , y_i−1, t_i, x_i+1, . . . , x_n) )

and

(4.2) g(x)−g(y) =

n

X

i=1

(_x_i₋₁ X

ti=yi

∆_ig(y₁, . . . , y_i−1, t_i, x_i+1, . . . , x_n) )

.

Multiplying both sides of (4.1) and (4.2) by g(x) and f(x) respectively, and adding we obtain

(4.3) 2f(x)g(x)−g(x)f(y)−f(x)g(y)

=g(x)

n

X

i=1

(_x_i₋₁ X

ti=yi

∆_if(y₁, . . . , y_i−1, t_i, x_i+1, . . . , x_n) )

+f(x)

n

X

i=1

(_x

i−1

X

ti=yi

∆_ig(y₁, . . . , yi−1, t_i, x_i+1, . . . , x_n) )

.

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Multiplying both sides of (4.3) byw(y)and summing both sides of the resulting identity with respect toyoverB,we have

(4.4) 2X

y

w(y)f(x)g(x)−g(x)X

y

w(y)f(y)−f(x)X

y

w(y)g(y)

=g(x)X

y n

X

i=1

(_x_i₋₁ X

ti=yi

∆_if(y₁, . . . , yi−1, t_i, x_i+1, . . . , x_n) )!

w(y)

+f(x)X

y n

X

i=1

(_x_i₋₁ X

ti=yi

∆_ig(y₁, . . . , yi−1, t_i, x_i+1, . . . , x_n) )!

w(y). Now, multiplying both sides of (4.4) byw(x)and summing the resulting identity with respect toxonBwe have

(4.5) 2 X

y

w(y)

! X

x

w(x)f(x)g(x)

− X

x

w(x)g(x)

! X

y

w(y)f(y)

!

− X

x

w(x)f(x)

! X

y

w(y)g(y)

!

=X

x

w(x)g(x)

"

X

y n

X

i=1

(_x

i−1

X

ti=yi

∆_if(y₁, . . . , yi−1, t_i, x_i+1, . . . , x_n) )!

w(y)

#

+X

x

w(x)f(x)

"

X

y n

X

i=1

(_x

i−1

X

ti=yi

∆_ig(y₁, . . . , yi−1, t_i, x_i+1, . . . , x_n) )!

w(y)

# .

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

|L(w, f, g)| |

≤ 1

2P

x

w(x)

"

X

x

w(x)|g(x)|

×X

y n

X

i=1

(_x_i₋₁ X

ti=yi

|∆_if(y₁, . . . , yi−1, t_i, x_i+1, . . . , x_n)|

)

! w(y)

+X

x

w(x)|f(x)|

× X

y n

X

i=1

(_x_i₋₁ X

ti=yi

|∆_ig(y₁, . . . , yi−1, t_i, x_i+1, . . . , x_n)|

)

! w(y)

#

≤ 1

2P

x

w(x) X

x

w(x)

"

|g(x)|X

y n

X

i=1

(

k∆_ifk_∞

xi−1

X

ti=yi

1 )

! w(y)

+|f(x)|X

y n

X

i=1

k∆_igk_∞|x_i−y_i|

! w(y)

#

= 1

2P

x

w(x) X

x

w(x)

"

X

x

w(x)|g(x)|X

y n

X

i=1

k∆ifk_∞|xi−yi|

! w(y)

+|f(x)|X

y n

X

i=1

k∆_igk_∞|x_i−y_i|

! w(y)

#

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= 1

2P

x

w(x) X

x

w(x)

"

|g(x)|X

y

M[f, x_i, y_i]w(y)

+|f(x)|X

y

M[g, x_i, y_i]w(y)

# ,

which is the required inequality in (2.3).

The proof of the inequality (2.4) can be completed by following the proof of (2.3) and closely looking at the proof of (2.2). Here we omit the details.

Remark 4.1. Multiplying the left sides and right sides of (4.1) and (4.2), then multiplying the resulting identity byw(y), summing it with respect toyoverB, again multiplying the resulting identity byw(x), summing it with respect tox overBand closely looking at the proof of the inequality (2.3) we get

(4.6) |L(w, f, g)|

≤ 1

2P

x

w(x) X

x

w(x) X

y

M[f, x_i, y_i]M[g, x_i, y_i]w(y)

! .

In concluding we note that in [2] Fink has given some Grüss type inequalities for measures other than the Lebesgue measure, including signed measures which provide different upper bounds. In addition, in [2] new proofs to some old results are also given. However, the inequalities established here are different and cannot be compared with those of given in [2].

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References

[1] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. Pure and Appl. Math., 31 (2000), 379–415.

[2] A.M. FINK, A treatise on Grüss inequality, Analytic and Geometric In- equalities and Applications, T.M. Rassias and H.M. Srivastava (eds.), Kluwer Academic Publishers, Dordrecht 1999, 93–113.

[3] 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.

[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 multidimensional Grüss type inequalities, J. In- equal. Pure and Appl. Math., 3(2) (2002), Art. 27. [ONLINE: http:

//jipam.vu.edu.au]

[6] B.G. PACHPATTE, On multidimensional Ostrowski and Grüss type finite difference inequalities, J. Inequal. Pure and Appl. Math., 4(2) (2003), Art.

7. [ONLINE:http://jipam.vu.edu.au]

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