JJ II

(1)

volume 4, issue 5, article 91, 2003.

Received 20 May, 2003;

accepted 18 September, 2003.

Communicated by:S.S. Dragomir

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

ON GRÜSS TYPE INEQUALITIES OF DRAGOMIR AND FEDOTOV

J.E. PE ˇCARI ´C AND B. TEPEŠ

Faculty of Textile Technology University of Zagreb, Pirottijeva 6, 1000 Zagreb, Croatia.

EMail:pecaric@mahazu.hazu.hr

URL:http://mahazu.hazu.hr/DepMPCS/indexJP.html Faculty of Philosophy,

I. Luˇci´ca 3, 1000 Zagreb, Croatia.

c

2000Victoria University ISSN (electronic): 1443-5756 065-03

(2)

On Grüss Type Inequalities of Dragomir and Fedotov J.E. Peˇcari´c and B. Tepeš

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of13

J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003

http://jipam.vu.edu.au

Abstract

Weighted versions of Grüss type inequalities of Dragomir and Fedotov are given. Some related results are also obtained.

2000 Mathematics Subject Classification:26D15.

Key words: Grüss type inequalities.

1. Introduction

In 1935, G. Grüss proved the following inequality:

(1.1)

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

6 1

4(Φ−ϕ) (Γ−γ), provided thatfandgare two integrable functions on[a, b]satisfying the condition (1.2) ϕ 6f(x)6Φandγ 6g(x)6Γfor allx∈[a, b].

The constant ¹₄ is best possible and is achieved for f(x) =g(x) = sgn

x− a+b 2

.

The following result of Grüss type was proved by S.S. Dragomir and I. Fedotov [1]:

Theorem 1.1. Letf, u : [a, b] →Rbe such that uisL−Lipschitzian on [a, b], i.e.,

(1.3) |u(x)−u(y)|6L|x−y| for all x∈[a, b],

f is Riemann integrable on[a, b]and there exist the real numbersm, M so that (1.4) m6f(x)6M for all x∈[a, b].

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of13

Then we have the inequality

(1.5)

Z b

a

f(x)du(x)− u(b)−u(a) b−a

Z b

a

f(t)dt 6 1

2L(M −m) (b−a), and the constant ¹₂ is sharp, in the sense that it cannot be replaced by a smaller one.

The following result of Grüss’ type was proved by S.S. Dragomir and I.

Fedotov [2]:

Theorem 1.2. Let f, u : [a, b] → Rbe such thatu isL−lipschitzian on[a, b], and f is a function of bounded variation on [a, b]. Denote by Wb

af the total variation off on[a, b]. Then the following inequality holds:

(1.6)

Z b

a

u(x)df(x)−f(b)−f(a) b−a ·

Z b

a

u(x)dx 6 1

2L(b−a)

b

_

a

f.

The constant ¹₂ is sharp, in the sense that it cannot be replaced by a smaller one.

Remark 1.1. For other related results see [3].

Let us also state that the weighted version of (1.1) is well known, that is we have with condition (1.2) the following generalization of (1.1):

(1.7) |D(f, g;w)|6 1

4(Φ−ϕ) (Γ−γ), where

D(f, g;w) = A(f, g;w)−A(f;w)A(g;w),

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of13

and

A(f;w) = Rb

a w(x)f(x)dx Rb

a w(x)dx .

So, in this paper we shall show that corresponding weighted versions of (1.5) and (1.6) are also valid. Some related results will be also given.

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of13

2. Results

Theorem 2.1. Let f, u : [a, b] → Rbe such that f is Riemann integrable on [a, b]anduisL−Lipschitzian on[a, b], i.e. (1.3) holds true. Ifw: [a, b]→Ris a positive weight function, then

(2.1) |T (f, u;w)|6L Z b

a

w(x)|f(x)−A(f;w)|dx, where

(2.2) T(f, u;w) = Z b

a

w(x)f(x)du(x)

− 1

Rb

a w(x)dx Z b

a

w(x)du(x) Z b

a

w(x)f(x)dx.

Moreover, if there exist the real numbersm, M such that (1.4) is valid, then

(2.3) |T(f, u;w)|6 L

2 (M−m) Z b

a

w(x)dx.

Proof. As in [1], we have

|T (f, u;w)|=

Z b

a

w(x) [f(x)−A(f;w)]du(x)

6L Z b

a

w(x)|f(x)−A(f;w)|dx.

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of13

That is, (2.1) is valid. Furthermore, from an application of Cauchy’s inequality we have:

(2.4) |T (f, u;w)|6L Z b

a

w(x)dx Z b

a

w(x) (f(x)−A(f;w))²dx

1 2

,

from where we obtain

(2.5) |T (f, u;w)|6L·(D(f, f;w))¹² · Z b

a

w(x)dx.

From (1.7) forg ≡f we get:

(2.6) (D(f, f;w))¹² 6 1

2(Φ−ϕ). Now, (2.4) and (2.5) give (2.3).

Now, we shall prove the following result.

Theorem 2.2. Letf : [a, b]→RbeM−Lipschitzian on[a, b]andu: [a, b]→ R beL−Lipschitzian on[a, b]. Ifw : [a, b] → Ris a positive weight function, then

(2.7) |T (f, u;w)|6L·M · Rb

a

Rb

aw(x)w(x)|x−y|dxdy Rb

a w(y)dy .

Proof. It follows from (2.1)

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of13

|T (f, u;w)|6L· Z b

a

w(x)

Rb

aw(y) (f(x)−f(y))dy Rb

aw(y)dy

dx

6L· Z b

a

w(x) Rb

aw(y)|f(x)−f(y)|dy Rb

a w(y)dy dx

6L·M · Rb

a

Rb

a w(x)w(x)|x−y|dxdy Rb

aw(y)dy .

If in the previous result we setw(x)≡ 1, then we can obtain the following corollary:

Corollary 2.3. Letf andube as in Theorem2.2, then,

Z b

a

f(x)du(x)− u(b)−u(a) b−a

Z b

a

f(t)dt

6 L·M·(b−a)²

3 .

Proof. The proof follows by the fact that Z b

a

Z b

a

|x−y|dxdy = Z b

a

Z b

a

|x−y|dx

dy

= Z b

a

Z y

a

(y−x)dx+ Z b

y

(x−y)dx

dy

= 1 2

Z b

a

(y−a)²+ (b−y)²

dy = 1

3(b−a)³.

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of13

Theorem 2.4. Letf, u : [a, b] →Rbe such that uisL−Lipschitzian on [a, b], andfis a function of bounded variation on[a, b]. Ifw: [a, b]→Ris a positive weight function, then the following inequality holds:

|T(u, f;w)|6M L

b

_

a

g 6W M L

b

_

a

f,

where T (u, f;w) is defined by (2.2), g : [a, b] → R is the function g(x) = Rx

a w(t)df(t),

W = sup_x∈[a,b]w(x), M = max ( Rb

aw(t) (b−t)dt Rb

aw(t)dt , Rb

a w(t) (t−a)dt Rb

a w(t)dt )

,

andWb

ag andWb

af denote the total variation ofg andf on[a, b], respectively.

Proof. We have

T(u, f;w)

= Z b

a

w(x)u(x)df(x)− 1 Rb

a w(x)dx Z b

a

w(x)df(x) Z b

a

w(x)u(x)dx

= Z b

a

w(x) u(x)− Rb

a w(t)u(t)dt Rb

a w(t)dt

! df(x)

= Z b

a

Rb

a w(t) (u(x)−u(t))dt Rb

a w(t)dt

!

w(x)df(x).

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of13

Using the fact thatuisL−Lipschitzian on[a, b], we can state that:

|T (u, f;w)|=

Z b

a

Rb

aw(t) (u(x)−u(t))dt Rb

aw(t)dt

!

w(x)df(x)

=

Z b

a

Rb

aw(t) (u(x)−u(t))dt Rb

aw(t)dt

! d

Z x

a

w(t)df(t)

6Lsup_x∈[a,b]

Rb

a w(t)|x−t|dt Rb

a w(t)dt

! _b _

a

Z x

a

w(t)df(t)

=M L

b

_

a

g.

The constantM has the value M = sup_x∈[a,b]

Rb

aw(t)|x−t|dt Rb

a w(t)dt

! .

If we denote a new functiony(x)as:

y(x) = Z b

a

w(t)|x−t|dt

= Z x

a

w(t) (x−t)dt+ Z b

x

w(t) (t−x)dt,

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of13

then the first derivative of this function is:

dy dx = d

dx

x Z x

a

w(t)tdt− Z x

a

tw(t)dt+ Z b

x

w(t)tdt−x Z b

x

w(t)dt

= Z x

a

w(t)dt+w(x)x−w(x)x−w(x)x− Z b

x

w(t)dt+w(x)x

= Z x

a

w(t)dt− Z b

x

w(t)dt;

and the second derivative is:

d²y

dx² =w(x) +w(x) = 2w(x)>0.

Obviouslyf is a convex function, so we have:

M = sup_x∈[a,b]

Rb

aw(t)|x−t|dt Rb

a w(t)dt

!

= sup_x∈[a,b] y(x) Rb

aw(t)dt

!

= max ( Rb

a w(t) (b−t)dt Rb

a w(t)dt , Rb

a w(t) (t−a)dt Rb

a w(t)dt )

.

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of13

That is:

|T(u, f;w)|=

Z b

a

Rb

aw(t)dt

!

w(x)df(x)

=

Z b

a

Rb

aw(t)dt

!

w(x)df(x)

6 Z b

a

Rb

aw(t)|u(x)−u(t)|dt Rb

aw(t)dt w(x)|df(x)|

6sup_x∈[a,b]w(x)Lsup_x∈[a,b]

Rb

a w(t)|x−t|dt Rb

a w(t)dt

! _b _

a

f

=W M L

b

_

a

f.

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of13

References

[1] S.S. DRAGOMIR AND I. FEDOTOV, An inequality of Grüss type for Riemann-Stieltjes integral and applications for special means, Tamkang J.

of Math., 29(4) (1998), 287–292.

[2] S.S. DRAGOMIR AND I. FEDOTOV, A Grüss type inequality for map- ping of bounded variation and applications to numerical analysis integral and applications for special means, RGMIA Res. Rep. Coll., 2(4) (1999).

[ONLINE:http://rgmia.vu.edu.au/v2n4.html].

[3] S.S. DRAGOMIR, New inequalities of Grüss type for the Stieltjes inte- gral, RGMIA Res. Rep. Coll., 5(4) (2002), Article 3. [ONLINE: http:

//rgmia.vu.edu.au/v5n4.html].