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volume 7, issue 1, article 11, 2006.

Received 15 August, 2005;

accepted 19 January, 2006.

Communicated by:J. Sándor

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Journal of Inequalities in Pure and Applied Mathematics

ON CEBYSEV-GRÜSS TYPE INEQUALITIES VIA PE ˇCARI ´C’S EXTENSION OF THE MONTGOMERY IDENTITY

B.G. PACHPATTE

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

EMail:bgpachpatte@gmail.com

c

2000Victoria University ISSN (electronic): 1443-5756

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On Cebysev-Grüss Type Inequalities via Pecari ´c’s Extension of the Montgomery

Identity B.G. Pachpatte

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J. Ineq. Pure and Appl. Math. 7(1) Art. 11, 2006

Abstract

In the present note we establish new ˇCebyšev-Grüss type inequalities by using Peˇcariˇc’s extension of the Montgomery identity.

2000 Mathematics Subject Classification:26D15, 26D20.

Key words: ˇCebyšev-Grüss type inequalities, Peˇcari´c’s extension, Montgomery identity.

1. Introduction

For two absolutely continuous functions f, g : [a, b] → R consider the func- tional

T (f, g) = 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

,

where the involved integrals exist. In 1882, ˇCebyšev [1] proved that iff⁰, g⁰ ∈ L∞[a, b], then

(1.1) |T (f, g)| ≤ 1

12(b−a)²kf⁰k_∞kg⁰k_∞. In 1935, Grüss [2] showed that

(1.2) |T (f, g)| ≤ 1

4(M −m) (N −n),

provided m, M, n, N are real numbers satisfying the condition −∞ < m ≤ M <∞,−∞< n≤N <∞forx∈[a, b].

Many researchers have given considerable attention to the inequalities (1.1), (1.2) and various generalizations, extensions and variants of these inequalities have appeared in the literature, to mention a few, see [4, 5] and the references cited therein. The aim of this note is to establish two new inequalities similar to those of ˇCebyšev and Grüss inequalities by using Peˇcariˇc’s extension of the

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2. Statement of Results

Letf : [a, b]→Rbe differentiable on[a, b]andf⁰ : [a, b]→Ris integrable on [a, b]. Then the Montgomery identity holds [3]:

(2.1) f(x) = 1

b−a Z b

a

f(t)dt+ Z b

a

P (x, t)f⁰(t)dt, whereP(x, t)is the Peano kernel defined by

(2.2) P (x, t) =

( t−a

b−a, a≤t≤x,

t−b

b−a, x < t≤b.

Let w : [a, b] → [0,∞) be some probability density function, that is, an integrable function satisfying Rb

a w(t)dt = 1, and W(t) = Rt

aw(x)dx for t ∈[a, b],W(t) = 0fort < a, andW(t) = 1fort > b. In [6] Peˇcari´c has given the following weighted extension of the Montgomery identity:

(2.3) f(x) =

Z b

a

w(t)f(t)dt+ Z b

a

P_w(x, t)f⁰(t)dt,

wherePw(x, t)is the weighted Peano kernel defined by

(2.4) Pw(x, t) =

( W(t), a ≤t ≤x, W(t)−1, x < t≤b.

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We use the following notation to simplify the details of presentation. For some suitable functonsw, f, g: [a, b]→R,we set

T (w, f, g) = Z b

a

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

− Z b

a

w(x)f(x)dx

Z b

a

w(x)g(x)dx

,

and define k·k_∞ as the usual Lebesgue norm on L∞[a, b] that is, khk_∞ :=

ess sup

t∈[a,b]

|h(t)|forh∈L∞[a, b].

Our main results are given in the following theorems.

Theorem 2.1. Let f, g : [a, b] → R be differentiable on [a, b] and f⁰, g⁰ : [a, b] → R are integrable on[a, b]. Let w : [a, b] → [0,∞) be an integrable function satisfyingRb

a w(t)dt= 1. Then (2.5) |T (w, f, g)| ≤ kf⁰k_∞kg⁰k_∞

Z b

a

w(x)H²(x)dx, where

(2.6) H(x) =

Z b

a

|P_w(x, t)|dt

forx∈[a, b]andP_w(x, t)is the weighted Peano kernel given by (2.4).

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Theorem 2.2. Letf, g, f⁰, g⁰, w be as in Theorem2.1. Then

(2.7) |T (w, f, g)| ≤ 1 2

Z b

a

w(x) [|g(x)| kf⁰k_∞+|f(x)| kg⁰k_∞]H(x)dx, whereH(x)is defined by (2.6).

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3. Proofs of Theorems 2.1 and 2.2

Proof of Theorem2.1. From the hypotheses the following identities hold [6]:

f(x) = Z b

a

w(t)f(t)dt+ Z b

a

P_w(x, t)f⁰(t)dt, (3.1)

g(x) = Z b

a

w(t)g(t)dt+ Z b

a

P_w(x, t)g⁰(t)dt, (3.2)

From (3.1) and (3.2) we observe that

f(x)− Z b

a

w(t)f(t)dt g(x)− Z b

a

w(t)g(t)dt

= Z b

a

Pw(x, t)f⁰(t)dt Z b

a

Pw(x, t)g⁰(t)dt

,

i.e.,

(3.3) f(x)g(x)−f(x) Z b

a

w(t)g(t)dt−g(x) Z b

a

w(t)f(t)dt

+ Z b

a

w(t)f(t)dt

Z b

a

w(t)g(t)dt

= Z b

a

P_w(x, t)f⁰(t)dt Z b

a

P_w(x, t)g⁰(t)dt

.

Multiplying both sides of (3.3) by w(x) and then integrating both sides of

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Rb

a w(t)dt= 1, we have (3.4) T(w, f, g) =

Z b

a

w(x) Z b

a

P_w(x, t)f⁰(t)dt

× Z b

a

P_w(x, t)g⁰(t)dt

dx.

From (3.4) and using the properties of modulus we observe that

|T (w, f, g)| ≤ Z b

a

w(x) Z b

a

|P_w(x, t)| |f⁰(t)|dt Z b

a

|P_w(x, t)| |g⁰(t)|dt

dx

≤ kf⁰k_∞kg⁰k_∞ Z b

a

w(x)H²(x)dx.

This completes the proof of Theorem2.1.

Proof of Theorem2.2. Multiplying both sides of (3.1) and (3.2) by w(x)g(x) andw(x)f(x), adding the resulting identities and rewriting we have

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

= 1 2

w(x)g(x) Z b

a

w(t)f(t)dt+w(x)f(x) Z b

a

w(t)g(t)dt

+1 2

w(x)g(x) Z b

a

Pw(x, t)f⁰(t)dt

+w(x)f(x) Z b

a

Pw(x, t)g⁰(t)dt

.

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Integrating both sides of (3.5) with respect to x from a tob and rewriting we have

(3.6) T(w, f, g) = 1 2

Z b

a

w(x)g(x) Z b

a

Pw(x, t)f⁰(t)dt

+w(x)f(x) Z b

a

Pw(x, t)g⁰(t)dt

dx.

From (3.6) and using the properties of modulus we observe that

|T(w, f, g)|

≤ 1 2

Z b

a

w(x)

|g(x)|

Z b

a

|P_w(x, t)| |f⁰(t)|dt

+|f(x)|

Z b

a

|P_w(x, t)| |g⁰(t)|dt

dx

≤ 1 2

Z b

a

w(x) [|g(x)| kf⁰(t)k_∞+|f(x)| kg⁰(t)k_∞]H(x)dx.

The proof of Theorem2.2is complete.

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References

[1] P.L. ˇCEBYŠEV, Sue les expressions approxmatives des intégrales définies par les autres prises entre les mêmes limites, Proc. Math. Soc. Charkov, 2 (1882), 93–98.

[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

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

[3] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and Their Integrals and Derivatives, Kluwer Academic Pub- lishers, Dordrecht, 1991.

[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, New weighted multivariate Grüss type inequalities, J.

Inequal. Pure and Appl. Math., 4(5) (2003), Art. 108. [ONLINE: http:

//jipam.vu.edu.au/article.php?sid=349].

[6] J.E. PE ˇCARI ´C, On the ˇCebyšev inequality, Bul. ¸Sti. Tehn. Inst. Politehn.

"Train Vuia" Timi¸sora, 25(39)(1) (1980), 5–9.