ON GENERALIZATION OF ˇCEBYŠEV TYPE INEQUALITIES

(1)

Generalization of ˇCebyšev Type Inequalities K. Boukerrioua and A. Guezane-Lakoud vol. 8, iss. 2, art. 55, 2007

Title Page

Contents

JJ II

J I

Page1of 9 Go Back Full Screen

Close

ON GENERALIZATION OF ˇ CEBYŠEV TYPE INEQUALITIES

K. BOUKERRIOUA AND A. GUEZANE-LAKOUD

Department of Mathematics University of Guelma Guelma, Algeria

EMail:khaledV2004@yahoo.fr and a_guezane@yahoo.fr

Received: 19 April, 2006

Accepted: 24 January, 2007 Communicated by: N.S. Barnett 2000 AMS Sub. Class.: Primary 30C45.

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

Abstract: A generalization of Peˇcari´c’s extension of Montgomery’s identity is established and used to derive new ˇCebyšev type inequalities.

(2)

Title Page Contents

JJ II

J I

Close

1. Introduction

In the present work we establish a generalization of Peˇcari´c’s extension of ‘Mont- gomery’s’ identity and use it to derive new ˇCebyšev type inequalities.

We recall the ˇCebyšev inequality [1], given by the following:

(1.1) |T (f, g)| ≤ 1

12(b−a)²kf⁰k_∞kg⁰k_∞,

wheref, g : [a, b] → Rare absolutely continuous functions, whose first derivatives f⁰ andg⁰ are bounded,

(1.2) 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

andk·k_∞denotes the norm inL∞[a, b]defined askpk_∞ =esssup

t∈[a,b]

|p(t)|.

Pachpatte in [6] established new inequalities of the ˇCebyšev type by using Peˇcari˙c’s extension of the Montgomery identity [7].

(4)

Title Page Contents

JJ II

J I

Close

2. Statement of Results

From [3], if f : [a, b] → R is differentiable on [a, b]with the first derivative f⁰(t) integrable on[a, b],then the Montgomery identity holds:

(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

P (x, t) =









 t−a

b−a, a≤t≤x t−b

b−a, x < t≤b.

We assume that w : [a, b] → [0,+∞[ is some probability density function, i.e.

Rb

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

aw(x)dx fora ≤ t ≤ b, W(t) = 0 fort < a and fort > b.We then have the following identity given by Peˇcari´c in [7], that is the weighted generalization of the Montgomery identity:

(2.2) f(x) =

Z b a

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

a

P_w(x, t)f⁰(t)dt, where the weighted Peano kernelP_w is:

(2.3) Pw(x, t) =

( W(t), a≤t ≤x W(t)−1, x < t≤b

Letϕ : [0,1] → Rbe a differentiable function on[0,1], withϕ(0) = 0, ϕ(1) 6= 0 andϕ⁰ integrable on[0,1]. To simplify the notation, for some given functionsw, f,

(5)

Title Page Contents

JJ II

J I

Close

g : [a, b]→R,we set (2.4) T (w, f, g, ϕ⁰) =

Z b a

w(x)ϕ⁰ Z x

a

w(t)dt

f(x)g(x)dx

− 1 ϕ(1)

Z b a

w(x)ϕ⁰ Z x

a

w(t)dt

f(x)dx

× Z b

a

w(x)ϕ⁰ Z x

a

w(t)dt

g(x)dx

. Theorem 2.1. Letf : [a, b] → R be differentiable and f⁰(t) integrable on [a, b], then,

(2.5) f(x) = 1 ϕ(1)

Z b a

w(t)ϕ⁰ Z t

a

w(s)ds

f(t)dt

+ 1

ϕ(1) Z b

a

P_w,ϕ(x, t)f⁰(t)dt, whereP_w,ϕis a generalization of the weighted Peano kernel defined by:

(2.6) P_w,ϕ(x, t) =

( ϕ(W(t)), a≤t ≤x;

ϕ(W(t))−ϕ(1), x < t≤b.

Proof. Using the hypothesis onϕ, Z b

a

P_w,ϕ(x, t)f⁰(t)dt (2.7)

= Z x

a

ϕ(W(t))f⁰(t)dt+ Z b

x

(ϕ(W(t))−ϕ(1))f⁰(t)dt

(6)

Title Page Contents

JJ II

J I

Close

= Z b

a

ϕ(W(t))f⁰(t)dt−ϕ(1) Z b

x

f⁰(t)dt

= [ϕ(W(t))f(t)]^b_a− Z b

a

w(t)ϕ⁰(W(t))f(t)dt−ϕ(1) [f(b)−f(x)]

=ϕ(1)f(x)− Z b

a

w(t)ϕ⁰ Z t

a

w(s)ds

f(t)dt.

Multiplying both sides by1/ϕ(1), we obtain, f(x) = 1

ϕ(1) Z b

a

w(t)ϕ⁰ Z t

a

w(s)ds

f(t)dt+ 1 ϕ(1)

Z b a

Pw,ϕ(x, t)f⁰(t)dt and this completes the proof.

Theorem 2.2. Letf, g[a, b]→ Rbe differentiable on[a, b]andf⁰, g⁰be integrable on[a, b]and letw, ϕbe as in Theorem2.1, then,

|T(w, f, g, ϕ⁰)| ≤ 1

ϕ²(1) kf⁰k_∞kg⁰k_∞kϕ⁰k_∞ Z b

a

w(x)H²(x)dx, whereH(x) = Rb

a |P_w,ϕ(x, t)|dtandkϕ⁰k_∞=esssup

t∈[0,1]

|ϕ⁰(t)|.

Since the functionsf andg satisfy the hypothesis of Theorem2.1, the following identities hold:

(2.8) f(x) = 1 ϕ(1)

Z b a

w(t)ϕ⁰ Z t

a

w(s)ds

f(t)dt

+ 1

ϕ(1) Z b

a

P_w,ϕ(x, t)f⁰(t)dt

(7)

Title Page Contents

JJ II

J I

Close

and

(2.9) g(x) = 1 ϕ(1)

Z b a

w(t)ϕ⁰ Z t

a

w(s)ds

g(t)dt

+ 1

ϕ(1) Z b

a

P_w,ϕ(x, t)g⁰(t)dt. Using (2.8) and (2.9) we obtain,

f(x)− 1 ϕ(1)

Z b a

w(t)ϕ⁰ Z t

a

w(s)ds

f(t)dt

×

g(x)− 1 ϕ(1)

Z b a

w(t)ϕ⁰ Z t

a

w(s)ds

g(t)dt

= 1

ϕ²(1) Z b

a

Pw,ϕ(x, t)f⁰(t)dt Z b

a

Pw,ϕ(x, t)g⁰(t)dt

. Consequently,

(2.10) f(x)g(x)− 1

ϕ(1)f(x) Z b

a

w(t)ϕ⁰ Z t

a

w(s)ds

g(t)dt

− 1

ϕ(1)g(x) Z b

a

w(t)ϕ⁰ Z t

a

w(s)ds

f(t)dt

+ 1

ϕ²(1) Z b

a

w(t)ϕ⁰ Z t

a

w(s)ds

f(t)dt

× Z b

a

w(t)ϕ⁰ Z t

a

w(s)ds

g(t)dt

(8)

Title Page Contents

JJ II

J I

Close

= 1

ϕ²(1) Z b

a

P_w,ϕ(x, t)f⁰(t)dt Z b

a

P_w,ϕ(x, t)g⁰(t)dt

. Multiplying both sides of (2.10) byw(x)ϕ⁰ Rx

a w(s)ds

and then integrating the resultant identity with respect toxfromatob, we get,

(2.11) T (w, f, g, ϕ⁰) = 1 ϕ²(1)

Z b a

w(x)ϕ⁰ Z x

a

w(t)dt

× Z b

a

P_w,ϕ(x, t)f⁰(t)dt Z b

a

P_w,ϕ(x, t)g⁰(t)dt

dx.

Finally,

|T(w, f, g, ϕ⁰)| ≤ 1

ϕ²(1) kf⁰k_∞kg⁰k_∞kϕ⁰k_∞ Z b

a

w(x)H²(x)dx.

(9)

Title Page Contents

JJ II

J I

Close

References

[1] P.L. ˇCEBYŠEV, Sur les expressions approximatives 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

af(x)dxRb

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

[3] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C AND A.M. FINK, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.

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

[5] B.G. PACHPATTE, New weighted multivariable 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] B.G. PACHPATTE, On ˇCebysev-Grüss type inequalities via Peˇcari´c’s extention of the Montgomery identity, J. Inequal. Pure and Appl. Math., 7(1) (2006), Art 108. [ONLINE: http://jipam.vu.edu.au/article.php?sid=

624].

[7] J.E. PE ˇCARI ´C, On the ˇCebysev inequality, Bul. Sti. Tehn. Inst. Politehn. "Tralan Vuia" Timi¸sora (Romania), 25(39) (1) (1980), 5–9.