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volume 7, issue 2, article 58, 2006.

Received 10 November, 2005;

accepted 16 November, 2005.

Communicated by:I. Gavrea

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Journal of Inequalities in Pure and Applied Mathematics

ON ˇCEBYŠEV TYPE INEQUALITIES INVOLVING FUNCTIONS WHOSE DERIVATIVES BELONG TO L_p SPACES

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 340-05

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On ˇCebyšev Type Inequalities Involving Functions Whose

Derivatives Belong toLp

Spaces B.G. Pachpatte

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J. Ineq. Pure and Appl. Math. 7(2) Art. 58, 2006

http://jipam.vu.edu.au

Abstract

In this note we establish new ˇCebyšev type integral inequalities involving functions whose derivatives belong toLpspaces via certain integral identities.

2000 Mathematics Subject Classification:26D15, 26D20.

Key words: ˇCebyšev type inequalities,Lpspaces, Integral identities, Absolutely continuous functions, Hölder’s integral inequality.

1. Introduction

One of the classical and important inequalities discovered by P.L. ˇCebyšev [1]

is the following integral inequality (see also [10, p. 207]):

(1.1) |T (f, g)| ≤ 1

12(b−a)²kf⁰k_∞kg⁰k_∞,

wheref, g : [a, b] → Rare absolutely continuous functions whose derivatives f⁰, g⁰ ∈L∞[a, b]and

(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

, which is called the ˇCebyšev functional, provided the integrals in (1.2) exist.

Because of fundamental importance of (1.1) in analysis and applications, many researchers have given considerable attention to it and a number of exten- sions, generalizations and variants have appeared in the literature, see [5], [6], [8] – [10] and the references given therein. The main purpose of the present note is to establish new inequalities similar to the inequality (1.1) involving functions whose derivatives belong toL_p spaces. The analysis used in the proofs is based on the integral identities proved in [3] and [2] and our results provide new esti- mates on these types of inequalities.

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2. Statement of Results

In what follows Rand ⁰ denote the set of real numbers and the derivative of a function. Let [a, b] ⊂ R, a < b; and as usual for any function h ∈ L_p[a, b], p > 1we define khk_p =

Rb

a |h(t)|^pdt¹_p

.We use the following notations to simplify the details of presentation. For suitable functionsf, g : [a, b]→ Rwe set

F = 1 3

f(a) +f(b)

2 + 2f

a+b 2

, G= 1

3

g(a) +g(b)

2 + 2g

a+b 2

,

S(f, g) = F G− 1 b−a

F

Z b

a

g(x)dx+G Z b

a

f(x)dx

+ 1

b−a Z b

a

f(x)dx 1 b−a

Z b

a

g(x)dx

,

H(f, g) = 1 b−a

Z b

a

[F g(x) +Gf(x)]dx

−2 1

b−a Z b

a

f(x)dx 1 b−a

Z b

a

g(x)dx

. Now, we state our main results as follows.

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Theorem 2.1. Let f, g : [a, b] → Rbe absolutely continuous functions whose derivativesf⁰, g⁰ ∈Lp[a, b],p >1. Then we have the inequalities

(2.1) |T(f, g)| ≤ 1

(b−a)³kf⁰k_pkg⁰k_p Z b

a

(B(x))²^q dx,

(2.2) |T (f, g)| ≤ 1 2 (b−a)²

Z b

a

h|g(x)| kf⁰k_p+|f(x)| kg⁰k_pi

(B(x))¹^q dx, where

(2.3) B(x) = (x−a)^q+1+ (b−x)^q+1

q+ 1 ,

forx∈[a, b]and ¹_p +¹_q = 1.

The following variants of the inequalities in (2.1) and (2.2) hold.

Theorem 2.2. Let f, g : [a, b] → Rbe absolutely continuous functions whose derivativesf⁰, g⁰ ∈L_p[a, b],p >1. Then we have the inequalities

(2.4) |S(f, g)| ≤ 1

(b−a)²M

2q

kf⁰k_pkg⁰k_p,

(2.5) |H(f, g)| ≤ 1 (b−a)²M

1q

Z b

a

h|g(x)| kf⁰k_p+|f(x)| kg⁰k_pi dx,

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where

(2.6) M = (2^q+1+ 1) (b−a)^q+1

3 (q+ 1) 6^q , and ¹_p +¹_q = 1.

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3. Proof of Theorem 2.1

From the hypotheses we have the following identities (see [3,7]):

(3.1) f(x)− 1 b−a

Z b

a

f(t)dt= 1 b−a

Z b

a

k(x, t)f⁰(t)dt,

(3.2) g(x)− 1 b−a

Z b

a

g(t)dt = 1 b−a

Z b

a

k(x, t)g⁰(t)dt, forx∈[a, b],where

(3.3) k(x, t) =

( t−a if t∈[a, x]

t−b if t ∈(x, b]

.

Multiplying the left sides and right sides of (3.1) and (3.2) we have (3.4) f(x)g(x)−f(x)

1 b−a

Z b

a

g(t)dt

−g(x) 1

b−a Z b

a

f(t)dt

+ 1

b−a Z b

a

f(t)dt 1 b−a

Z b

a

g(t)dt

= 1

(b−a)² Z b

a

k(x, t)f⁰(t)dt

Z b

a

k(x, t)g⁰(t)dt

.

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Integrating both sides of (3.4) with respect to x over [a, b] and dividing both sides of the resulting identity by(b−a)we get

(3.5) T(f, g)

= 1

(b−a)³ Z b

a

Z b

a

k(x, t)f⁰(t)dt

Z b

a

k(x, t)g⁰(t)dt

dx.

From (3.5) and using the properties of modulus and Hölder’s integral inequality we have

|T (f, g)| ≤ 1 (b−a)³

Z b

a

Z b

a

|k(x, t)| |f⁰(t)|dt

Z b

a

|k(x, t)| |g⁰(t)|dt

dx

≤ 1

(b−a)³ Z b

a

Z b

a

|k(x, t)|^q

¹q Z b

a

|f⁰(t)|^pdt ¹p!

×

Z b

a

|k(x, t)|^qdt

1

q Z b

a

|g⁰(t)|^pdt

1 p!

dx

= 1

(b−a)³ kf⁰k_pkg⁰k_p Z b

a

Z b

a

|k(x, t)|^q

1 q!² (3.6) dx.

A simple calculation shows that (see [4]) Z b

a

|k(x, t)|^qdt = Z x

a

|t−a|^qdt+ Z b

x

|t−b|^qdt

= Z x

a

(t−a)^qdt+ Z b

x

(b−t)^qdt

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= (x−a)^q+1+ (b−x)^q+1

q+ 1 =B(x). (3.7)

Using (3.7) in (3.6) we get (2.1).

Multiplying both sides of (3.1) and (3.2) byg(x)andf(x)respectively and adding the resulting identities we get

(3.8) 2f(x)g(x)

−

g(x) 1

b−a Z b

a

f(t)dt

+f(x) 1

b−a Z b

a

g(t)dt

=g(x) 1

b−a Z b

a

k(x, t)f⁰(t)dt

+f(x) 1

b−a Z b

a

k(x, t)g⁰(t)dt

. Integrating both sides of (3.8) with respect to x over [a, b] and rewriting we obtain

(3.9) T(f, g)

= 1

2 (b−a)² Z b

a

g(x)

Z b

a

k(x, t)f⁰(t)dt+f(x) Z b

a

k(x, t)g⁰(t)dt

dx.

From (3.9) and using the properties of modulus, Hölder’s integral inequality and (3.7) we have

|T (f, g)| ≤ 1 2 (b−a)²

Z b

a

|g(x)|

Z b

a

|k(x, t)| |f⁰(t)|dt +|f(x)|

Z b

a

|k(x, t)| |g⁰(t)|dt

dx

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≤ 1

2 (b−a)² Z b

a

"

|g(x)|

Z b

a

|k(x, t)|^qdt

¹_q Z b

a

|f⁰(t)|^pdt ¹_p

+|f(x)|

Z b

a

|k(x, t)|^qdt

1

q Z b

a

|g⁰(t)|^pdt

1 p#

dx

= 1

2 (b−a)² Z b

a

h|g(x)| kf⁰k_p +|f(x)| kg⁰k_piZ b a

|k(x, t)|^qdt ¹q

dx

= 1

2 (b−a)² Z b

a

h|g(x)| kf⁰k_p +|f(x)| kg⁰k_pi

(B(x))¹^q dx.

This is the required inequality in (2.2). The proof is complete.

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4. Proof of Theorem 2.2

From the hypotheses we have the following identities (see [2]):

(4.1) F − 1

b−a Z b

a

f(x)dx= 1 b−a

Z b

a

m(x)f⁰(x)dx,

(4.2) G− 1

b−a Z b

a

g(x)dx= 1 b−a

Z b

a

m(x)g⁰(x)dx, where

(4.3) m(x) =

( x−^5a+b₆ if x∈

a,^a+b₂ x− ^a+5b₆ if x∈_a+b

2 , b . Multiplying the left sides and right sides of (4.1) and (4.2) we get (4.4) S(f, g) = 1

(b−a)² Z b

a

m(x)f⁰(x)dx

Z b

a

m(x)g⁰(x)dx

. From (4.4) and using the properties of modulus and Hölder’s integral inequality, we have

|S(f, g)| ≤ 1 (b−a)²

Z b

a

|m(x)| |f⁰(x)|dx

Z b

a

|m(x)| |g⁰(x)|dx

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≤ 1

(b−a)²

Z b

a

|m(x)|^qdx

¹_q Z b

a

|f⁰(x)|^pdx ¹_p!

×

Z b

a

|m(x)|^qdx

1

q Z b

a

|g⁰(x)|^pdx

1 p!

= 1

(b−a)²

Z b

a

|m(x)|^qdx ¹q!²

kf⁰k_pkg⁰k_p. (4.5)

A simple computation gives (see [2]) Z b

a

|m(x)|^qdx= Z ^a+b₂

a

x− 5a+b 6

q

dx+ Z b

a+b 2

x−a+ 5b 6

q

dx

= Z ^5a+b₆

a

5a+b 6 −x

q

dx+ Z ^a+b₂

5a+b 6

x− 5a+b 6

q

dx +

Z ^a+5b₆

a+b 2

a+ 5b 6 −x

q

dx+ Z b

a+5b 6

x−a+ 5b 6

q

dx

= 1

q+ 1

"

5a+b 6 −a

q+1

+

a+b

2 − 5a+b 6

q+1

+

a+ 5b

6 − a+b 2

q+1

+

b− a+ 5b 6

q+1#

= (2^q+1+ 1) (b−a)^q+1 3 (q+ 1) 6^q =M.

(4.6)

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Using (4.6) in (4.5) we get the required inequality in (2.4).

Multiplying both sides of (4.1) and (4.2) byg(x)andf(x)respectively and adding the resulting identities we get

(4.7) F g(x) +Gf(x)

−

g(x) 1

b−a Z b

a

f(x)dx

+f(x) 1

b−a Z b

a

g(x)dx

=g(x) 1

b−a Z b

a

m(x)f⁰(x)dx

+f(x) 1

b−a Z b

a

m(x)g⁰(x)dx

. Integrating both sides of (4.7) with respect to x over [a, b] and dividing both sides of the resulting identity by(b−a)we get

(4.8) H(f, g)

= 1

(b−a)² Z b

a

g(x)

Z b

a

m(x)f⁰(x)dx+f(x) Z b

a

m(x)g⁰(x)dx

dx.

From (4.8) and using the properties of modulus, Hölder’s integral inequality and (4.6) we have

|H(f, g)| ≤ 1 (b−a)²

Z b

a

|g(x)|

Z b

a

|m(x)| |f⁰(x)|dx +|f(x)|

Z b

a

|m(x)| |g⁰(x)|dx

dx

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≤ 1

(b−a)² Z b

a

"

|g(x)|

Z b

a

|m(x)|^qdx

¹_q Z b

a

|f⁰(x)|^pdx ¹_p

+|f(x)|

Z b

a

|m(x)|^qdx

1

q Z b

a

|g⁰(x)|^pdx

1 p#

dx

= 1

(b−a)² Z b

a

h|g(x)| kf⁰k_p+|f(x)| kg⁰k_piZ b a

|m(x)|^qdx ¹q

= 1

(b−a)²M¹^q Z b

a

h|g(x)| kf⁰k_p+|f(x)| kg⁰k_pi dx.

This is the desired inequality in (2.5). The proof is complete.

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References

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

[2] S.S. DRAGOMIR, On Simpson’s quadrature formula for differentiable mappings whose derivatives belong toLpspaces and applications, RGMIA Res. Rep. Coll., 1(2) (1998), 89–96.

[3] S.S. DRAGOMIRANDN.S. BARNETT, An Ostrowski type inequality for mappings whose second derivatives are bounded and applications, RGMIA Res. Rep. Coll., 1(2) (1998), 69–77.

[4] S.S. DRAGOMIR AND S. WANG, A new inequality of Ostrowski’s type inLp norm, Indian J. Math., 40(3) (1998), 299–304.

[5] H.P. HEINIGANDL. MALIGRANDA, Chebyshev inequality in function spaces, Real Analysis Exchange, 17 (1991-92), 211–247.

[6] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.

[7] 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, 1994.

[8] B.G. PACHPATTE, On Trapezoid and Grüss like integral inequalities, Tamkang J. Math., 34(4) (2003), 365–369.

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[9] B.G. PACHPATTE, On Ostrowski-Grüss- ˇCebyšev type inequalities for functions whose modulus of derivatives are convex, J. Inequal. Pure and Appl. Math., 6(4) (2005), Art. 128. [ONLINE: http://jipam.vu.

edu.au/article.php?sid=602].

[10] J.E. PE ˇCARI ´C, F. PORCHANANDY. TONG, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, San Diego, 1992.