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volume 7, issue 3, article 112, 2006.

Received 22 February, 2005;

accepted 04 May, 2005.

Communicated by:A. Sofo

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Journal of Inequalities in Pure and Applied Mathematics

AN OSTROWSKI TYPE INEQUALITY FOR p−NORMS

A. RAFIQ AND NAZIR AHMAD MIR

Center for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University

Multan, Pakistan.

EMail:caspam@bzu.edu.pk

c

2000Victoria University ISSN (electronic): 1443-5756 050-05

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An Ostrowski Type Inequality forp−norms A. Rafiq and Nazir Ahmad Mir

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J. Ineq. Pure and Appl. Math. 7(3) Art. 112, 2006

http://jipam.vu.edu.au

Abstract

In this paper, we establish general form of an inequality of Ostrowski type for twice differentiable mappings in terms ofL_p−norm, with first derivative absolutely continuous. The integral inequality of similar type already pointed out in literature is a special case of ours. The already established inequality contains a mistake and as a result incorrect consequences and applications. The corrected version of the inequality is pointed out and the inequality is also applied to special means and numerical integration.

2000 Mathematics Subject Classification:26D15.

Key words: Ostrowski inequality, Numerical integration, Special means.

1. Introduction

We establish here the general form of an inequality of Ostrowski type, differ- ent to that of Cerone, Dragomir and Roumeliotis [1], for twice differentiable mappings in terms ofL_p−norm. The integral inequality of similar type already pointed out by N.S. Barnett, P. Cerone, S.S. Dragomir, J. Roumeliotis and A.

Sofo [2], contains a mistake which has already been reported by N.A. Mir and A. Rafiq in their research work [3]. The same mistake has been carried out in their other research article, namely Theorem 20 of [2] and as a result incorrect consequences and applications of this theorem. The corrected form of the theorem is as follows:

Theorem 1.1. Let g : [a, b] −→ R be a mapping whose first derivative is absolutely continuous on [a, b]. If we assume that the second derivativeg⁰⁰ ∈ L_p(a, b),1< p <∞,then we have the inequality

(1.1)

Z b a

g(t)dt− 1 2

g(x) + g(a) +g(b) 2

(b−a) +1

2(b−a)

x−a+b 2

g⁰(x)

≤ 1 2

b−a 2

2+¹_q

kg⁰⁰k_p

×







[B(q+ 1, q+ 1) +B_x₁(q+ 1, q+ 1) + Ψ_x₂(q+ 1, q+ 1)]¹^q forx∈

a,^a+b₂ , [B(q+ 1, q+ 1) +B_x₃(q+ 1, q+ 1) +B_x₄(q+ 1, q+ 1)]¹^q forx∈(^a+b₂ , b], where ¹_p +¹_q = 1, p > 1, q > 1,andB(·,·)is the Beta function of Euler given

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by

B(l, s) = Z 1

0

t^l−1(1−t)^s−1dt, l, s >0.

Further

B_r(l, s) = Z r

0

t^l−1(1−t)^s−1dt is the incomplete Beta function,

Ψr(l, s) = Z r

0

t^l−1(1 +t)^s−1dt is the real positive valued integral,

x₁ = 2(x−a)

b−a , x₂ = 1−x₁, x₃ =x₁−1, x₄ = 2−x₁ and

kg⁰⁰k_p :=

Z b a

|g⁰⁰(t)|^pdt ¹_p

. If we assume thatg⁰⁰ ∈L₁(a, b),then we have

(1.2)

Z b a

g(t)dt− 1 2

g(x) + g(a) +g(b) 2

(b−a) +1

2(b−a)

x− a+b 2

g⁰(x)

≤ kg⁰⁰k₁

8 (b−a)², where

kg⁰⁰k₁ :=

Z b a

|g⁰⁰(t)|dt.

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2. Main Results

The following theorem is now proved and subsequently applied to numerical integration and special means.

Theorem 2.1. Let g : [a, b] −→ R be a mapping whose first derivative is absolutely continuous on [a, b]. If we assume that the second derivative g⁰⁰ ∈ Lp(a, b),1< p <∞,then we have the inequality

(2.1)

1 α+β

α x−a

Z x a

g(t)dt+ β b−x

Z b x

g(t)dt

− 1

2g(x)− 1 2(α+β)

x− a+b 2

g(x)

α

x−a − β b−x

+ (b−a) 2

α

x−ag(a) + β b−xg(b)

−(α+β)

x− a+b 2

g⁰(x)

≤

b−a 2

2+¹_q

kg⁰⁰k_p

×









 h β

α+β 1 b−x

q

B(q+ 1, q+ 1) + α

α+β 1 x−a

q

Bx1(q+ 1, q+ 1) +

β α+β

1 b−x

q

Ψ_x₂(q+ 1, q+ 1)i¹_q

forx∈ a,^a+b₂

, h α

α+β 1 x−a

q

B(q+ 1, q+ 1) + α

α+β 1 x−a

q

Bx3(q+ 1, q+ 1) +

β α+β

1 b−x

q

B_x₄(q+ 1, q+ 1)i¹_q

forx∈ ^a+b₂ , b ,

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where ¹_p +¹_q = 1, p > 1, q > 1,andB(·,·)is the Beta function of Euler given by

B(l, s) = Z 1

0

t^l−1(1−t)^s−1dt, l, s >0.

Further,

B_r(l, s) = Z r

0

t^l−1(1−t)^s−1dt is the incomplete Beta function,

Ψ_r(l, s) = Z r

0

t^l−1(1 +t)^s−1dt is a real positive valued integral,

x₁ = 2(x−a)

b−a , x₂ = 1−x₁, x₃ =x₁−1, x₄ = 2−x₁ and

kg⁰⁰k_p :=

Z b a

|g⁰⁰(t)|^pdt ¹p

. If we assume thatg⁰⁰ ∈L₁(a, b),then we have

(2.2)

1 α+β

α x−a

Z x a

g(t)dt+ β b−x

Z b x

g(t)dt

−1 2g(x)

− 1 2(α+β)

x− a+b 2

g(x)

α

x−a − β b−x

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+(b−a) 2

α

−(α+β)

x− a+b 2

g⁰(x)

≤ 1

2kg⁰⁰k₁kK(x, t)k_∞, where

kg⁰⁰k₁ = Z b

a

|g⁰⁰(t)|dt, and

kK(x, t)k_∞ = 1

α+β max α

x−a, β b−x

(b−a)²

4 forx∈[a, b].

Proof. We begin by recalling the following integral equality proved by N.A.

Mir and A. Rafiq [3] which is generalization of an integral equality proved by Dragomir and Wang [4].

(2.3)

1 α+β

α x−a

Z x a

g(t)dt+ β b−x

Z b x

g(t)dt

−1 2g(x)

− 1 2(α+β)

x− a+b 2

g(x)

α

x−a − β b−x

+(b−a) 2

α

−(α+β)

x− a+b 2

g⁰(x)

= 1 2

Z b a

p(x, t)

t− a+b 2

g⁰⁰(t)dt

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whose left hand side is equivalent to that of (2.1). From the right hand side of (2.3) we have, by Hölder’s inequality, that

Z b a

p(x, t)

t− a+b 2

g⁰⁰(t)dt

≤ Z b

a

|g⁰⁰(t)|^pdt

¹p Z b a

|p(x, t)|^q

t−a+b 2

q

dt ¹q

=kg⁰⁰k_p Z b

a

|p(x, t)|^q

t− a+b 2

q

dt ¹q

, and from (2.3) we obtain the inequality

(2.4)

1 α+β

α x−a

Z x a

g(t)dt+ β b−x

Z b x

g(t)dt

− 1

2g(x)− 1 2(α+β)

x− a+b 2

g(x)

α

x−a − β b−x

+ (b−a) 2

α

−(α+β)

x− a+b 2

g⁰(x)

≤ 1 2kg⁰⁰k_p

Z b a

|p(x, t)|^q

t− a+b 2

q

dt ¹_q

.

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From the right hand side of (2.4) we may define I :=

Z b a

|p(x, t)|^q

t− a+b 2

q

dt

= α

α+β · 1 x−a

qZ x a

(t−a)^q

t− a+b 2

q

dt +

β

α+β · 1 b−x

qZ b x

|t−b|^q

t− a+b 2

q

dt (2.5)

such that we can identify two distinct cases.

(a) Forx∈ a,^a+b₂ I_A=

α α+β

1 x−a

qZ x a

(t−a)^q

a+b 2 −t

q

dt +

β α+β

1 b−x

qZ ^a+b₂

x

(b−t)^q

a+b 2 −t

q

dt +

β α+β

1 b−x

qZ b

a+b 2

(b−t)^q

t− a+b 2

q

dt.

Investigating the three separate integrals, we may evaluate as follows:

I₁ = Z x

a

(t−a)^q

a+b 2 −t

q

dt,

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making the change of variablet=a+ ^b−a₂

w,we arrive at I₁ =

b−a 2

2q+1Z x1

0

w^q(1−w)^qdw,

=

b−a 2

2q+1

Bx₁(q+ 1, q+ 1), whereB_x₁(·,·)is the incomplete Beta function andx₁ = ^2(x−a)_b−a .

I₂ = Z ^a+b₂

x

(b−t)^q

a+b 2 −t

q

dt, making the change of variablet= ^a+b₂ − ^b−a₂

w,we obtain I₂ =

b−a 2

2q+1Z x2

0

w^q(1 +w)^qdw=

b−a 2

2q+1

Ψ_x₂(q+ 1, q+ 1), where

Ψ_x₂ :=

Z x2

0

w^q(1 +w)^qdw andx₂ = ^a+b−2x_b−a = 1−x_1.

I₃ = Z b

a+b 2

(b−t)^q

t−a+b 2

q

dt, making the change of variablet= ^a+b₂ + ^b−a₂

w,we get I₃ =

b−a 2

2q+1Z 1 0

w^q(1−w)^qdw=

b−a 2

2q+1

B(q+ 1, q+ 1),

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whereB(·,·)is the Beta function.

We may now write IA =I1+I2+I3

=

b−a 2

2q+1 α α+β

1 x−a

q

Bx₁(q+ 1, q+ 1) +

β α+β

1 b−x

q

Ψx2(q+ 1, q+ 1) +

β α+β

1 b−x

q

B(q+ 1, q+ 1)

forx∈ a,^a+b₂

.

(b) Forx∈ a,^a+b₂ I_B =

α α+β

1 x−a

qZ ^a+b₂

a

(t−a)^q

a+b 2 −t

q

dt +

α α+β

1 x−a

qZ x

a+b 2

(t−a)^q

t− a+b 2

q

dt +

β α+β

1 b−x

qZ b x

(b−t)^q

t− a+b 2

q

dt.

In a similar fashion to the previous case, we have I₄ =

Z ^a+b₂

a

(t−a)^q

a+b 2 −t

q

dt.

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Lettingt=a+ ^b−a₂

w,we obtain I₄ =

b−a 2

2q+1Z 1 0

w^q(1−w)^qdw=

b−a 2

2q+1

B(q+ 1, q+ 1), whereB(·,·)is the Beta function.

I₅ = Z x

a+b 2

(t−a)^q

t−a+b 2

q

dt, making the change of variablet= ^a+b₂ + ^b−a₂

w,we arrive at I5 =

b−a 2

2q+1Z x3

0

w^q(1−w)^qdw=

b−a 2

2q+1

Bx3(q+ 1, q+ 1), whereB_x₃(·,·)is the incomplete Beta function andx₃ =x₁−1.

I₆ = Z b

x

(b−t)^q

t− a+b 2

q

dt, making the change of variablet=b− ^b−a₂

w,we get I₆ =

b−a 2

2q+1Z x4

0

w^q(1−w)^qdw=

b−a 2

2q+1

B_x₄(q+ 1, q+ 1),

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whereB_x₄(·,·)is the incomplete Beta function andx₄ = 2−x₁. I_B =I₄+I₅+I₆

=

b−a 2

2q+1 α α+β

1 x−a

q

B(q+ 1, q+ 1) +

α α+β

1 x−a

q

B_x₃(q+ 1, q+ 1) +

β α+β

1 b−x

q

B_x₄(q+ 1, q+ 1)

forx∈(^a+b₂ , b].

Also from (2.5) I =I_A+I_B

=

b−a 2

2q+1









 α

α+β 1 x−a

q

B_x₁(q+ 1, q+ 1) +

β α+β

1 b−x

q

Ψ_x₂(q+ 1, q+ 1) +

β α+β

1 b−x

q

B(q+ 1, q+ 1)forx∈ a,^a+b₂

, α

α+β 1 x−a

q

B(q+ 1, q+ 1) +

α α+β

1 x−a

q

B_x₃(q+ 1, q+ 1) +

β α+β

1 b−x

q

B_x₄(q+ 1, q+ 1)forx∈_a+b

2 , b .

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Using (2.4), we obtain the result (2.1). Using the inequality (2.3), we can also state that

1 α+β

α x−a

Z x a

g(t)dt+ β b−x

Z b x

g(t)dt

− 1

2g(x)− 1 2(α+β)

x− a+b 2

g(x)

α

x−a − β b−x

+ (b−a) 2

α

−(α+β)

x− a+b 2

g⁰(x)

≤ 1

2kg⁰⁰k₁kK(x, t)k_∞, where

kK(x, t)k_∞ =p(x, t)

t−a+b 2

. As it is easy to see that

kK(x, t)k_∞ = 1

α+β ·max α

x−a, β b−x

· (b−a)²

4 forx∈[a, b], we deduce (2.2).

Remark 1. Putting α = x−a andβ = b −xin (2.1) and (2.2), we get the inequalities (1.1) and (1.2).

Remark 2. Simple manipulation of (2.1) will allow for the corrected result of (1.1) and (1.2), owing to a missing factor of ¹₂ in the third term of the original result (1.1) of the Barnett, Cerone, Dragomir, Roumeliotis and Sofo, this will not be done here.

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References

[1] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, An inequality of Ostrowski-Grüss type for twice differentiable mappings and applicatios in numerical integration, Kyungpook Mathematical Journal, 39(2) (1999), 331–341.

[2] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR, J. ROUMELIOTISAND

A. SOFO, A survey on Ostrowski type inequalities for twice differentiable mappings and applications, Inequality Theory and Applications, 1 (2001), 33–86.

[3] N.A. MIR AND A. RAFIQ, An integral inequality for twice differentiable bounded mappings with first derivative absolutely continuous and applications, submitted.

[4] S.S. DRAGOMIR ANDS. WANG, Applications of Ostrowski’s inequality for the estimation of error bounds for some special means and some numer- ical quadrature rules, Appl. Math. Lett., 11 (1998), 105–109.