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volume 7, issue 4, article 124, 2006.

Received 15 September, 2005;

accepted 10 May, 2006.

Communicated by:N.S. Barnett

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Journal of Inequalities in Pure and Applied Mathematics

A GENERALIZED OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE MAPPINGS AND APPLICATIONS

A. RAFIQ, N.A. MIR AND FIZA ZAFAR

COMSATS Institute of Information Technology Islamabad, Pakistan

EMail:arafiq@comsats.edu.pk EMail:namir@comsats.edu.pk

CASPAM, B. Z. University Multan, Pakistan

EMail:fizazafar@gmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 275-05

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A Generalized Ostrowski-Grüss Type Inequality for Twice Differentiable Mappings and

Applications A. Rafiq, N.A. Mir and Fiza Zafar

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Abstract

A generalized Ostrowski type inequality for twice differentiable mappings in terms of the upper and lower bounds of the second derivative is established.

The inequality is applied to numerical integration.

2000 Mathematics Subject Classification:26D15.

Key words: Ostrowski inequality, Grüss inequality.

The authors express thanks to Prof. N.S. Barnett for giving valuable suggestions during the preparation of this manuscript.

1. Introduction

The integral inequality that establishes a connection between the integral of the product of two functions and the product of the integrals is known in the literature as the Grüss inequality. The inequality is as follows:

Theorem 1.1. Let f, g : [a, b] −→ R be integrable functions such that Ψ ≤ f(x) ≤ ϕ and γ ≤ g(x) ≤ Γ for all x ∈ [a, b], where Ψ, ϕ, γ and Γ are constants. It follows that,

(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

≤ 1

4(ϕ−Ψ) (Γ−γ), where the constant ¹₄ is sharp.

In [2], S.S. Dragomir and S. Wang proved the following Ostrowski type inequality in terms of lower and upper bounds of the first derivative.

Theorem 1.2. Letf : [a, b]−→Rbe continuous on[a, b]and differentiable on (a, b)and where the first derivative satisfies the condition,

γ ≤f⁰(x)≤Γ for allx∈[a, b], then,

(1.2)

f(x)− 1 b−a

Z b a

f(t)dt− f(b)−f(a) b−a

x− a+b 2

≤ 1

4(b−a) (Γ−γ)

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for allx∈[a, b].

In [1], S.S. Dragomir and N.S. Barnett, proved the following inequality.

Theorem 1.3. Let f : [a, b] −→ R be continuous on [a, b] and twice differ- entiable on (a, b), where the second derivativef⁰⁰ : (a, b) −→ R satisfies the condition,

ϕ ≤f⁰⁰(x)≤Φ for allx∈(a, b), then,

(1.3)

f(x) +

"

(b−a)² 24 +1

2

x− a+b 2

2#

f⁰(b)−f⁰(a) b−a

−

x−a+b 2

f⁰(x)− 1 b−a

Z b a

f(t)dt

≤ 1

8(Φ−ϕ) 1

2(b−a) +

x− a+b 2

2

for allx∈[a, b].

In this paper we establish a more general form of (1.3) and apply the result to numerical integration.

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2. Main Results

Theorem 2.1. Let f : [a, b] −→ R be a continuous mapping on [a, b], and twice differentiable on(a, b)with second derivativef⁰⁰ : (a, b)−→Rsatisfying the condition:

ϕ≤f⁰⁰(x)≤Φ, for allx∈

a+hb−a

2 , b−hb−a 2

. It follows that,

(2.1)

(1−h)

f(x)−

x− a+b 2

f⁰(x)

+hf(a) +f(b) 2 +

"

1

2(1−h)

x− a+b 2

2

− (3h−1) (b−a)² 24

#

− 1 b−a

Z b a

f(t)dt

≤ 1

8(Φ−ϕ) 1

2(b−a) (1−h) +

x− a+b 2

2

,

for allx∈[a+h^b−a₂ , b−h^b−a₂ ]andh∈[0,1]. Proof. The proof uses the following identity, (2.2)

Z b a

f(t)dt = (b−a) (1−h)f(x)

−(b−a) (1−h)

x− a+b 2

f⁰(x) +hb−a

2 (f(a) +f(b))

− h²(b−a)²

8 (f⁰(b)−f⁰(a)) + Z b

a

K(x, t)f⁰⁰(t)dt.

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for allx∈[a+h^b−a₂ , b−h^b−a₂ ],where the kernelK : [a, b]² →Ris defined by

(2.3) K(x, t) =







1 2

t− a+h^b−a₂ 2

ift∈[a, x]

1 2

t− b−h^b−a₂ 2

ift∈(x, b].

This is a particular form of the identity given in [3, p. 59; Corollary 2.3].

Observe that the kernelKsatisfies the estimation

(2.4) 0≤K(x, t)≤







1 2

b−h^b−a₂

−x2

, x∈

a+h^b−a₂ ,^a+b₂

1 2

x− a+h^b−a₂ 2

, x∈_a+b

2 , b−h^b−a₂ . Applying the Grüss inequality for the mappingsf⁰⁰(·)andK(x,·) we get,

(2.5)

1 b−a

Z b a

K(x, t)f⁰⁰(t)dt− 1 b−a

Z b a

K(x, t)dt 1 b−a

Z b a

f⁰⁰(t)dt

≤ 1

4(Φ−ϕ)×







1 2

b−h^b−a₂

−x2

, x∈

1 2

x− a+h^b−a₂ 2

, x∈_a+b

2 , b−h^b−a₂ . Observe that,

(2.6) Z b

a

K(x, t)dt= Z x

a

t− a+h^b−a₂ 2

2 dt+

Z b x

t− b−h^b−a₂ 2

2 dt

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= 1 6

"

x−

a+hb−a 2

3

+

b−hb−a 2

−x 3

+h³(b−a)³ 4

#

= (b−a) (1−h)

"

(b−a)²(1−h)²

24 +1

2

x− a+b 2

2#

+h³(b−a)³ 24 . Using(2.6)in(2.5),we get

1 b−a

Z b a

K(x, t)f⁰⁰(t)dt−

"

(b−a)²(1−h)³

24 + 1

2(1−h)

x−a+b 2

2

+ h³(b−a)² 24

#

≤ 1

4(Φ−ϕ)×







1 2

b−h^b−a₂

−x2

, x∈

1 2

x− a+h^b−a₂ 2

, x∈_a+b

2 , b−h^b−a₂ .

Also, by using identity(2.2),the above inequality reduces to,

(1−h)

f(x)−

x−a+b 2

f⁰(x)

+hf(a) +f(b) 2 +

"

1

2(1−h)

x− a+b 2

2

− (3h−1) (b−a)² 24

#

− 1 b−a

Z b a

f(t)dt

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≤ 1

4(Φ−ϕ)×







1 2

b−h^b−a₂

−x2

, x∈

;

1 2

x− a+h^b−a₂ 2

, x∈_a+b

2 , b−h^b−a₂ . Since,

max

( b−h^b−a₂

−x2

2 ,

x− a+h^b−a₂ 2

2

)

=







1 2

b−h^b−a₂

−x2

, x∈

1 2

x− a+h^b−a₂ 2

, x∈_a+b

2 , b−h^b−a₂ , but on the other hand,

max

( b−h^b−a₂

−x2

2 ,

x− a+h^b−a₂ 2

2

)

= 1 2

1

2(b−a) (1−h) +

x− a+b 2

2

,

inequality (2.1) is proved.

Remark 1. Forh= 0in (2.1), we obtain (1.3).

Corollary 2.2. Iff is as in Theorem2.1, then we have the following perturbed

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midpoint inequality:

(2.7)

(1−h)f

a+b 2

+hf(a) +f(b) 2

− (3h−1) (b−a)

24 (f⁰(b)−f⁰(a))− 1 b−a

Z b a

f(t)dt

≤ 1

32(Φ−ϕ) (b−a)²(1−h)², giving,

(2.8) f

a+b 2

+(b−a)

24 (f⁰(b)−f⁰(a))− 1 b−a

Z b a

f(t)dt

≤ 1

32(Φ−ϕ) (b−a)², forh= 0.

Remark 2. The classical midpoint inequality states that (2.9)

f

a+b 2

− 1 b−a

Z b a

f(t)dt

≤ 1

24(b−a)²kf⁰⁰k_∞.

If Φ−ϕ ≤ ⁴₃kf⁰⁰k_∞, then the estimation provided by(2.7)is better than the estimation in the classical midpoint inequality (2.9). A sufficient condition for Φ −ϕ ≤ ⁴₃ kf⁰⁰k_∞ to be true is 0 ≤ ϕ ≤ Φ.Indeed, if 0 ≤ ϕ ≤ Φ, then Φ−ϕ≤ kf⁰⁰k_∞< ⁴₃ kf⁰⁰k_∞.

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Corollary 2.3. Letf be as in Theorem2.1, then,

(2.10)

f(a) +f(b)

2 − (b−a)

12 (f⁰(b)−f⁰(a))− 1 b−a

Z b a

f(t)dt

≤ 1

32(Φ−ϕ) (2−h)²(b−a)². Proof. Putx=aandx=bin turn in (2.1) and use the triangle inequality.

Corollary 2.4. Let f be as in Theorem 2.1, then we have the following per- turbed Trapezoid inequality:

(2.11)

f(a) +f(b)

2 − (b−a)

12 (f⁰(b)−f⁰(a))− 1 b−a

Z b a

f(t)dt

≤ 1

32(Φ−ϕ) (b−a)². Proof. Puth= 1in (2.10).

Remark 3. The classical Trapezoid inequality states that (2.12)

f(a) +f(b)

2 − 1

b−a Z b

a

f(t)dt

≤ 1

12(b−a)²kf⁰⁰k_∞.

If we assume thatΦ−ϕ ≤ ²₃kf⁰⁰k_∞,then the estimation provided by(2.10)is better than the estimation in the classical Trapezoid inequality(2.12).

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3. Applications in Numerical Integration

Let In : a = x0 < x1 < · · · < xn−1 < xn = b be a division of the interval [a, b], ξ_i ∈ [x_i, x_i+1], (i= 0,1, . . . , n−1) a sequence of intermediate points andh_i :=x_i+1 −x_i(i= 0,1, . . . , n−1).Following the approach taken in [1]

we have the following:

Theorem 3.1. Letf : [a, b]−→ Rbe continuous on[a, b]and a twice differen- tiable function on(a, b),whose second derivative,f⁰⁰: (a, b)−→Rsatisfies:

ϕ ≤f⁰⁰(x)≤Φ, for allx∈(a, b), then,

(3.1)

Z b a

f(t)dt =A(f, f⁰, I_n, ξ, δ) +R(f, f⁰, I_n, ξ, δ), where

(3.2) A(f, f⁰, I_n, ξ, δ)

= (1−δ)

n−1

X

i=0

h_if(ξ_i)−(1−δ)

n−1

X

i=0

h_i

ξ_i− x_i+xi−1

2

f⁰(ξ_i)

+δ

n−1

X

i=0

hi

f(xi) +f(xi+1) 2

+

n−1

X

i=0

"

1

2(1−δ)

ξi− xi+xi+1

2 2

−(3δ−1)h²_i 24

(f⁰(xi+1)−f⁰(xi))

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and the remainderR(f, f⁰, I_n, ξ, δ)satisfies the estimation:

|R(f, f⁰, I_n, ξ, δ)|

≤ 1

8(Φ−ϕ)

n−1

X

i=0

h_i

(1−δ) 2 h_i+

ξ_i− x_i+x_i+1 2

2

≤ 1

32(Φ−ϕ) (1−δ)²

n−1

X

i=0

h³_i, (3.3)

whereδ∈[0,1]andx_i+δ^h₂ⁱ ≤ξ_i ≤x_i+1−δ^h₂ⁱ.

Proof. Applying Theorem2.1on the interval[x_i, x_i+1] (i= 0, . . . , n−1)gives:

(1−δ)

h_if(ξ_i)−h_i

ξ_i−x_i+x_i+1 2

f⁰(ξ_i)

+δh_i

f(x_i) +f(x_i+1) 2

+

"

1

2(1−δ)

ξi− xi+xi+1

2 2

− (3δ−1)h²_i 24

#

(f⁰(xi+1)−f⁰(xi))

− Z xi+1

xi

f(t)dt

≤ 1

8(Φ−ϕ)h_i 1

2(1−δ)h_i+

ξ_i−x_i+x_i+1 2

2

,

≤ 1

8(Φ−ϕ) (1−δ)²h³_i

as

ξ_i− x_i+x_i+1 2

≤(1−δ)h_i

2 for alli∈ {0,1, ..., n−1}

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for any choiceξ_iof the intermediate points.

Summing the above inequalities overifrom0ton−1, and using the generalized triangle inequality, we get the desired estimation(3.3).

Corollary 3.2. The following perturbed midpoint rule holds:

(3.4)

Z b a

f(x)dx=M(f, f⁰, In) +RM(f, f⁰, In), where

(3.5) M(f, f⁰, I_n) =

n−1

X

i=0

h_if

x_i+x_i+1 2

+ 1

24

n−1

X

i=0

h²_i (f⁰(x_i+1)−f⁰(x_i)) and the remainder termR_M(f, f⁰, I_n)satisfies the estimation:

(3.6) |R_M(f, f⁰, I_n)| ≤ 1

32(Φ−ϕ)

n−1

X

i=0

h³_i.

Corollary 3.3. The following perturbed trapezoid rule holds (3.7)

Z b a

f(x)dx =T (f, f⁰, I_n) +R_T (f, f⁰, I_n), where

(3.8) T (f, f⁰, I_n) =

n−1

X

i=0

h_if(x_i) +f(x_i+1)

2 − 1

12

n−1

X

i=0

h²_i (f⁰(x_i+1)−f⁰(x_i))

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and the remainder termR_T (f, f⁰, I_n)satisfies the estimation:

(3.9) |R_T (f, f⁰, I_n)| ≤ 1

32(Φ−ϕ)

n−1

X

i=0

h³_i.

Remark 4. Note that the above mentioned perturbed midpoint formula (3.5) and perturbed trapezoid formula (3.8)can offer better approximations of the integralRb

a f(x)dxfor general classes of mappings as discussed in Remarks1 and2.

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References

[1] 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.

[2] S.S. DRAGOMIR AND S. WANG, An inequality of Ostrwski-Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Computers Math. Appl., 33 (1997), 15–20.

[3] S.S. DRAGOMIR ANDN.S. BARNETT, An Ostrowski type inequality for mappings whose second derivatives are bounded and applications, RGMIA Research Report of Collection, 1(2) (1998), 61–63.

[4] S.S. DRAGOMIR AND Th.M. RASSIAS (Eds.), Ostrowski Type Inequali- ties and Applications in Numerical Integration, Kluwer Academic Publish- ers 2002.