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Weighted Ostrowski-Gruss type inequality for differentiable mappings.

Abstract

We establish weighted Ostrowski-Gruss type inequality for differentiable mappings in terms of the upper and lower bounds of first derivative. The inequality is then applied to numerical integration.

AMS Subject Classification: Primary 65D30; Secondary 65D32.

Keywords: Ostrowski inequality, Gruss inequality.

1. Introduction

Let the weight w : [a, b] [right arrow] [0,8) be non-negative, integrable and

[[integral].sup.b.sub.a] w(t)dt < [infinity].

The domain of w may be finite or infinite and w may vanish at the boundary points. We denote the moments to be m, M, N and notation [micro] as:

m(a, b) = [[integral].sup.b.sub.a] w(t)dt

M(a, b) = [[integral].sup.b.sub.a] tw(t)dt

N(a, b) = [[integral].sup.b.sub.a] [t.sup.2]w(t)dt

and

[mu](a, b) = M(a, b)/m(a, b).

The following integral inequality is well known in the literature as weighted Gruss inequality [3].

Theorem 1.1. Let f, g : [a, b] [right arrow] [??] be two integrable functions such that [phi] [less than or equal to] f (x) [less than or equal to] [PHI] and [gamma] [less than or equal to] g(x) [less than or equal to] [GAMMA] for all x [member of] [a, b] and [phi], [PHI], [gamma] and [GAMMA] are constants. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

The constant 1/4 is sharp.

In [5] Dragomir and Wang, pointed out Ostrowski-Gruss type inequality for single differentiable mappings in terms of the upper and lower bounds of first derivative. The inequality then applied to numerical integration and for special means, given in the form of the following theorem:

Theorem 1.2. Let f : [a, b] [right arrow] R be continuous on [a, b] and differentiable on (a, b), whose first derivative satisfies the condition:

[gamma] [less than or equal to] f'(x) [less than or equal to] [GAMMA],

for all x [member of] (a, b).

Then, we have the inequality:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.2)

for all x [member of] (a, b).

The main aim of this paper is to point out new estimation of (1.2) and to apply it in numerical integration. It turns out that these new estimations can give much better results than estimations based on (1.2). Some closely related new results are also given.

2. Main Results

We now give our main theorem:

Theorem 2.1. Let f : [a, b] [right arrow] [??] be continuous on [a, b] and differentiable on (a, b), whose first derivative f : (a, b) [right arrow] [??], satisfies the condition [phi] [less than or equal to] f'(x) [less than or equal to] [PHI] for all x [member of] (a, b). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

for all x [member of] [a, b].

Proof. Let us define the mapping P(., .) : [[a, b].sup.2] [right arrow] [??], given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

Using integration by parts techniques, Dragomir and Rassias proved the following identity in [4]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

From (2.2) we prove

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

Now, let us observe that the Kernel P satisfies the estimation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

Applying weighted Gruss integral inequality (1.1) for the mappings f (x) = f'(x), g(x) = P(x, x)/w(x) and using (2.5), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

Using (2.3) and (2.4) in (2.6), we obtain the desired inequality (2.1).

Remark 2.2. If we put w(t) = 1 in (2.1), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Corollary 2.3. Under the assumptions of theorem 2.1 and putting x = a + b/2 in (2.1), we have the mid-point like inequality:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.5)

Corollary 2.4. Under the assumptions of theorem 2.1, we have the following trapezoidal like inequality:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.6)

Proof. The inequality (2.8) can be drawn from (2.1) with x = a and x = b, adding the results, using the triangular inequality and then dividing by 2.

3. Applications in Numerical Integration

Let [I.sub.n] : a = [x.sub.0] < [x.sub.1] < [x.sub.2] < ... < [x.sub.n-1] < [x.sub.n] = b be the division of the interval [a, b], [[xi].sub.i] [member of] [[x.sub.i], [x.sub.i]+1], i = 1, 2, ..., n - 1. We have the following quadrature formula:

Theorem 3.1. Let f : [a, b] [right arrow] [??] be continuous on [a, b] and differentiable on (a, b) and f' : (a, b) [right arrow] [??], satisfies the condition [phi] [less than or equal to] f' (x) [less than or equal to] [PHI] for all x [member of] (a, b). Then, we have the following perturbed Riemann's type quadrature formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the remainder term satisfies the estimation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

for all [[xi].sub.i] [member of] [[x.sub.i], [x.sub.i+1]] where [h.sub.i] = [x.sub.i+1] - [x.sub.i] , for i = 1, 2,..., n - 1.

Proof. Apply theorem 2.1 on the interval [[x.sub.i], [x.sub.i+1]], [[xi].sub.i] [member of] [[x.sub.i], [x.sub.i+1], where [h.sub.i] = [x.sub.i+1-[x.sub.i]], for i = 1, 2,..., n - 1, to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Summing over i from 0 to n - 1 and using the generalized triangular inequality, we deduce the desired estimation (3.2).

Corollary 3.2. Under the assumption of Theorem 3.1, by choosing [[xi].sub.i] = [x.sub.i] + [x.sub.i+1]/2, we recapture the midpoint quadrature formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the remainder term satisfies the estimation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Corollary 3.3. Under the assumption of Theorem 3.1, the following perturbed trapezoidal rule also holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the remainder term satisfies the estimation:

|R(f, f', [xi], [I.sub.n])| [less than or equal to] [PHI] - [phi]/4 [n-1.summation over (i=0)] m([x.sub.i]).

References

[1] Barnett N.S., Cerone P., Dragomir S.S., Roumeliotis, J., and Sofo A., 2001,A survey on Ostrowski type inequalities for twice differentiable mappings and applications, Inequality, Theory and Applications, Nova Science Publishers, Inc, Huntington, NY, pp. 33-86.

[2] Cerone P., Dragomir S.S., and Roumeliotis J., 1999, An inequality of Ostrowski-Gruss type for twice differentiable mappings and applications in numerical integration. Kyungpook Mathematical Journal, 39(2), pp. 331-341.

[3] Dragomir S.S., 2000, Some integral inequalities of Gruss type, Indian J. Pure Appl. Math., 31(4), pp. 397-415.

[4] Dragomir S.S. and Rassias T.M., 2002, Ostrowski Type Inequalities and Applications in Numerical Integration, Springer, USA, 2002.

[5] Dragomir S.S. and Wang S., 1997, An inequality of Ostrowski-Gruss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules. Comp. Math. Appl., 33, pp. 15-22.

Farooq Ahmad

Centre for Advanced Studies in Pure and Applied Mathematics,

Bahauddin Zakariya University, Multan 60800, Pakistan

E-mail: farooqgujar@gmail.com

Arif Rafiq and Nazir Ahmad Mir

Mathematics Department, COMSATS Institute of Information Technology,

Plot # 30, Sector H-8/1, Islamabad 44000, Pakistan

E-mail: arafiq@comsats.edu.pk, namir@comsats.edu.pk
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Author:Ahmad, Farooq; Rafiq, Arif; Mir, Nazir Ahmad
Publication:Global Journal of Pure and Applied Mathematics
Geographic Code:9PAKI
Date:Aug 1, 2006
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