General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

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Volume 2012, Article ID 343191,12pages doi:10.1155/2012/343191

Research Article

Sharp Integral Inequalities Based on a

General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

J. Pe ˇcari´c

¹

and M. Ribi ˇci ´c Penava

²

1Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia

2Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 000 Osijek, Croatia

Correspondence should be addressed to M. Ribiˇci´c Penava,[email protected] Received 28 March 2012; Accepted 10 July 2012

Academic Editor: Marianna Shubov

Copyrightq2012 J. Peˇcari´c and M. Ribiˇci´c Penava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider families of general four-point quadrature formulae using a generalization of the Mont- gomery identity via Taylor’s formula. The results are applied to obtain some sharp inequalities for functions whose derivatives belong toLpspaces. Generalizations of Simpson’s 3/8 formula and the Lobatto four-point formula with related inequalities are considered as special cases.

1. Introduction

The most elementary quadrature rules in four nodes are Simpson’s 3/8 rule based on the following four point formula

_b

a

ftdt b−a 8

fa 3f

2ab 3

3f

a2b 3

fb

− b−a⁵

6480 f⁴ξ, 1.1

whereξ∈a, b, and Lobatto rule based on the following four point formula

₁

−1ftdt 1 6

f−1 5f

−

√5

5 5f

√ 5

5 f1

− 2 23625f⁶

η

, 1.2

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where η ∈ −1,1. Formula 1.1 is valid for any function f with a continuous fourth derivativef⁴ona, band formula1.2is valid for any functionfwith a continuous sixth derivativef⁶on−1,1.

Letf:a, b → Rbe diﬀerentiable ona, bandf:a, b → Rintegrable ona, b.

Then the Montgomery identity holdssee1

fx 1 b−a

_b

a

ftdt _b

a

Px, tftdt, 1.3

where the Peano kernel is

Px, t

⎧⎪

⎪⎨

⎪⎪

⎩ t−a

b−a, a≤t≤x, t−b

b−a, x < t≤b.

1.4

In2, Peˇcari´c proved the following weighted Montgomery identity

fx _b

a

wtftdt

_b

a

Pwx, tftdt, 1.5

wherew :a, b → 0,∞is some probability density function, that is, integrable function, satisfying_b

awtdt1, andWt _t

awxdxfort∈a, b,Wt 0 fort < aandWt 1 fort > bandPwx, tis the weighted Peano kernel defined by

Pwx, t

Wt, a≤t≤x,

Wt−1, x < t≤b. 1.6

Now, let us suppose thatIis an open interval inR,a, b⊂I,f :I → Ris such thatfⁿ⁻¹is absolutely continuous for somen≥ 2,w :a, b → 0,∞is a probability density function.

Then the following generalization of the weighted Montgomery identity via Taylor’s formula statesgiven by Aglić Aljinović and Peˇcarić in3

fx _b

a

wtftdt−ⁿ⁻²

i0

fⁱ¹x i1!

_b

a

wss−xⁱ¹ds

1 n−1!

_b

a

Tw,nx, sfⁿsds,

1.7

wherex∈a, band

Tw,nx, s

⎧⎪

⎪⎨

⎪⎪

⎩ _s

a

wuu−sⁿ⁻¹du, a≤s≤x,

− _b

s

wuu−sⁿ⁻¹du, x < s≤b.

1.8

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If we takewt 1/b−a,t∈a, b, equality1.7reduces to

fx 1

b−a _b

a

ftdt−ⁿ⁻²

i0

fⁱ¹xb−xⁱ²−a−xⁱ² i2!b−a 1

n−1!

_b

a

Tnx, sfⁿsds,

1.9

wherex∈a, band

Tnx, s

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−a−sⁿ

nb−a, a≤s≤x,

−b−sⁿ

nb−a, x < s≤b.

1.10

Forn1,1.9reduces to the Montgomery identity1.3.

In this paper, we generalize the results from4. Namely, we use identities1.7and 1.9to establish for each numberx∈a,ab/2a general four-point quadrature formula of the type

_b

a

wtftdt

1

2−Ax

fa fb

Ax

fx fab−x R

f, w;x ,

1.11

whereRf, w;xis the remainder andA:a,ab/2 → Ris a real function. The obtained formula is used to prove a number of inequalities which give error estimates for the general four-point formula for functions whose derivatives are fromLp-spaces. These inequalities are generally sharp. As special cases of the general non-weighted four-point quadrature formula, we obtain generalizations of the well-known Simpson’s 3/8 formula and Lobatto four-point formula with related inequalities.

2. General Weighted Four-Point Formula

Let f : a, b → Rbe such that fⁿ⁻¹ exists on a, bfor some n ≥ 2. We introduce the following notation for eachx∈a,ab/2:

Dx 1

2 −Ax

fa fb

Ax

fx fab−x ,

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tw,nx Ax _n−2

i0

fⁱ¹x i1!

_b

a

wss−xⁱ¹ds

ⁿ⁻²

i0

fⁱ¹ab−x i1!

_b

a

wss−a−bxⁱ¹ds

1

2−Ax _n−2

i0

fⁱ¹a i1!

_b

a

wss−aⁱ¹ds

ⁿ⁻²

i0

fⁱ¹b i1!

_b

a

wss−bⁱ¹ds

,

Tw,nx, s − 1

2−Ax

Tw,na, s Tw,nb, s−AxTw,nx, s Tw,nab−x, s

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

− 1

2Ax ^s

a

wuu−sⁿ⁻¹du

1

2−Ax ^b

s

wuu−sⁿ⁻¹du, a≤s≤x,

−1 2

_s

a

wuu−sⁿ⁻¹du− _b

s

wuu−sⁿ⁻¹du

, x < s≤ab−x,

− 1

2−Ax ^s

a

wuu−sⁿ⁻¹du

1

2Ax ^b

s

wuu−sⁿ⁻¹du, ab−x < s≤b.

2.1

In the next theorem we establish the general weighted four-point formula.

Theorem 2.1. LetI be an open interval inR,a, b ⊂ I, and letw : a, b → 0,∞ be some probability density function. Letf : I → Rbe such thatfⁿ⁻¹ is absolutely continuous for some n≥2. Then for eachx∈a,ab/2the following identity holds

_b

a

wtftdtDx tw,nx 1 n−1!

_b

a

Tw,nx, sfⁿsds. 2.2

Proof. We putx≡a, x≡x, x≡ab−xandx≡bin1.7to obtain four new formulae. After multiplying these four formulae by 1/2−Ax, Ax, Axand 1/2−Ax, respectively, and adding, we get2.2.

Remark 2.2. Identity 2.2 holds true in the casen 1. It can also be obtained by taking x ≡ a, x ≡ x, x ≡ ab−xandx ≡ bin1.5, multiplying these four formulae by 1/2− Ax, Ax, Axand 1/2−Ax, respectively, and adding. In this special case we have

_b

a

wtftdtDx

_b

a

Tw,1x, sfsds, 2.3

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where

Tw,1x, s − 1

2−Ax

Tw,1a, s Tw,1b, s−AxTw,1x, s Tw,1ab−x, s

− 1

2−Ax

Pwa, s Pwb, s−AxPwx, s Pwab−x, s

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 1

2−Ax−Ws, a≤s≤x, 1

2−Ws, x < s≤ab−x, 1

2Ax−Ws, ab−x < s≤b.

2.4 Theorem 2.3. Suppose that all assumptions ofTheorem 2.1hold. Additionally, assume thatp, qis a pair of conjugate exponents, that is, 1 ≤ p, q ≤ ∞, 1/p1/q 1, letfⁿ ∈L^pa, bfor some n≥1. Then for eachx∈a,ab/2we have

_b

a

wtftdt−Dx−tw,nx ≤ 1

n−1!Tw,nx,·

q

fⁿ

p. 2.5

Inequality2.5is sharp for 1< p≤ ∞.

Proof. By applying the H ¨older inequality we have

1 n−1!

_b

a

Tw,nx, sfⁿsds ≤ 1

n−1!Tw,nx,·

q

fⁿ

p. 2.6

By using the above inequality from2.2 we obtain estimate2.5. Let us denoteU^x_ns Tw,nx, s. For the proof of sharpness, we will find a functionfsuch that

_b

a

U^x_nsfⁿsds

U^x_n_qfⁿ

p. 2.7

For 1< p <∞, takefto be such that

fⁿs signU^x_ns· |U^xns|^1/p−1, 2.8

where forp∞we put

fⁿs signU^x_ns. 2.9

Remark 2.4. Inequality2.5forAx 1/4 was proved by Agli´c Aljinovi´c et al. in4.

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3. Non-Weighted Four-Point Formula and Applications

Here we define tnx Axⁿ⁻²

i0

fⁱ¹x −1ⁱ¹fⁱ¹ab−xb−xⁱ²−a−xⁱ² i2!b−a

1

2−Ax _n−2

i0

fⁱ¹a −1ⁱ¹fⁱ¹bb−aⁱ¹ i2! ,

3.1

Tnx, s −n 1

2−Ax

Tna, s Tnb, s AxTnx, s Tnab−x, s

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ 1

2Ax

a−sⁿ b−a

1 2−Ax

b−sⁿ

b−a, a≤s≤x, a−sⁿ b−sⁿ

2b−a , x < s≤ab−x,

1

2−Ax

a−sⁿ b−a

1

2Ax

b−sⁿ

b−a, ab−x < s≤b.

3.2

Theorem 3.1. LetIbe an open interval inR,a,b⊂ I, and letf :I → Rbe such thatfⁿ⁻¹ is absolutely continuous for somen≥1. Then for eachx∈a,ab/2the following identity holds

1 b−a

_b

a

ftdtDx tnx 1 n!

_b

a

Tnx, sfⁿsds. 3.3

Proof. We takewt 1/b−a,t∈a, bin2.2.

Theorem 3.2. Suppose that all assumptions ofTheorem 3.1hold. Additionally, assume thatp, qis a pair of conjugate exponents, that is, 1 ≤p, q ≤ ∞, 1/p1/q 1 andfⁿ ∈ L^pa, bfor some n≥1. Then for eachx∈a,ab/2we have

1 b−a

_b

a

ftdt−Dx−tnx ≤ 1

n!

Tnx,·

q

fⁿ

p. 3.4

Inequality3.4is sharp for 1< p≤ ∞.

Proof. We takewt 1/b−a,t∈a, bin2.5.

Now, we set

Ax b−a²

12x−ab−x, x∈

a,ab 2

. 3.5

This special choice of the functionAenables us to consider generalizations of the well-known Simpson’s 3/8 formula1.1and Lobatto formula1.2

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3.1.x 2ab/3

Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of Simpson’s 3/8 formula reads

1 b−a

_b

a

ftdtD

2ab 3

tn

2ab 3

1

n!

_b

a

Tn

2ab 3 , s

fⁿsds, 3.6

where

D

2ab 3

1

8

fa 3f

2ab 3

3f

a2b 3

fb

,

tn

2ab 3

1

8

n−2

i0

fⁱ¹

2ab 3

−1ⁱ¹fⁱ¹

a2b 3

2ⁱ² −1ⁱ¹

b−aⁱ¹ 3ⁱ¹i2!

1 8

n−2 i0

fⁱ¹a −1ⁱ¹fⁱ¹bb−aⁱ¹ i2! , Tn

2ab 3 , s

−n

8

Tna, s 3Tn

2ab 3 , s

3Tn

a2b 3 , s

Tnb, s

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

7a−sⁿ b−sⁿ

8b−a a≤s≤ 2ab 3 , a−sⁿ b−sⁿ

2b−a , 2ab

3 < s≤ a2b 3 , a−sⁿ7b−sⁿ

8b−a

a2b

3 < s≤b.

3.7

In the next corollaries we will use the beta function and the incomplete beta function of Euler type defined by

B x, y

₁

0

t^x−11−t^y−1dt, Br

x, y

_r

0

t^x−11−t^y−1dt, x, y >0. 3.8

Corollary 3.3. Suppose that all assumptions ofTheorem 3.1hold. Additionally, assume thatp, qis a pair of conjugate exponents andn∈N.

aIffⁿ∈L^∞a, b, then

1 b−a

_b

a

ftdt−D

2ab 3

≤ 25

288b−af

∞,

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1 b−a

_b

a

ftdt−D

2ab 3

−tn

2ab 3

≤ 1 n1!

3ⁿ¹3·2ⁿ¹3−1ⁿ b−aⁿ 4·3ⁿ¹

− b−a

2 n

−1ⁿ¹1

2 fⁿ

∞, n≥2.

3.9

bIffⁿ∈L²a, b, then

1 b−a

_b

a

ftdt−D

2ab 3

−tn

2ab 3

≤ 1 n!

3²ⁿ5·2²ⁿ¹11

b−a²ⁿ⁻¹

32·3²ⁿ2n1 −1ⁿb−a²ⁿ⁻¹ 32

×7Bn1, n1 9B2/3n1, n1−9B1/3n1, n1

1/2fⁿ

2. 3.10

cIffⁿ∈L¹a, b, then

1 b−a

_b

a

ftdt−D

2ab 3

−tn

2ab 3

≤ 1 n!Kn

2ab 3

fⁿ

1, 3.11

whereK12ab/3 5/24,K22ab/3 5/18 b−a,K32ab/3 7/54b−a² andKn2ab/3 1/8b−aⁿ⁻¹, forn≥4.

The first and the second inequality are sharp.

Proof. We apply3.4withx 2ab/3 andp∞ _b

a

Tn

2ab 3 , s

ds

_2ab/3

a

7a−sⁿ b−sⁿ 8b−a

ds

_a2b/3

2ab/3

a−sⁿ b−sⁿ 2b−a

ds _b

a2b/3

a−sⁿ7b−sⁿ 8b−a

ds 2

3ⁿ¹−2ⁿ¹7·−1ⁿ b−aⁿ 8·3ⁿ¹n1

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2ⁿ¹ −1ⁿ¹ b−aⁿ

3ⁿ¹n1 −

1 −1ⁿ¹ b−aⁿ 2ⁿ¹n1

3ⁿ¹3·2ⁿ¹3−1ⁿ b−aⁿ 4·3ⁿ¹n1 −

b−a 2

n

−1ⁿ¹1 2n1

,

3.12

forn≥2 and

_b

a

T1

2ab 3 , s

ds 25

288b−a. 3.13

To obtain the second inequality we takep2 _b

a

Tn

2ab 3 , s

²ds

_2ab/3

a

²ds

_a2b/3

2ab/3

a−sⁿb−sⁿ 2b−a

²ds _b

a2b/3

²ds

3²ⁿ5·2²ⁿ¹11

b−a²ⁿ⁻¹

32·3²ⁿ2n1 −1ⁿb−a²ⁿ⁻¹ 32

×7Bn1, n1 9B2/3n1, n1−9B1/3n1, n1.

3.14 Ifp1, we have

sup

s∈a,b

Tn

2ab 3 , s

max

sup

s∈a,2ab/3

,

sup

s∈2ab/3,a2b/3

sup

s∈a2b/3,b

.

3.15

By an elementary calculation we get

sup

s∈a,2ab/3

7a−s b−s 8b−a

sup

s∈a2b/3,b

a−s 7b−s 8b−a

5

24b−a,

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sup

s∈a,2ab/3

7a−s² b−s² 8b−a

sup

s∈a2b/3,b

a−s²7b−s² 8b−a

11

72b−a,

sup

s∈a,2ab/3

sup

s∈a2b/3,b

b−aⁿ⁻¹

8 ,

3.16 forn≥3. The functiony:a, b → R,yx a−xⁿb−xⁿ, is decreasing ona,ab/2 and increasing onab/2, bifnis even, and decreasing ona, bifnis odd. Thus

sup

s∈2ab/3,a2b/3

−1ⁿ2ⁿ

b−aⁿ⁻¹

2·3ⁿ . 3.17

Finally,

sup

s∈a,b

T1

2ab 3 , s

5

24 3.18

and forn≥2

sup

s∈a,b

Tn

2ab 3 , s

b−aⁿ⁻¹max 1

8,2ⁿ −1ⁿ 2·3ⁿ

. 3.19

3.2.a, b −1,1,x−√ 5/5

Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of Lobatto formula reads

1 2

₁

−1ftdtD

−

√5 5 tn

−

√5

5 1

n!

₁

−1

Tn

−

√5

5 , s fⁿsds, 3.20 where

D

−

√5

5 1

12

f−1 5f

−

√5

5 5f

√ 5

5 f1 ,

tn

−

√5

5 5

12 n−2

i0

fⁱ¹

−

√5

5 −1ⁱ¹fⁱ¹ √

5 5

×

5√ 5_i2

−1ⁱ¹ 5−√

5_i2 2·5ⁱ²i2!

1 12

n−2

i0

fⁱ¹−1 −1ⁱ¹fⁱ¹1 2ⁱ¹ i2!,

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Tn

−

√5

5 , s − n 12

Tn−1, s 5Tn

−

√5

5 , s 5Tn

√5

5 , s Tn1, s

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

11−1−sⁿ 1−sⁿ

24 −1≤s≤ −

√5 5 ,

−1−sⁿ 1−sⁿ

4 −

√5 5 < s≤

√5 5 ,

−1−sⁿ111−sⁿ 24

√5

5 < s≤1.

3.21

Corollary 3.4. Suppose that all assumptions ofTheorem 3.1hold. Additionally, assume thatp, qis a pair of conjugate exponents andn∈N.

aiffⁿ∈L^∞−1,1, then

1 2

₁

−1ftdt−D

−

√5 5 ≤

101 180 −

√5

6 f_∞,

1 2

₁

−1ftdt−D

−

√5 5 −tn

−

√5

5

≤ 1 n1!

⎛

⎜⎝2ⁿ¹·5ⁿ 5√

5_n1

−

−5√ 5_n1 12·5ⁿ

−1 −1ⁿ¹

2 fⁿ

∞, n≥2.

3.22

biffⁿ∈L²−1,1, then

1 2

₁

−1ftdt−D

−

√5 5 −tn

−

√5

5

≤ 1 n!·2ⁿ⁻²

3

⎛

⎜⎝35 5√

5_2n1 85

5−√ 5_2n1

10²ⁿ¹ 10²ⁿ¹2n1

−1ⁿ

11Bn1, n1 25B5√5/10n1, n1

−25B₅₋^√_5/10n1, n1

⎞

⎟⎠

1/2fⁿ

2.

3.23

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ciffⁿ∈L¹−1,1, then

1 2

₁

−1ftdt−D

−

√5 5 −tn

−

√5

5 ≤ 1 n!Kn

−

√5

5 fⁿ

1, 3.24

whereK1−√

5/5 1/2√

5,K2−√

5/5 3/5,K3−√

5/5 8/5√

5,K4−√

5/5 28/25, K5−√

5/5 88/25√

5,Kn−√

5/5 2ⁿ⁻³/3, forn≥6.

The first and the second inequality are sharp.

Proof. Applying3.4witha, b −1,1,x −√

5/5 andp ∞, p 2, p 1 and carrying out the same analysis as in Corollay3.3we obtain the above inequalities.

References

1 D. S. Mitrinovi´c, J. E. Peˇcari´c, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, vol. 53 of Mathematics and its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.

2 J. E. Peˇcari´c, “On the ˇCebyˇsev inequality,” Buletinul S¸tiint¸ific al Universit˘at¸ii Politehnica din Timis¸oara.

Seria Matematic˘a-Fizic˘a, vol. 25, no. 39, pp. 10–11, 1980.

3 A. Aglić Aljinović and J. Peˇcarić, “On some Ostrowski type inequalities via Montgomery identity and Taylor’s formula,” Tamkang Journal of Mathematics, vol. 36, no. 3, pp. 199–218, 2005.

4 A. Aglić Aljinović, J. Peˇcarić, and M. Ribiˇcić Penava, “Sharp integral inequalities based on general two-point formulae via an extension of Montgomery’s identity,” The ANZIAM Journal, vol. 51, no. 1, pp. 67–101, 2009.

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General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

Research Article

Sharp Integral Inequalities Based on a

General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

J. Pe ˇcari´c

and M. Ribi ˇci ´c Penava

1. Introduction

2. General Weighted Four-Point Formula

3. Non-Weighted Four-Point Formula and Applications

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

Stochastic Analysis