IJMMS International Journal of Mathematics and Mathematical Sciences 1687-0425 0161-1712 Hindawi Publishing Corporation 343191 10.1155/2012/343191 343191 Research Article Sharp Integral Inequalities Based on a General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity Pečarić J. 1 Ribičić Penava M. 2 Shubov Marianna 1 Faculty of Textile Technology University of Zagreb Pierottijeva 6, 10000 Zagreb Croatia unizg.hr 2 Department of Mathematics University of Osijek Trg Ljudevita Gaja 6, 31 000 Osijek Croatia unios.hr 2012 17 06 2012 2012 28 03 2012 10 07 2012 2012 Copyright © 2012 J. Pečarić and M. Ribičić 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 Montgomery identity via Taylor’s formula. The results are applied to obtain some sharp inequalities for functions whose derivatives belong to Lp spaces. 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 (1.1)abf(t)dt=b-a8[f(a)+3f(2a+b3)+3f(a+2b3)+f(b)]-(b-a)56480f(4)(ξ), where ξ[a,b], and Lobatto rule based on the following four point formula (1.2)-11f(t)dt=16[f(-1)+5f(-55)+5f(55)+f(1)]-223625f(6)(η), where η[-1,1]. Formula (1.1) is valid for any function f with a continuous fourth derivative f(4) on [a,b] and formula (1.2) is valid for any function f with a continuous sixth derivative f(6) on [-1,1].

Let f:[a,b] be differentiable on [a,b] and f:[a,b] integrable on [a,b].

Then the Montgomery identity holds (see ) (1.3)f(x)=1b-aabf(t)dt+abP(x,t)f(t)dt, where the Peano kernel is (1.4)P(x,t)={t-ab-a,atx,t-bb-a,x<tb.

In , Pečarić proved the following weighted Montgomery identity (1.5)f(x)=abw(t)f(t)dt+abPw(x,t)f(t)dt, where w:[a,b][0, is some probability density function, that is, integrable function, satisfying abw(t)dt=1, and W(t)=atw(x)dx for t[a,b], W(t)=0 for t<a and W(t)=1 for t>b and Pw(x,t) is the weighted Peano kernel defined by (1.6)Pw(x,t)={W(t),atx,W(t)-1,x<tb. Now, let us suppose that I is an open interval in , [a,b]I, f:I is such that f(n-1) is absolutely continuous for some n2, w:[a,b][0, is a probability density function. Then the following generalization of the weighted Montgomery identity via Taylor’s formula states (given by Aglić Aljinović and Pečarić in ) (1.7)f(x)=abw(t)f(t)dt-i=0n-2f(i+1)(x)(i+1)!abw(s)(s-x)i+1ds+1(n-1)!abTw,n(x,s)f(n)(s)ds, where x[a,b] and (1.8)Tw,n(x,s)={asw(u)(u-s)n-1du,asx,-sbw(u)(u-s)n-1du,x<sb. If we take w(t)=1/(b-a), t[a,b], equality (1.7) reduces to (1.9)f(x)=1b-aabf(t)dt-i=0n-2f(i+1)(x)(b-x)i+2-(a-x)i+2(i+2)!(b-a)+1(n-1)!abTn(x,s)f(n)(s)ds, where x[a,b] and (1.10)Tn(x,s)={-(a-s)nn(b-a),asx,-(b-s)nn(b-a),x<sb. For n=1, (1.9) reduces to the Montgomery identity (1.3).

In this paper, we generalize the results from . Namely, we use identities (1.7) and (1.9) to establish for each number x(a,(a+b)/2] a general four-point quadrature formula of the type (1.11)abw(t)f(t)dt=(12-A(x))[f(a)+f(b)]+A(x)[f(x)+f(a+b-x)]+R(f,w;x), where R(f,w;x) is the remainder and A:(a,(a+b)/2] is 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 from Lp-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] be such that f(n-1) exists on [a,b] for some n2. We introduce the following notation for each x(a,(a+b)/2]: (2.1)D(x)=(12-A(x))[f(a)+f(b)]+A(x)[f(x)+f(a+b-x)],tw,n(x)=A(x)[i=0n-2f(i+1)(x)(i+1)!abw(s)(s-x)i+1ds+i=0n-2f(i+1)(a+b-x)(i+1)!abw(s)(s-a-b+x)i+1ds]+(12-A(x))[i=0n-2f(i+1)(a)(i+1)!abw(s)(s-a)i+1ds+i=0n-2f(i+1)(b)(i+1)!abw(s)(s-b)i+1ds],T^w,n(x,s)=-(12-A(x))[Tw,n(a,s)+Tw,n(b,s)]-A(x)[Tw,n(x,s)+Tw,n(a+b-x,s)]={-(12+A(x))asw(u)(u-s)n-1du+(12-A(x))sbw(u)(u-s)n-1du,asx,-12[asw(u)(u-s)n-1du-sbw(u)(u-s)n-1du],x<sa+b-x,-(12-A(x))asw(u)(u-s)n-1du+(12+A(x))sbw(u)(u-s)n-1du,a+b-x<sb.

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

Theorem 2.1.

Let I be an open interval in , [a,b]I, and let w:[a,b][0, be some probability density function. Let f:I be such that f(n-1) is absolutely continuous for some n2. Then for each x(a,(a+b)/2] the following identity holds (2.2)abw(t)f(t)dt=D(x)+tw,n(x)+1(n-1)!abT^w,n(x,s)f(n)(s)ds.

Proof.

We put xa,xx,xa+b-x and xb in (1.7) to obtain four new formulae. After multiplying these four formulae by 1/2-A(x),A(x),A(x)and  1/2-A(x), respectively, and adding, we get (2.2).

Remark 2.2.

Identity (2.2) holds true in the case n=1. It can also be obtained by taking xa,xx,xa+b-x and xb in (1.5), multiplying these four formulae by 1/2-A(x),A(x),A(x)and1/2-A(x), respectively, and adding. In this special case we have (2.3)abw(t)f(t)dt=D(x)+abT^w,1(x,s)f(s)ds, where (2.4)T^w,1(x,s)=-(12-A(x))[Tw,1(a,s)+Tw,1(b,s)]-A(x)[Tw,1(x,s)+Tw,1(a+b-x,s)]=-(12-A(x))[Pw(a,s)+Pw(b,s)]-A(x)[Pw(x,s)+Pw(a+b-x,s)]={12-A(x)-W(s),asx,12-W(s),x<sa+b-x,12+A(x)-W(s),a+b-x<sb.

Theorem 2.3.

Suppose that all assumptions of Theorem 2.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents, that is, 1p,q, 1/p+1/q=1, let f(n)Lp[a,b] for some n1. Then for each x(a,(a+b)/2] we have (2.5)|abw(t)f(t)dt-D(x)-tw,n(x)|1(n-1)!T^w,n(x,)qf(n)p. Inequality (2.5) is sharp for 1<p.

Proof.

By applying the Hölder inequality we have (2.6)|1(n-1)!abT^w,n(x,s)f(n)(s)ds|1(n-1)!T^w,n(x,)qf(n)p. By using the above inequality from (2.2) we obtain estimate (2.5). Let us denote Unx(s)=T^w,n(x,s). For the proof of sharpness, we will find a function f such that (2.7)|abUnx(s)f(n)(s)ds|=Unxqf(n)p. For 1<p<, take f to be such that (2.8)f(n)(s)=signUnx(s)|Unx(s)|1/(p-1), where for p= we put (2.9)f(n)(s)=signUnx(s).

Remark 2.4.

Inequality (2.5) for A(x)=1/4 was proved by Aglić Aljinović et al. in .

3. Non-Weighted Four-Point Formula and Applications

Here we define (3.1)t^n(x)=A(x)i=0n-2[f(i+1)(x)+(-1)i+1f(i+1)(a+b-x)](b-x)i+2-(a-x)i+2(i+2)!(b-a)+(12-A(x))i=0n-2[f(i+1)(a)+(-1)i+1f(i+1)(b)](b-a)i+1(i+2)!,(3.2)T^n(x,s)=-n{(12-A(x))[Tn(a,s)+Tn(b,s)]+A(x)[Tn(x,s)+Tn(a+b-x,s)]}={(12+A(x))(a-s)n(b-a)+(12-A(x))(b-s)n(b-a),asx,(a-s)n+(b-s)n2(b-a),x<sa+b-x,(12-A(x))(a-s)n(b-a)+(12+A(x))(b-s)n(b-a),a+b-x<sb.

Theorem 3.1.

Let I be an open interval in , [a,b]I, and let f:I be such that f(n-1) is absolutely continuous for some n1. Then for each x(a,(a+b)/2] the following identity holds (3.3)1b-aabf(t)dt=D(x)+t^n(x)+1n!abT^n(x,s)f(n)(s)ds.

Proof.

We take w(t)=1/(b-a), t[a,b] in (2.2).

Theorem 3.2.

Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents, that is, 1p,q, 1/p+1/q=1 and f(n)Lp[a,b] for some n1. Then for each x(a,(a+b)/2] we have (3.4)|1b-aabf(t)dt-D(x)-t^n(x)|1n!T^n(x,)qf(n)p. Inequality (3.4) is sharp for 1<p.

Proof.

We take w(t)=1/(b-a), t[a,b] in (2.5).

Now, we set (3.5)A(x)=(b-a)212(x-a)(b-x),x(a,a+b2]. This special choice of the function A enables us to consider generalizations of the well-known Simpson’s 3/8 formula (1.1) and Lobatto formula (1.2)

3.1. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M117"><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>a</mml:mi><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>

Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of Simpson’s 3/8 formula reads (3.6)1b-aabf(t)dt=D(2a+b3)+t^n(2a+b3)+1n!abT^n(2a+b3,s)f(n)(s)ds, where (3.7)D(2a+b3)=18(f(a)+3f(2a+b3)+3f(a+2b3)+f(b)),t^n(2a+b3)=18i=0n-2[f(i+1)(2a+b3)+(-1)i+1f(i+1)(a+2b3)][2i+2+(-1)i+1](b-a)i+13i+1(i+2)!+18i=0n-2[f(i+1)(a)+(-1)i+1f(i+1)(b)](b-a)i+1(i+2)!,T^n(2a+b3,s)=-n8[Tn(a,s)+3Tn(2a+b3,s)+3Tn(a+2b3,s)+Tn(b,s)]={7(a-s)n+(b-s)n8(b-a)as2a+b3,(a-s)n+(b-s)n2(b-a),2a+b3<sa+2b3,(a-s)n+7(b-s)n8(b-a)a+2b3<sb.

In the next corollaries we will use the beta function and the incomplete beta function of Euler type defined by (3.8)B(x,y)=01tx-1(1-t)y-1dt,Br(x,y)=0rtx-1(1-t)y-1dt,x,y>0.

Corollary 3.3.

Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents and n.

If f(n)L[a,b], then (3.3)|1b-aabf(t)dt-D(2a+b3)|25288(b-a)f,55555555|1b-aabf(t)dt-D(2a+b3)-t^n(2a+b3)|555555555551(n+1)!([3n+1+32n+1+3(-1)n](b-a)n43n+11(n+1)!55555555555-(b-a2)n[(-1)n+1+12])f(n),n2.

If f(n)L2[a,b], then (3.10)|1b-aabf(t)dt-D(2a+b3)-t^n(2a+b3)|5551n!([32n+522n+1+11](b-a)2n-13232n(2n+1)+(-1)n(b-a)2n-132(b-a)2n-132×[7B(n+1,n+1)+9B2/3(n+1,n+1)-9B1/3(n+1,n+1)][32n+522n+1+11](b-a)2n-13232n(2n+1))1/2f(n)2.

If f(n)L1[a,b], then (3.11)55555|1b-aabf(t)dt-D(2a+b3)-t^n(2a+b3)|1n!Kn(2a+b3)f(n)1,

where K1((2a+b)/3)=5/24, K2((2a+b)/3)=(5/18)  (b-a), K3((2a+b)/3)=(7/54)(b-a)2 and Kn((2a+b)/3)=(1/8)(b-a)n-1, for n4.

The first and the second inequality are sharp.

Proof.

We apply (3.4) with x=(2a+b)/3 and p=(3.12)ab|T^n(2a+b3,s)|ds=a(2a+b)/3|7(a-s)n+(b-s)n8(b-a)|ds+(2a+b)/3(a+2b)/3|(a-s)n+(b-s)n2(b-a)|ds+(a+2b)/3b|(a-s)n+7(b-s)n8(b-a)|ds=2[3n+1-2n+1+7(-1)n](b-a)n83n+1(n+1)+(2n+1+(-1)n+1)(b-a)n3n+1(n+1)-(1+(-1)n+1)(b-a)n2n+1(n+1)=[3n+1+32n+1+3(-1)n](b-a)n43n+1(n+1)-(b-a2)n[(-1)n+1+12(n+1)], for n2 and (3.13)ab|T^1(2a+b3,s)|ds=25288(b-a). To obtain the second inequality we take p=2(3.14)ab|T^n(2a+b3,s)|2ds=a(2a+b)/3|7(a-s)n+(b-s)n8(b-a)|2ds+(2a+b)/3(a+2b)/3|(a-s)n+(b-s)n2(b-a)|2ds+(a+2b)/3b|(a-s)n+7(b-s)n8(b-a)|2ds=[32n+522n+1+11](b-a)2n-13232n(2n+1)+(-1)n(b-a)2n-132×[7B(n+1,n+1)+9B2/3(n+1,n+1)-9B1/3(n+1,n+1)]. If p=1, we have (3.15)sups[a,b]|T^n(2a+b3,s)|=max{sups[a,(2a+b)/3]|7(a-s)n+(b-s)n8(b-a)|,max222sups[(2a+b)/3,(a+2b)/3]|(a-s)n+(b-s)n2(b-a)|max222sups[(a+2b)/3,b]|(a-s)n+7(b-s)n8(b-a)|}.

By an elementary calculation we get (3.16)sups[a,(2a+b)/3]|7(a-s)+(b-s)8(b-a)|=sups[(a+2b)/3,b]|(a-s)+7(b-s)8(b-a)|=524(b-a),sups[a,(2a+b)/3]|7(a-s)2+(b-s)28(b-a)|=sups[(a+2b)/3,b]|(a-s)2+7(b-s)28(b-a)|=1172(b-a),sups[a,(2a+b)/3]|7(a-s)n+(b-s)n8(b-a)|=sups[(a+2b)/3,b]|(a-s)n+7(b-s)n8(b-a)|=(b-a)n-18, for n3. The function y:[a,b], y(x)=(a-x)n+(b-x)n, is decreasing on a,(a+b)/2 and increasing on (a+b)/2,b if n is even, and decreasing on a,b if n is odd. Thus (3.17)sups[(2a+b)/3,(a+2b)/3]|(a-s)n+(b-s)n2(b-a)|=((-1)n+2n)(b-a)n-123n. Finally, (3.18)sups[a,b]|T^1(2a+b3,s)|=524 and for n2(3.19)sups[a,b]|T^n(2a+b3,s)|=(b-a)n-1max{18,2n+(-1)n23n}.

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Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of Lobatto formula reads (3.20)12-11f(t)dt=D(-55)+t^n(-55)+1n!-11T^n(-55,s)f(n)(s)ds, where (3.21)D(-55)=112(f(-1)+5f(-55)+5f(55)+f(1)),t^n(-55)=512i=0n-2[f(i+1)(-55)+(-1)i+1f(i+1)(55)]×(5+5)i+2+(-1)i+1(5-5)i+225i+2(i+2)!+112i=0n-2[f(i+1)(-1)+(-1)i+1f(i+1)(1)]2i+1(i+2)!,T^n(-55,s)=-n12[Tn(-1,s)+5Tn(-55,s)+5Tn(55,s)+Tn(1,s)]={11(-1-s)n+(1-s)n24-1s-55,(-1-s)n+(1-s)n4-55<s55,(-1-s)n+11(1-s)n2455<s1.

Corollary 3.4.

Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents and n.

if f(n)L[-1,1], then (3.22)|12-11f(t)dt-D(-55)|(101180-56)f,|12-11f(t)dt-D(-55)-t^n(-55)|1(n+1)!(2n+15n+(5+5)n+1-(-5+5)n+1125n1(n+1)!55-1+(-1)n+122n+15n+(5+5)n+1-(-5+5)n+1125n)f(n),n2.

if f(n)L2[-1,1], then (3.23)|12-11f(t)dt-D(-55)-t^n(-55)|1n!2n-23(35(5+5)2n+1+85(5-5)2n+1+102n+1102n+1(2n+1)1n!2n-23555+(-1)n[11B(n+1,n+1)+25B(5+5)/10(n+1,n+1)5555555555555-25B(5-5)/10(n+1,n+1)](5-5)2n+1102n+1)1/2f(n)2.

if f(n)L1[-1,1], then (3.24)|12-11f(t)dt-D(-55)-t^n(-55)|1n!Kn(-55)f(n)1,

where K1(-5/5)=1/(25), K2(-5/5)=3/5, K3(-5/5)=8/(55), K4(-5/5)=28/25, K5(-5/5)=88/(255), Kn(-5/5)=2n-3/3, for n6.

The first and the second inequality are sharp.

Proof.

Applying (3.4) with [a,b]=[-1,1], x=-5/5 and p=,p=2,p=1 and carrying out the same analysis as in Corollay 3.3 we obtain the above inequalities.

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