# Sharp Integral Inequalities Based on a General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

Authors: J. Pečarić; M. Ribičić Penava
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

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 toLp spaces. Generalizations of Simpson’s
3/8 formula and the Lobatto four-point formula with related inequalities are
considered as special cases.


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## Body

## 1. Introduction

The most elementary quadrature rules in four nodes are Simpson’s3/8 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].Letf:[a,b]→ℝ be differentiable on [a,b] and f′:[a,b]→ℝ integrable on [a,b].Then the Montgomery identity holds (see [1])
(1.3)f(x)=1b-a∫ab‍f(t)dt+∫ab‍P(x,t)f′(t)dt,
where the Peano kernel is
(1.4)P(x,t)={t-ab-a,a≤t≤x,t-bb-a,x<t≤b.In [2], 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),a≤t≤x,W(t)-1,x<t≤b.
Now, let us suppose that I is an open interval in ℝ, [a,b]⊂I, f:I→ℝ is such that f(n-1) is absolutely continuous for some n≥2, 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 [3])
(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,a≤s≤x,-∫sb‍w(u)(u-s)n-1du,x<s≤b.
If we take w(t)=1/(b-a), t∈[a,b], equality (1.7) reduces to
(1.9)f(x)=1b-a∫ab‍f(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),a≤s≤x,-(b-s)nn(b-a),x<s≤b.
For n=1, (1.9) reduces to the Montgomery identity (1.3).In this paper, we generalize the results from [4]. 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]→ℝ 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

Letf:[a,b]→ℝ be such that f(n-1) exists on [a,b] for some n≥2. 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,a≤s≤x,-12[∫as‍w(u)(u-s)n-1du-∫sb‍w(u)(u-s)n-1du],x<s≤a+b-x,-(12-A(x))∫as‍w(u)(u-s)n-1du+(12+A(x))∫sb‍w(u)(u-s)n-1du,a+b-x<s≤b.In the next theorem we establish the general weighted four-point formula.Theorem 2.1.
LetI be an open interval in ℝ, [a,b]⊂I, and let w:[a,b]→[0,∞〉 be some probability density function. Let f:I→ℝ be such that f(n-1) is absolutely continuous for some n≥2. 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 putx≡a,x≡x,x≡a+b-x and x≡b in (1.7) to obtain four new formulae. After multiplying these four formulae by 1/2-A(x),A(x),A(x)and1/2-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 x≡a,x≡x,x≡a+b-x and x≡b 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),a≤s≤x,12-W(s),x<s≤a+b-x,12+A(x)-W(s),a+b-x<s≤b.Theorem 2.3.
Suppose that all assumptions of Theorem2.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents, that is, 1≤p,q≤∞, 1/p+1/q=1, let f(n)∈Lp[a,b] for some n≥1. 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,⋅)‖q‖f(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,⋅)‖q‖f(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|=‖Unx‖q‖f(n)‖p.
For 1<p<∞, take f to be such that
(2.8)f(n)(s)=sign⁡Unx(s)⋅|Unx(s)|1/(p-1),
where for p=∞ we put
(2.9)f(n)(s)=sign⁡Unx(s).Remark 2.4.
Inequality (2.5) for A(x)=1/4 was proved by Aglić Aljinović et al. in [4].


## 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),a≤s≤x,(a-s)n+(b-s)n2(b-a),x<s≤a+b-x,(12-A(x))(a-s)n(b-a)+(12+A(x))(b-s)n(b-a),a+b-x<s≤b.Theorem 3.1.
LetI be an open interval in ℝ, [a,b]⊂I, and let f:I→ℝ be such that f(n-1) is absolutely continuous for some n≥1. Then for each x∈(a,(a+b)/2] the following identity holds
(3.3)1b-a∫ab‍f(t)dt=D(x)+t^n(x)+1n!∫ab‍T^n(x,s)f(n)(s)ds.Proof.
We takew(t)=1/(b-a), t∈[a,b] in (2.2).Theorem 3.2.
Suppose that all assumptions of Theorem3.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents, that is, 1≤p,q≤∞, 1/p+1/q=1 and f(n)∈Lp[a,b] for some n≥1. Then for each x∈(a,(a+b)/2] we have
(3.4)|1b-a∫ab‍f(t)dt-D(x)-t^n(x)|≤1n!‖T^n(x,⋅)‖q‖f(n)‖p.
Inequality (3.4) is sharp for 1<p≤∞.Proof.
We takew(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.x=(2a+b)/3

Suppose that all assumptions of Theorem3.1 hold. Then the following generalization of Simpson’s 3/8 formula reads
(3.6)1b-a∫ab‍f(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)=18∑i=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)!+18∑i=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)a≤s≤2a+b3,(a-s)n+(b-s)n2(b-a),2a+b3<s≤a+2b3,(a-s)n+7(b-s)n8(b-a)a+2b3<s≤b.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 Theorem3.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents and n∈ℕ. (a)
Iff(n)∈L∞[a,b], then
(3.3)|1b-a∫ab‍f(t)dt-D(2a+b3)|≤25288(b-a)‖f′‖∞,55555555|1b-a∫ab‍f(t)dt-D(2a+b3)-t^n(2a+b3)|55555555555≤1(n+1)!([3n+1+3⋅2n+1+3(-1)n](b-a)n4⋅3n+1≤1(n+1)!55555555555≤-(b-a2)n[(-1)n+1+12])‖f(n)‖∞,n≥2.(b)
Iff(n)∈L2[a,b], then
(3.10)≤|1b-a∫ab‍f(t)dt-D(2a+b3)-t^n(2a+b3)|555≤1n!([32n+5⋅22n+1+11](b-a)2n-132⋅32n(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+5⋅22n+1+11](b-a)2n-132⋅32n(2n+1))1/2‖f(n)‖2.(c)
Iff(n)∈L1[a,b], then
(3.11)55555|1b-a∫ab‍f(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 n≥4.
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)n8⋅3n+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+3⋅2n+1+3(-1)n](b-a)n4⋅3n+1(n+1)-(b-a2)n[(-1)n+1+12(n+1)],
for n≥2 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+5⋅22n+1+11](b-a)2n-132⋅32n(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)sup⁡s∈[a,b]|T^n(2a+b3,s)|=max⁡{sup⁡s∈[a,(2a+b)/3]|7(a-s)n+(b-s)n8(b-a)|,max⁡222sup⁡s∈[(2a+b)/3,(a+2b)/3]|(a-s)n+(b-s)n2(b-a)|max⁡222sup⁡s∈[(a+2b)/3,b]|(a-s)n+7(b-s)n8(b-a)|}.
By an elementary calculation we get(3.16)sup⁡s∈[a,(2a+b)/3]|7(a-s)+(b-s)8(b-a)|=sup⁡s∈[(a+2b)/3,b]|(a-s)+7(b-s)8(b-a)|=524(b-a),sup⁡s∈[a,(2a+b)/3]|7(a-s)2+(b-s)28(b-a)|=sup⁡s∈[(a+2b)/3,b]|(a-s)2+7(b-s)28(b-a)|=1172(b-a),sup⁡s∈[a,(2a+b)/3]|7(a-s)n+(b-s)n8(b-a)|=sup⁡s∈[(a+2b)/3,b]|(a-s)n+7(b-s)n8(b-a)|=(b-a)n-18,
for n≥3. The function y:[a,b]→ℝ, 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)sup⁡s∈[(2a+b)/3,(a+2b)/3]|(a-s)n+(b-s)n2(b-a)|=((-1)n+2n)(b-a)n-12⋅3n.
Finally,
(3.18)sup⁡s∈[a,b]|T^1(2a+b3,s)|=524
and for n≥2(3.19)sup⁡s∈[a,b]|T^n(2a+b3,s)|=(b-a)n-1max⁡{18,2n+(-1)n2⋅3n}.


### 3.2.[a,b]=[-1,1],x=-5/5

Suppose that all assumptions of Theorem3.1 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)=512∑i=0n-2‍[f(i+1)(-55)+(-1)i+1f(i+1)(55)]×(5+5)i+2+(-1)i+1(5-5)i+22⋅5i+2(i+2)!+112∑i=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)n24-1≤s≤-55,(-1-s)n+(1-s)n4-55<s≤55,(-1-s)n+11(1-s)n2455<s≤1.Corollary 3.4.
Suppose that all assumptions of Theorem3.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents and n∈ℕ. (a)
iff(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+1⋅5n+(5+5)n+1-(-5+5)n+112⋅5n≤1(n+1)!55≤-1+(-1)n+122n+1⋅5n+(5+5)n+1-(-5+5)n+112⋅5n)‖f(n)‖∞,n≥2.(b)
iff(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)1/2‖f(n)‖2.(c)
iff(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 n≥6.
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.


## 3.1.x=(2a+b)/3

Suppose that all assumptions of Theorem3.1 hold. Then the following generalization of Simpson’s 3/8 formula reads
(3.6)1b-a∫ab‍f(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)=18∑i=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)!+18∑i=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)a≤s≤2a+b3,(a-s)n+(b-s)n2(b-a),2a+b3<s≤a+2b3,(a-s)n+7(b-s)n8(b-a)a+2b3<s≤b.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 Theorem3.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents and n∈ℕ. (a)
Iff(n)∈L∞[a,b], then
(3.3)|1b-a∫ab‍f(t)dt-D(2a+b3)|≤25288(b-a)‖f′‖∞,55555555|1b-a∫ab‍f(t)dt-D(2a+b3)-t^n(2a+b3)|55555555555≤1(n+1)!([3n+1+3⋅2n+1+3(-1)n](b-a)n4⋅3n+1≤1(n+1)!55555555555≤-(b-a2)n[(-1)n+1+12])‖f(n)‖∞,n≥2.(b)
Iff(n)∈L2[a,b], then
(3.10)≤|1b-a∫ab‍f(t)dt-D(2a+b3)-t^n(2a+b3)|555≤1n!([32n+5⋅22n+1+11](b-a)2n-132⋅32n(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+5⋅22n+1+11](b-a)2n-132⋅32n(2n+1))1/2‖f(n)‖2.(c)
Iff(n)∈L1[a,b], then
(3.11)55555|1b-a∫ab‍f(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 n≥4.
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)n8⋅3n+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+3⋅2n+1+3(-1)n](b-a)n4⋅3n+1(n+1)-(b-a2)n[(-1)n+1+12(n+1)],
for n≥2 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+5⋅22n+1+11](b-a)2n-132⋅32n(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)sup⁡s∈[a,b]|T^n(2a+b3,s)|=max⁡{sup⁡s∈[a,(2a+b)/3]|7(a-s)n+(b-s)n8(b-a)|,max⁡222sup⁡s∈[(2a+b)/3,(a+2b)/3]|(a-s)n+(b-s)n2(b-a)|max⁡222sup⁡s∈[(a+2b)/3,b]|(a-s)n+7(b-s)n8(b-a)|}.
By an elementary calculation we get(3.16)sup⁡s∈[a,(2a+b)/3]|7(a-s)+(b-s)8(b-a)|=sup⁡s∈[(a+2b)/3,b]|(a-s)+7(b-s)8(b-a)|=524(b-a),sup⁡s∈[a,(2a+b)/3]|7(a-s)2+(b-s)28(b-a)|=sup⁡s∈[(a+2b)/3,b]|(a-s)2+7(b-s)28(b-a)|=1172(b-a),sup⁡s∈[a,(2a+b)/3]|7(a-s)n+(b-s)n8(b-a)|=sup⁡s∈[(a+2b)/3,b]|(a-s)n+7(b-s)n8(b-a)|=(b-a)n-18,
for n≥3. The function y:[a,b]→ℝ, 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)sup⁡s∈[(2a+b)/3,(a+2b)/3]|(a-s)n+(b-s)n2(b-a)|=((-1)n+2n)(b-a)n-12⋅3n.
Finally,
(3.18)sup⁡s∈[a,b]|T^1(2a+b3,s)|=524
and for n≥2(3.19)sup⁡s∈[a,b]|T^n(2a+b3,s)|=(b-a)n-1max⁡{18,2n+(-1)n2⋅3n}.


## 3.2.[a,b]=[-1,1],x=-5/5

Suppose that all assumptions of Theorem3.1 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)=512∑i=0n-2‍[f(i+1)(-55)+(-1)i+1f(i+1)(55)]×(5+5)i+2+(-1)i+1(5-5)i+22⋅5i+2(i+2)!+112∑i=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)n24-1≤s≤-55,(-1-s)n+(1-s)n4-55<s≤55,(-1-s)n+11(1-s)n2455<s≤1.Corollary 3.4.
Suppose that all assumptions of Theorem3.1 hold. Additionally, assume that (p,q) is a pair of conjugate exponents and n∈ℕ. (a)
iff(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+1⋅5n+(5+5)n+1-(-5+5)n+112⋅5n≤1(n+1)!55≤-1+(-1)n+122n+1⋅5n+(5+5)n+1-(-5+5)n+112⋅5n)‖f(n)‖∞,n≥2.(b)
iff(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)1/2‖f(n)‖2.(c)
iff(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 n≥6.
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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*Source: 343191-2012-08-13.xml*