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volume 3, issue 2, article 21, 2002.

Received December, 2000;

accepted 16 November, 2001.

Communicated by:T. Mills

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Journal of Inequalities in Pure and Applied Mathematics

INTEGRAL INEQUALITIES OF THE OSTROWSKI TYPE

A. SOFO

School of Communications and Informatics Victoria University of Technology

PO Box 14428 Melbourne City MC 8001 Victoria, Australia EMail:sofo@matilda.vu.edu.au URL:http://rgmia.vu.edu.au/sofo/

c

2000Victoria University ISSN (electronic): 1443-5756 083-01

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Integral Inequalities of the Ostrowski Type

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J. Ineq. Pure and Appl. Math. 3(2) Art. 21, 2002

http://jipam.vu.edu.au

Abstract

Integral inequalities of Ostrowski type are developed forn−times differentiable mappings, with multiple branches, on theL∞norm. Some particular inequalities are also investigated, which include explicit bounds for perturbed trapezoid, midpoint, Simpson’s, Newton-Cotes and left and right rectangle rules. The results obtained provide sharper bounds than those obtained by Dragomir [5] and Cerone, Dragomir and Roumeliotis [2].

2000 Mathematics Subject Classification:26D15, 41A55.

Key words: Ostrowski Integral Inequality, Quadrature Formulae.

I wish to acknowledge the useful comments of an anonymous referee which led to an improvement of this work.

1. Introduction

In 1938 Ostrowski [18] obtained a bound for the absolute value of the difference of a function to its average over a finite interval. The theorem is as follows.

Theorem 1.1. Let f : [a, b] → Rbe a differentiable mapping on [a, b]and let

|f⁰(t)| ≤M for allt ∈(a, b), then the following bound is valid (1.1)

f(x)− 1 b−a

Z b a

f(t)dt

≤(b−a)M

"

1

4+ x− ^a+b₂ 2

(b−a)²

#

for allx∈[a, b].

The constant ¹₄ is sharp in the sense that it cannot be replaced by a smaller one.

Dragomir and Wang [12, 13] extended the result (1.1) and applied the extended result to numerical quadrature rules and to the estimation of error bounds for some special means. Dragomir [8,9,10] further extended the result (1.1) to incorporate mappings of bounded variation, Lipschitzian mappings and mono- tonic mappings.

Cerone, Dragomir and Roumeliotis [3] as well as Dedić, Matić and Peˇcarić [4] and Pearce, Peˇcarić, Ujević and Varošanec [19] further extended the result (1.1) by consideringn−times differentiable mappings on an interior pointx∈ [a, b].

In particular, Cerone and Dragomir [1] proved the following result.

Theorem 1.2. Let f : [a, b] → Rbe a mapping such that f⁽ⁿ⁻¹⁾ is absolutely continuous on [a, b], (AC[a, b]for short). Then for allx ∈ [a, b]the following

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bound is valid:

(1.2)

Z b a

f(t)dt−

n−1

X

j=0

(b−x)^j+1+ (−1)^j(x−a)^j+1 (j + 1)!

!

f^(j)(x)

≤

f⁽ⁿ⁾ _∞

(n+ 1)! (x−a)ⁿ⁺¹+ (b−x)ⁿ⁺¹

if f⁽ⁿ⁾ ∈L_∞[a, b], where

f⁽ⁿ⁾

_∞:= sup

t∈[a,b]

f⁽ⁿ⁾(t) <∞.

Dragomir also generalised the Ostrowski inequality fork points,x₁, . . . , x_k and obtained the following theorem.

Theorem 1.3. Let I_k := a = x₀ < x₁ < ... < xk−1 < x_k = b be a division of the interval [a, b], α_i (i= 0, ..., k+ 1) be “k + 2” points so that α₀ = a, α_i ∈[x_i−1, x_i] (i= 1, ..., k)andα_k+1 =b. Iff : [a, b]→Ris AC[a, b], then we have the inequality

Z b a

f(x)dx−

k

X

i=0

(α_i+1−α_i)f(x_i) (1.3)

≤

"

1 4

k−1

X

i=0

h²_i +

k−1

X

i=0

α_i+1− x_i+x_i+1 2

2# kf⁰k_∞

≤ 1 2kf⁰k_∞

k−1

X

i=0

h²_i ≤ 1

2(b−a)kf⁰k_∞ν(h) ,

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whereh_i :=x_i+1−x_i(i= 0, ..., k−1)andν(h) := max{h_i|i= 0, ..., k−1}.

The constant ¹₄ in the first inequality and the constant ¹₂ in the second and third inequalities are the best possible.

The main aim of this paper is to develop the upcoming Theorem 3.1 (see page 10) of integral inequalities forn−times differentiable mappings. The mo- tivation for this work has been to improve the order of accuracy of the results given by Dragomir [5], and Cerone and Dragomir [1]. In the case of Dragomir [5], the bound of (1.3) is of order 1, whilst in this paper we improve the bound of (1.3) to ordern.

We begin the process by obtaining the following integral equalities.

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2. Integral Identities

Theorem 2.1. Let I_k : a = x₀ < x₁ < · · · < xk−1 < x_k = bbe a division of the interval [a, b]and αi (i= 0, . . . , k+ 1) be ‘k + 2’ points so that α0 = a, α_i ∈[xi−1, x_i] (i= 1, . . . , k)andα_k+1 =b. Iff : [a, b]→Ris a mapping such thatf⁽ⁿ⁻¹⁾is AC[a, b], then for allx_i ∈[a, b]we have the identity:

(2.1) Z b

a

f(t)dt+

n

X

j=1

(−1)^j j!

k−1

X

i=0

n

(x_i+1−α_i+1)^jf^(j−1)(x_i+1)

−(x_i−α_i+1)^jf^(j−1)(x_i)o

= (−1)ⁿ Z b

a

K_n,k(t)f⁽ⁿ⁾(t)dt, where the Peano kernel

(2.2) Kn,k(t) :=











(t−α₁)ⁿ

n! , t ∈[a, x₁) (t−α₂)ⁿ

n! , t ∈[x₁, x₂) ...

(t−αk−1)ⁿ

n! , t ∈[xk−2, xk−1) (t−αk)ⁿ

n! , t ∈[xk−1, b], nandkare natural numbers,n ≥1,k ≥1andf⁽⁰⁾(x) =f(x).

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Proof. The proof is by mathematical induction. Forn = 1, from (2.1) we have the equality

(2.3) Z b

a

f(t)dt =

k−1

X

i=0

h

(x_i+1−α_i+1)^jf(x_i+1)−(x_i−α_i+1)^jf(x_i)i

− Z b

a

K1,k(t)f⁰(t)dt, where

K_1,k(t) :=











(t−α1), t∈[a, x1) (t−α₂), t∈[x₁, x₂)

...

(t−αk−1), t∈[xk−2, xk−1) (t−α_k), t∈[xk−1, b]. To prove (2.3), we integrate by parts as follows

Z b a

K_1,k(t)f⁰(t)dt

=

k−1

X

i=0

Z xi+1

xi

(t−α_i+1)f⁰(t)dt

=

k−1

X

i=0

(t−α_i+1)f(t)

xi+1

xi − Z xi+1

xi

f(t)dt

=

k−1

X

i=0

[(x_i+1−α_i+1)f(x_i+1)−(x_i−α_i+1)f(x_i)]−

k−1

X

i=0

Z xi+1

xi

f(t)dt,

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Z b a

f(t)dt+ Z b

a

K_1,k(t)f⁰(t)dt

=

k−1

X

i=0

[(xi+1−αi+1)f(xi+1)−(xi−αi+1)f(xi)]. Hence (2.3) is proved.

Assume that (2.1) holds for ‘n’ and let us prove it for ‘n+ 1’. We need to prove the equality

(2.4) Z b

a

f(t)dt+

k−1

X

i=0 n+1

X

j=1

(−1)^j j!

×n

(x_i+1−α_i+1)^jf^(j−1)(x_i+1)−(x_i−α_i+1)^jf^(j−1)(x_i)o

= (−1)ⁿ⁺¹ Z b

a

K_n+1,k(t)f⁽ⁿ⁺¹⁾(t)dt, where from (2.2)

K_n+1,k(t) :=











(t−α₁)ⁿ⁺¹

(n+1)! , t∈[a, x₁)

(t−α₂)ⁿ⁺¹

(n+1)! , t∈[x₁, x₂) ...

(t−α_k−1)ⁿ⁺¹

(n+1)! , t∈[xk−2, xk−1)

(t−α_k)ⁿ⁺¹

(n+1)! , t∈[xk−1, b].

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Consider Z b a

K_n+1,k(t)f⁽ⁿ⁺¹⁾(t)dt=

k−1

X

i=0

Z xi+1

xi

(t−α_i+1)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁺¹⁾(t)dt and upon integrating by parts we have

Z b a

K_n+1,k(t)f⁽ⁿ⁺¹⁾(t)dt

=

k−1

X

i=0

"

(t−α_i+1)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁾(t)

xi+1

xi

− Z xi+1

xi

(t−α_i+1)ⁿ

n! f⁽ⁿ⁾(t)dt

#

=

k−1

X

i=0

((x_i+1−α_i+1)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁾(xi+1)− (x_i−α_i+1)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁾(xi) )

− Z b

a

K_n,k(t)f⁽ⁿ⁾(t)dt.

Upon rearrangement we may write Z b

a

K_n,k(t)f⁽ⁿ⁾(t)dt

=

k−1

X

i=0

((x_i+1−α_i+1)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁾(x_i+1)− (x_i−α_i+1)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁾(x_i) )

− Z b

a

K_n+1,k(t)f⁽ⁿ⁺¹⁾(t)dt.

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Now substitute Rb

a K_n,k(t)f⁽ⁿ⁾(t)dt from the induction hypothesis (2.1) such that

(−1)ⁿ Z b

a

f(t)dt+ (−1)ⁿ

k−1

X

i=0

" _n X

j=1

(−1)^j j!

n

(x_i+1−α_i+1)^jf^(j−1)(x_i+1)

− (x_i−α_i+1)^jf^(j−1)(x_i)oi

=

k−1

X

i=0

((x_i+1−α_i+1)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁾(x_i+1)− (x_i−α_i+1)ⁿ⁺¹

(n+ 1)! f⁽ⁿ⁾(x_i) )

− Z b

a

K_n+1,k(t)f⁽ⁿ⁺¹⁾(t)dt.

Collecting the second and third terms and rearranging, we can state Z b

a

f(t)dt+

k−1

X

i=0 n+1

X

j=1

(−1)^j j!

×n

(x_i+1−α_i+1)^jf^(j−1)(x_i+1) −(x_i−α_i+1)^jf^(j−1)(x_i)o

= (−1)ⁿ⁺¹ Z b

a

K_n+1,k(t)f⁽ⁿ⁺¹⁾(t)dt, which is identical to (2.4), hence Theorem2.1is proved.

The following corollary gives a slightly different representation of Theorem 2.1, which will be useful in the following work.

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Corollary 2.2. From Theorem2.1, the equality (2.1) may be represented as (2.5)

Z b a

f(t)dt +

n

X

j=1

(−1)^j j!

" _k X

i=0

n

(x_i−α_i)^j−(x_i−α_i+1)^jo

f^(j−1)(x_i)

#

= (−1)ⁿ Z b

a

Proof. From (2.1) consider the second term and rewrite it as

(2.6) S₁+S₂ :=

k−1

X

i=0

{−(x_i+1−α_i+1)f(x_i+1) + (x_i−α_i+1)f(x_i)}

+

k−1

X

i=0

" _n X

j=2

(−1)^j j!

n

(x_i+1−α_i+1)^jf^(j−1)(x_i+1)

− (x_i−α_i+1)^jf^(j−1)(x_i)o . Now

S₁ = (a−α₁)f(a) +

k−1

X

i=1

(x_i−α_i+1)f(x_i)

+

k−2

X

i=0

{−(x_i+1−α_i+1)f(x_i+1)} −(b−α_k)f(b)

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= (a−α₁)f(a) +

k−1

X

i=1

(x_i−α_i+1)f(x_i)

+

k−1

X

i=1

{−(x_i−α_i)f(x_i)} −(b−α_k)f(b)

=−(α1−a)f(a)−

k−1

X

i=1

(αi+1−αi)f(xi)−(b−αk)f(b). Also,

S₂ =

k−2

X

i=0

" _n X

j=2

(−1)^j j!

n

(x_i+1−α_i+1)^jf^(j−1)(x_i+1)o

#

+

n

X

j=2

(−1)^j j!

n

(x_k−α_k)^jf^(j−1)(x_k)o

−

n

X

j=2

(−1)^j j!

n

(x0−α1)^jf^(j−1)(x0) o

−

k−1

X

i=1

" _n X

j=2

(−1)^j j!

n

(x_i−α_i+1)^jf^(j−1)(x_i)o

#

=

n

X

j=2

(−1)^j

j! (b−α_k)^jf^(j−1)(b)−

n

X

j=2

(−1)^j

j! (a−α₁)^jf^(j−1)(a)

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+

k−1

X

i=1

" _n X

j=2

(−1)^j j!

n

(x_i −α_i)^j−(x_i−α_i+1)^jo

f^(j−1)(x_i)

# . From (2.6)

S1+S2

:=− (

(b−αk)f(b) + (α1−a)f(a) +

k−1

X

i=1

(αi+1−αi)f(xi) )

+

n

X

j=2

(−1)^j j!

n

(b−α_k)^jf^(j−1)(b)−(a−α₁)^jf^(j−1)(a)o

+

k−1

X

i=1

" _n X

j=2

(−1)^j j!

n

(x_i−α_i)^j −(x_i−α_i+1)^jo

f^(j−1)(x_i)

#

=− (

(α₁−a)f(a) +

k−1

X

i=1

(α_i+1−α_i)f(x_i) + (b−α_k)f(b) )

+

n

X

j=2

(−1)^j j!

"

−(a−α₁)^jf^(j−1)(a)

+

k−1

X

i=1

n

f^(j−1)(x_i) + (b−α_k)^jf^(j−1)(b)

# .

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Keeping in mind thatx₀ =a,α₀ = 0,x_k =bandα_k+1 =bwe may write S₁+S₂ =−

k

X

i=0

(α_i+1−α_i)f(x_i)

+

n

X

j=2

(−1)^j j!

" _k X

i=0

n

f^(j−1)(x_i)

#

=

n

X

j=1

(−1)^j j!

" _k X

i=0

n

(xi −αi)^j−(xi−αi+1)^j o

f^(j−1)(xi)

# . And substituting S1 +S2 into the second term of (2.1) we obtain the identity (2.5).

If we now assume that the points of the divisionIk are fixed, we obtain the following corollary.

Corollary 2.3. LetI_k :a =x₀ < x₁ <· · · < x_k−1 < x_k =b be a division of the interval[a, b]. Iff : [a, b]→ Ris as defined in Theorem2.1, then we have the equality

(2.7) Z b

a

f(t)dt+

n

X

j=1

1 2^jj!

" _k X

i=0

n−h^j_i + (−1)^jh^j_i−1o

f^(j−1)(x_i)

#

= (−1)ⁿ Z b

a

K_n,k(t)f⁽ⁿ⁾(t)dt, wherehi :=xi+1−xi,h−1 := 0andhk := 0.

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Proof. Choose

α₀ = a, α₁ = a+x₁

2 , α₂ = x₁+x₂ 2 , . . . , αk−1 = xk−2+xk−1

2 ,α_k = xk−1+x_k

2 and α_k+1 =b.

From Corollary2.2, the term

(b−α_k)f(b) + (α₁−a)f(a) +

k−1

X

i=1

(α_i+1−α_i)f(x_i)

= 1 2

(

h₀f(a) +

k−1

X

i=1

(h_i+h_i−1)f(x_i) +h_k−1f(b) )

, the term

n

X

j=2

(−1)^j j!

n

(b−α_k)^jf^(j−1)(b)−(a−α₁)^jf^(j−1)(a)o

=

n

X

j=2

(−1)^j j!2^j

n

h^j_k−1f^(j−1)(b)−(−1)^jh^j₀f^(j−1)(a)o and the term

k−1

X

i=1

" _n X

j=2

(−1)^j j!

n

f^(j−1)(x_i)

#

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=

k−1

X

i=1

" _n X

j=2

(−1)^j j!2^j

n

h^j_i−1−(−1)^jh^j_io

f^(j−1)(x_i)

# . Putting the last three terms in (2.5) we obtain

Z b a

f(t)dt− 1 2

(

h0f(a) +

k−1

X

i=1

(hi+hi−1)f(xi) +hk−1f(b) )

+

n

X

j=2

(−1)^j j!2^j

n

h^j_k−1f^(j−1)(b)−(−1)^jh^j₀f^(j−1)(a)o

+

k−1

X

i=1

" _n X

j=2

(−1)^j j!2^j

n

h^j_i−1−(−1)^jh^j_io

f^(j−1)(x_i)

#

= (−1)ⁿ Z b

a

Collecting the inner three terms of the last expression, we have Z b

a

f(t)dt+

n

X

j=1

(−1)^j j!2^j

k

X

i=0

n

h^j_i−1−(−1)^jh^j_i o

f^(j−1)(xi)

= (−1)ⁿ Z b

a

K_n,k(t)f⁽ⁿ⁾(t)dt, which is equivalent to the identity (2.7).

The case of equidistant partitioning is important in practice, and with this in mind we obtain the following corollary.

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Corollary 2.4. Let

(2.8) I_k:x_i =a+i

b−a k

, i= 0, . . . , k

be an equidistant partitioning of[a, b], andf : [a, b] → Rbe a mapping such thatf⁽ⁿ⁻¹⁾is AC[a, b],then we have the equality

(2.9) Z b

a

f(t)dt+

n

X

j=1

b−a 2k

j

1 j!

"

−f^(j−1)(a)

+

k−1

X

i=1

n(−1)^j −1o

f^(j−1)(x_i) + (−1)^jf^(j−1)(b)

#

= (−1)ⁿ Z b

a

It is of some interest to note that the second term of (2.9) involves only even derivatives at all interior pointsxi,i= 1, . . . , k−1.

Proof. Using (2.8) we note that

h0 = x1−x0 = b−a

k , hk−1 = (xk−xk−1) = b−a k , hi = xi+1−xi = b−a

k and h_i−1 = x_i−x_i−1 = b−a

k , (i= 1, . . . , k−1)

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and substituting into (2.7) we have Z b

a

f(t)dt+

n

X

j=1

1 j!2^j

"

−

b−a 2k

j

f^(j−1)(a) +

k−1

X

i=0

(

−

b−a k

j

+ (−1)^j

b−a k

j)

f^(j−1)(x_i) + (−1)^j

b−a k

j

f^(j−1)(b)

#

= (−1)ⁿ Z b

a

K_n,k(t)f⁽ⁿ⁾(t)dt, which simplifies to (2.9) after some minor manipulation.

The following Taylor-like formula with integral remainder also holds.

Corollary 2.5. Letg : [a, y]→Rbe a mapping such thatg⁽ⁿ⁾is AC[a, y]. Then for allx_i ∈[a, y]we have the identity

(2.10) g(y) =g(a)−

k−1

X

i=0

" _n X

j=1

(−1)^j j!

n

(x_i+1−α_i+1)^jg^(j)(x_i+1)

− (x_i−α_i+1)^jg^(j)(x_i)o

+ (−1)ⁿ Z y

a

K_n,k(y, t)g⁽ⁿ⁺¹⁾(t)dt or

(2.11) g(y)

=g(a)−

n

X

j=1

(−1)^j j!

" _k X

i=0

n

g^(j)(x_i)

#

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+ (−1)ⁿ Z y

a

K_n,k(y, t)g⁽ⁿ⁺¹⁾(t)dt.

The proof of (2.10) and (2.11) follows directly from (2.1) and (2.5) respec- tively upon choosingb=yandf =g⁰.

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3. Integral Inequalities

In this section we utilise the equalities of Section2and develop inequalities for the representation of the integral of a function with respect to its derivatives at a multiple number of points within some interval.

Theorem 3.1. Let I_k : a = x₀ < x₁ < · · · < xk−1 < x_k = bbe a division of the interval [a, b]and αi (i= 0, . . . , k+ 1) be ‘k + 2’ points so that α0 = a, α_i ∈[xi−1, x_i] (i= 1, . . . , k)andα_k+1 =b. Iff : [a, b]→Ris a mapping such thatf⁽ⁿ⁻¹⁾is AC[a, b], then for allx_i ∈[a, b]we have the inequality:

Z b a

f(t)dt+

n

X

j=1

(−1)^j j!

" _k X

i=0

n

(x_i −α_i)^j−(x_i−α_i+1)^jo

f^(j−1)(x_i)

# (3.1)

≤

f⁽ⁿ⁾ _∞ (n+ 1)!

k−1

X

i=0

(α_i+1−x_i)ⁿ⁺¹+ (x_i+1−α_i+1)ⁿ⁺¹

≤

f⁽ⁿ⁾ _∞ (n+ 1)!

k−1

X

i=0

hⁿ⁺¹_i

≤

f⁽ⁿ⁾ _∞

(n+ 1)! (b−a)νⁿ(h) if f⁽ⁿ⁾ ∈L∞[a, b], where

f⁽ⁿ⁾

_∞ := sup

t∈[a,b]

f⁽ⁿ⁾(t) <∞, h_i :=x_i+1−x_i and

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ν(h) := max{h_i|i= 0, . . . , k−1}. Proof. From Corollary2.2we may write

(3.2)

Z b a

f(t)dt +

n

X

j=1

(−1)^j j!

" _k X

i=0

n

f^(j−1)(x_i)

#

=

(−1)ⁿ Z b

a

K_n,k(t)f⁽ⁿ⁾(t)dt ,

and

(−1)ⁿ Z b

a

K_n,k(t)f⁽ⁿ⁾(t)dt

≤ f⁽ⁿ⁾

∞

Z b a

|K_n,k(t)|dt,

Z b a

|K_n,k(t)|dt =

k−1

X

i=0

Z xi+1

xi

|t−α_i+1|ⁿ

n! dt

=

k−1

X

i=0

Z αi+1

xi

(α_i+1−t)ⁿ n! dt+

Z xi+1

αi+1

(t−α_i+1)ⁿ

n! dt

= 1

(n+ 1)!

k−1

X

i=0

(α_i+1−x_i)ⁿ⁺¹+ (x_i+1−α_i+1)ⁿ⁺¹

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and thus

(−1)ⁿ Z b

a

Kn,k(t)f⁽ⁿ⁾(t)dt

≤

f⁽ⁿ⁾ ∞

(n+ 1)!

k−1

X

i=0

(α_i+1−x_i)ⁿ⁺¹+ (x_i+1−α_i+1)ⁿ⁺¹ . Hence, from (3.2), the first part of the inequality (3.1) is proved. The second and third lines follow by noting that

f⁽ⁿ⁾ _∞ (n+ 1)!

k−1

X

i=0

(α_i+1−x_i)ⁿ⁺¹+ (x_i+1−α_i+1)ⁿ⁺¹ ≤

f⁽ⁿ⁾ _∞ (n+ 1)!

k−1

X

i=0

hⁿ⁺¹_i , since for0< B < A < C it is well known that

(3.3) (A−B)ⁿ⁺¹+ (C−A)ⁿ⁺¹ ≤(C−B)ⁿ⁺¹. Also

f⁽ⁿ⁾ _∞ (n+ 1)!

k−1

X

i=0

hⁿ⁺¹_i ≤

f⁽ⁿ⁾ _∞ (n+ 1)!νⁿ(h)

k−1

X

i=0

hi =

f⁽ⁿ⁾ _∞

(n+ 1)! (b−a)νⁿ(h), where ν(h) := max{hi|i= 0, . . . , k−1} and therefore the third line of the inequality (3.1) follows, hence Theorem3.1is proved.

When the points of the divisionI_kare fixed, we obtain the following inequality.

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Corollary 3.2. Let f : [a, b] → R be a mapping such that f⁽ⁿ⁻¹⁾ is AC[a, b], andI_kbe defined as in Corollary2.3, then

(3.4)

Z b a

f(t)dt+

n

X

j=1

(−1)^j 2^jj!

" _k X

i=0

n−h^j_i + (−1)^jh^j_i+1o

f^(j−1)(x_i)

#

≤

f⁽ⁿ⁾ _∞ (n+ 1)!2ⁿ

k−1

X

i=0

hⁿ⁺¹_i forf⁽ⁿ⁾ ∈L∞[a, b].

Proof. From Corollary2.3we choose α₀ = a, α₁ = a+x₁

2 , . . . , αk−1 = xk−2+xk−1

2 , αk = xk−1+xk

2 andαk+1 =b.

Now utilising the first line of the inequality (3.1), we may evaluate

k−1

X

i=0

(α_i+1−x_i)ⁿ⁺¹+ (x_i+1−α_i+1)ⁿ⁺¹ =

k−1

X

i=0

2 h_i

2 n+1

and therefore the inequality (3.4) follows.

For the equidistant partitioning case we have the following inequality.

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Corollary 3.3. Letf : [a, b]→Rbe a mapping such thatf⁽ⁿ⁻¹⁾ is AC[a, b]and letI_kbe defined by (2.8). Then

(3.5)

Z b a

f(t)dt+

n

X

j=1

b−a 2k

j

1 j!

×

"

−f^(j−1)(a) +

k−1

X

i=1

n(−1)^j−1o

f^(j−1)(x_i) + (−1)^jf^(j−1)(b)

#

≤

f⁽ⁿ⁾ _∞

(n+ 1)! (2k)ⁿ (b−a)ⁿ⁺¹ forf⁽ⁿ⁾ ∈L∞[a, b].

Proof. We may utilise (2.9) and from (3.1), note that

h₀ =x₁−x₀ = b−a k and h_i =x_i+1−x_i = b−a

k , i= 1, . . . , k−1 in which case (3.5) follows.

The following inequalities for Taylor-like expansions also hold.

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Corollary 3.4. Letgbe defined as in Corollary2.5. Then we have the inequality (3.6)

g(y)−g(a) +

n

X

j=1

(−1)^j j!

×

" _k X

i=0

n

g^(j)(x_i)

#

≤

g⁽ⁿ⁺¹⁾ _∞ (n+ 1)!

k−1

X

i=0

(α_i+1−x_i)ⁿ⁺¹+ (x_i+1−α_i+1)ⁿ⁺¹ ifg⁽ⁿ⁺¹⁾∈L∞[a, b]

for allx_i ∈[a, y]where g⁽ⁿ⁺¹⁾

_∞ := sup

t∈[a,y]

g⁽ⁿ⁺¹⁾(t) <∞.

Proof. Follows directly from (2.11) and using the norm as in (3.1).

When the points of the divisionI_kare fixed we obtain the following.

Corollary 3.5. Let g be defined as in Corollary2.5 and I_k : a = x₀ < x₁ <

· · · < x_k−1 < x_k = y be a division of the interval [a, y]. Then we have the inequality

(3.7)

g(y)−g(a) +

n

X

j=1

1 2^jj!

" _k X

i=0

n

−h^j_i + (−1)^jh^j_i+1 o

g^(j)(xi)

#

≤

g⁽ⁿ⁺¹⁾ _∞ (n+ 1)!2ⁿ

k−1

X

i=0

hⁿ⁺¹_i ifg⁽ⁿ⁺¹⁾ ∈L∞[a, y].

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Proof. The proof follows directly from using (2.7).

For the equidistant partitioning case we have:

Corollary 3.6. Letg be defined as in Corollary2.5and I_k :x_i =a+i·

y−a k

, i= 0, . . . , k

be an equidistant partitioning of[a, y], then we have the inequality:

(3.8)

g(y)−g(a) +

n

X

j=1

y−a 2k

j

1 j!

×

"

−g^(j)(a) +

k−1

X

i=1

n(−1)^j−1o

g^(j)(x_i) + (−1)^jg^(j)(y)

#

≤

g⁽ⁿ⁺¹⁾ _∞

(n+ 1)! (2k)ⁿ(y−a)ⁿ⁺¹ ifg⁽ⁿ⁺¹⁾ ∈L∞[a, y]. Proof. The proof follows directly upon using (2.9) withf⁰ =g andb =y.

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4. The Convergence of a General Quadrature For- mula

Let

∆m :a=x^(m)₀ < x^(m)₁ <· · ·< x^(m)_m−1 < x^(m)_m =b

be a sequence of division of[a, b]and consider the sequence of real numerical integration formula

(4.1) I_m f, f⁰, . . . , f⁽ⁿ⁾,∆_m, w_m :=

m

X

j=0

w_j^(m)f x^(m)_j

−

n

X

r=2

(−1)^r r!

" _m X

j=0

(

x^(m)_j −a−

j−1

X

s=0

w_s^(m)

!^r

− x^(m)_j −a−

j

X

s=0

w^(m)_s

!^r) f^(r−1)

x^(m)_j

# ,

wherew_j(j = 0, . . . , m)are the quadrature weights and assume thatPm

j=0w^(m)_j = b−a.

The following theorem contains a sufficient condition for the weights w_j^(m)

so that

Im f, f⁰, . . . , f⁽ⁿ⁾,∆m, wm

approximates the integral Rb

af(x)dx with an error expressed in terms of

f⁽ⁿ⁾ _∞.

Theorem 4.1. Let f : [a, b] → Rbe a continuous mapping on[a, b] such that

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f⁽ⁿ⁻¹⁾ is AC[a, b]. If the quadrature weights,w^(m)_j satisfy the condition

(4.2) x^(m)_i −a≤

i

X

j=0

w^(m)_j ≤x^(m)_i+1 −a for alli= 0, . . . , m−1

then we have the estimation

Im f, f⁰, . . . , f⁽ⁿ⁾,∆m, wm

− Z b

a

f(t)dt (4.3)

≤

f⁽ⁿ⁾ _∞ (n+ 1)!

m−1

X

i=0



 a+

i

X

j=0

w_j^(m)−x^(m)_i

!n+1

− x^(m)_i+1−a−

i

X

j=0

w^(m)_j

!n+1



≤

f⁽ⁿ⁾ _∞ (n+ 1)!

m−1

X

i=0

h^(m)_i

n+1

≤

f⁽ⁿ⁾ _∞ (n+ 1)!

ν h^(m)

(b−a), where f⁽ⁿ⁾ ∈L∞[a, b]. Also

ν h^(m)

:= max

i=0,...,m−1

n h^(m)_i o h^(m)_i := x^(m)_i+1 −x^(m)_i .

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In particular, if f⁽ⁿ⁾

_∞<∞, then lim

ν(^h^(m))^→0I_m f, f⁰, . . . , f⁽ⁿ⁾,∆_m, w_m

= Z b

a

f(t)dt uniformly by the influence of the weightsw_m.

Proof. Define the sequence of real numbers

α^(m)_i+1 :=a+

i

X

j=0

w^(m)_j , i= 0, . . . , m.

Note that

α^(m)_i+1 =a+

m

X

j=0

w^(m)_j =a+b−a=b.

By the assumption (4.2), we have α^(m)_i+1 ∈h

x^(m)_i , x^(m)_i+1i

for all i= 0, . . . , m−1.

Defineα^(m)₀ =aand compute α^(m)₁ −α^(m)₀ = w₀^(m) α^(m)_i+1 −α^(m)_i = a+

i

X

j=0

w^(m)_j −a−

i−1

X

j=0

w^(m)_j =w_i^(m) (i= 0, . . . , m−1)