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volume 3, issue 1, article 13, 2002.

Received 03 May, 2001;

accepted 26 October, 2001.

Communicated by:P. Cerone

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Journal of Inequalities in Pure and Applied Mathematics

A GENERALIZATION OF THE PRE-GRÜSS INEQUALITY AND APPLICATIONS TO SOME QUADRATURE FORMULAE

NENAD UJEVI ´C

Department of Mathematics University of Split

Teslina 12/III 21000 Split, Croatia.

EMail:ujevic@pmfst.hr

c

2000Victoria University ISSN (electronic): 1443-5756 038-01

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A Generalization of the Pre-Grüss Inequality and

Applications to some Quadrature Formulae

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J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002

http://jipam.vu.edu.au

Abstract

A generalization of the pre-Grüss inequality is presented. It is applied to estimations of remainders of some quadrature formulas.

2000 Mathematics Subject Classification:26D10, 41A55.

Key words: Pre-Grüss inequality, Generalization, Quadrature formulae.

1. Introduction

In recent years a number of authors have written about generalizations of Os- trowski’s inequality. For example, this topic is considered in [1], [2], [5], [7], [9] and [12]. In this way some new types of inequalities are formed, such as inequalities of Ostrowski-Grüss type, inequalities of Ostrowski-Chebyshev type, etc. An important role in forming these inequalities is played by the pre- Grüss inequality. This paper develops a new approach to the topic obtaining better results than the approach using the pre-Grüss inequality. It presents new, improved versions of the mid-point and trapezoidal inequality. The mid-point inequality is considered in [1], [2], [3], [7] and [9], while the trapezoidal inequality is considered in [4], [5], [7] and [9].

In [11] we can find the pre-Grüss inequality:

(1.1) T(f, g)² ≤T(f, f)T(g, g),

wheref, g∈L₂(a, b)andT(f, g)is the Chebyshev functional:

(1.2) T(f, g) = 1 b−a

Z b a

f(t)g(t)dt− 1 (b−a)²

Z b a

f(t)dt Z b

a

g(t)dt.

If there exist constantsγ, δ,Γ,∆∈R such that

δ≤f(t)≤∆andγ ≤g(t)≤Γ,t ∈[a, b]

then, using (1.1), we get the Grüss inequality:

(1.3) |T(f, g)| ≤ (∆−δ)(Γ−γ)

4 .

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Specially, we have

(1.4) T(f, f)≤ (∆−δ)²

4 .

Using the above inequalities we get the following inequalities:

f

a+b 2

(b−a)− Z b

a

f(t)dt (1.5)

≤ (b−a)² 2√

3

"

1

b−akf⁰k²₂−

f(b)−f(a) b−a

2#¹₂

≤ (b−a)² 4√

3 (Γ−γ)

where f : [a, b] → R is an absolutely continuous function whose derivative f⁰ ∈ L₂(a, b) and γ ≤ f⁰(t) ≤ Γ, t ∈ [a, b]. As usual, k·k₂ is the norm in L₂(a, b).Further,

f(a) +f(b)

2 (b−a)− Z b

a

f(t)dt (1.6)

≤ (b−a)² 2√

3

"

1

b−akf⁰k²₂−

f(b)−f(a) b−a

2#¹₂

≤ (b−a)² 4√

3 (Γ−γ)

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and

f(a) + 2f ^a+b₂

+f(b)

4 (b−a)−

Z b a

f(t)dt (1.7)

≤ (b−a)² 4√

3

"

1

b−akf⁰k²₂−

f(b)−f(a) b−a

2#¹₂

≤ (b−a)² 8√

3 (Γ−γ)

where the functionf satisfies the above conditions. The inequalities (1.5)-(1.7) are considered (and proved) in [2], [9] and [12].

In this paper we generalize (1.1). We use the generalization to improve the above inequalities.

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2. Main Results

Lemma 2.1. Letf, g,Ψ_i ∈L₂(a, b),i= 0,1,2, ..., n, whereΨ⁰_i = Ψ_i(t)/kΨ_ik₂ are orthonormal functions. IfSn(f, g)is defined by

S_n(f, g) = Z b

a

f(t)g(t)dt−

n

X

i=0

Z b a

f(s)Ψ⁰_i(s)ds Z b

a

g(s)Ψ⁰_i(s)ds

then we have

|S_n(f, g)| ≤S_n(f, f)¹²S_n(g, g)¹².

The proof follows by the known inequality holding in inner product spaces (H,h·,·i)

hx, yi −

n

X

i=0

hx, l_ii hl_i, yi

2

≤ kxk²−

n

X

i=0

|hx, l_ii|²

!

kyk²−

n

X

i=0

|hl_i, yi|²

! , wherex, y ∈ H and{l_i}_i=0,n is an orthonormal family inH,i.e., (l_i, l_j) = δ_ij fori, j ∈ {0, . . . , n}.

We here use only the casen = 1. We chooseΨ⁰₀(t) = 1/√

b−a, Ψ1(t) = Ψ(t)and denoteS₁(g, h) =S_Ψ(g, h)such that

(2.1) S_Ψ(g, h) = Z b

a

g(t)h(t)dt− 1 b−a

Z b a

g(t)dt Z b

a

h(t)dt

− Z b

a

g(t)Ψ₀(t)dt Z b

a

h(t)Ψ₀(t)dt

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whereg, h,Ψ∈L₂(a, b),Ψ₀(t) = Ψ(t)/kΨk₂ and (2.2)

Z b a

Ψ(t)dt= 0.

Lemma 2.2. With the above notations we have

(2.3) |S_Ψ(g, h)| ≤S_Ψ(g, g)¹²S_Ψ(h, h)¹². It is obvious that

(2.4) S_Ψ(g, h) = (b−a)T(g, h)− Z b

a

g(t)Ψ₀(t)dt Z b

a

h(t)Ψ₀(t)dt so thatS_Ψ(g, h)is a generalization of the Chebyshev functional.

We also define the functions:

(2.5) Φ(t) =







t− ^2a+b₃ , t ∈

a,^a+b₂

t− ^a+2b₃ , t ∈ ^a+b₂ , b and

(2.6) χ(t) =







t− ^5a+b₆ , t∈ a,^a+b₂

t− ^a+5b₆ , t∈ ^a+b₂ , b .

It is not difficult to verify that (2.7)

Z b a

Φ(t)dt = Z b

a

χ(t)dt = 0

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and

(2.8) kΦk²₂ =kχk²₂ = (b−a)³ 36 . We define

(2.9) Φ0(t) = Φ(t)

kΦk₂, χ0(t) = χ(t) kχk₂. Integrating by parts, we have

Q(f;a, b) = Z b

a

Φ₀(t)f⁰(t)dt (2.10)

= 2

√b−a

f(a) +f

a+b 2

+f(b)− 3 b−a

Z b a

f(t)dt

and

P(f;a, b) (2.11)

= Z b

a

χ0(t)f⁰(t)dt

= 1

√b−a

f(a) + 4f

a+b 2

+f(b)− 6 b−a

Z b a

f(t)dt

. Remark 2.1. It is obvious that

(2.12) SΨ(g, g) = (b−a)T(g, g)− Z b

a

g(t)Ψ0(t)dt ²

≤(b−a)T(g, g).

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Theorem 2.3. (Mid-point inequality) LetI ⊂Rbe a closed interval anda, b∈ Int I, a < b.Iff :I →Ris an absolutely continuous function whose derivative f⁰ ∈L₂(a, b)then we have

(2.13)

f

a+b 2

(b−a)− Z b

a

f(t)dt

≤ (b−a)³² 2√

3 C1, where

(2.14) C₁ =

(

kf⁰k²₂− [f(b)−f(a)]²

b−a −[Q(f;a, b)]² )¹₂

andQ(f;a, b)is defined by (2.10).

Proof. We define

(2.15) p(t) =







t−a, t ∈ a,^a+b₂

t−b, t ∈ ^a+b₂ , b .

Then we have (2.16)

Z b a

p(t)dt= 0

and

(2.17) kpk²₂ =

Z b a

p(t)²dt= (b−a)³ 12 .

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We now calculate (2.18)

Z b a

p(t)Φ(t)dt

= Z ^a+b₂

a

(t−a)

t− 2a+b 3

dt+

Z b

a+b 2

(t−b)

t− a+ 2b 3

dt= 0.

Integrating by parts, we have Z b

a

p(t)f⁰(t)dt =

Z ^a+b₂

a

(t−a)f⁰(t)dt+ Z b

a+b 2

(t−b)f⁰(t)dt (2.19)

= f

a+b 2

(b−a)− Z b

a

f(t)dt.

Using (2.16), (2.18) and (2.19) we get S_Φ(p, f⁰) =

Z b a

p(t)f⁰(t)dt− 1 b−a

Z b a

p(t)dt Z b

a

f⁰(t)dt (2.20)

− Z b

a

f⁰(t)Φ₀(t)dt Z b

a

p(t)Φ₀(t)dt

= f

a+b 2

(b−a)− Z b

a

f(t)dt.

From (2.20) and (2.3) it follows that (2.21)

f

a+b 2

(b−a)− Z b

a

f(t)dt

≤S_Φ(f⁰, f⁰)¹²S_Φ(p, p)¹².

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From (2.16)-(2.18) we get

S_Φ(p, p) =kpk²₂− 1 b−a

Z b a

p(t)dt ²

− Z b

a

p(t)Φ₀(t)dt ² (2.22)

= (b−a)³ 12 . We also have

(2.23) C₁² =S_Φ(f⁰, f⁰).

From (2.21)-(2.23) we easily find that (2.13) holds.

Remark 2.2. It is not difficult to see that (2.13) is better than the first estimation in (1.5).

Theorem 2.4. (Trapezoidal inequality) Under the assumptions of Theorem2.3 we have

(2.24)

f(a) +f(b)

2 (b−a)− Z b

a

f(t)dt

≤ (b−a)³² 2√

3 C₂, where

(2.25) C₂ =

(

kf⁰k²₂− [f(b)−f(a)]²

b−a −[P(f;a, b)]² )¹₂

andP(f;a, b)is defined by (2.11).

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Proof. Letp(t)be defined by (2.15). We calculate Z b

a

p(t)χ(t)dt (2.26)

= Z ^a+b₂

a

(t−a)

t− 5a+b 6

dt+

Z b

a+b 2

(t−b)

t−a+ 5b 6

dt

= (b−a)³ 24 . Integrating by parts, we have

Z b a

f⁰(t)χ(t)dt|

(2.27)

= Z ^a+b₂

a

t− 5a+b 6

f⁰(t)dt+ Z b

a+b 2

t− a+ 5b 6

f⁰(t)dt

= f(a) + 4f ^a+b₂

+f(b)

6 (b−a)−

Z b a

f(t)dt.

Using (2.16), (2.19), (2.26), (2.27) and (2.8) we get (2.28) S_χ(f⁰, p) =

Z b a

f⁰(t)dt Z b

a

p(t)dt

− Z b

a

p(t)χ0(t)dt Z b

a

f⁰(t)χ0(t)dt

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=f

a+b 2

(b−a)− Z b

a

f(t)dt

− 3 2

"

f(a) + 4f(^a+b₂ ) +f(b)

6 (b−a)−

Z b a

f(t)dt

#

=−1

2(b−a)f(a) +f(b)

2 + 1

2 Z b

a

f(t)dt.

f(a) +f(b)

2 (b−a)− Z b

a

f(t)dt

≤2S_χ(f⁰, f⁰)¹²S_χ(p, p)¹².

We have

S_χ(p, p) =kpk²₂− 1 b−a

Z b a

p(t)dt ²

− Z b

a

p(t)χ₀(t)dt ² (2.30)

= (b−a)³ 48 and

(2.31) C₂² =S_χ(f⁰, f⁰).

From (2.29)-(2.31) we easily get (2.24).

Remark 2.3. We see that (2.24) is better than the first estimation in (1.6).

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We now consider a simple quadrature rule of the form (2.32) f(a) + 2f ^a+b₂

+f(b)

4 (b−a)−

Z b a

f(t)dt

= 1 2

f

a+b 2

+ f(a) +f(b) 2

(b−a)− Z b

a

f(t)dt=R(f).

It is not difficult to see that (2.32) is a convex combination of the mid-point quadrature rule and the trapezoidal quadrature rule. In [5] it is shown that (2.32) has a better estimation of error than the well-known Simpson quadrature rule (when we estimate the error in terms of the first derivativef⁰ of integrand f). We here have a similar case.

Theorem 2.5. Under the assumptions of Theorem2.3we have

(2.33)

f(a) + 2f ^a+b₂

+f(b)

4 (b−a)−

Z b a

f(t)dt

≤ (b−a)³² 4√

3 C₃, where

(2.34) C₃ =

"

kf⁰k²₂− [f(b)−f(a)]² b−a

− 1 b−a

f(a)−2f

a+b 2

+f(b) 2#¹₂

.

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Proof. We define

(2.35) η(t) =







1, t∈ a,^a+b₂

−1, t∈ ^a+b₂ , b ,

(2.36) η₀(t) = η(t)

kηk₂.

We easily find that (2.37)

Z b a

η(t)dt= 0, kηk²₂ =b−a.

Letp(t)be defined by (2.15). Then we have Z b

a

p(t)η(t)dt= Z ^a+b₂

a

(t−a)dt− Z b

a+b 2

(t−b)dt (2.38)

= (b−a)²

4 .

We also have (2.39)

Z b a

f⁰(t)η(t)dt=−f(a) + 2f

a+b 2

−f(b).

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From (2.37)-(2.39) we get S_η(f⁰, p) =

Z b a

f⁰(t)dt Z b

a

p(t)dt (2.40)

− Z b

a

p(t)η₀(t)dt Z b

a

f⁰(t)η₀(t)dt

=f

a+b 2

(b−a)− Z b

a

f(t)dt

− b−a 4

−f(a) + 2f

a+b 2

−f(b)

= f(a) + 2f ^a+b₂

+f(b)

4 (b−a)−

Z b a

f(t)dt.

f(a) + 2f ^a+b₂

+f(b)

4 (b−a)−

Z b a

f(t)dt

≤S_η(f⁰, f⁰)¹²S_η(p, p)¹².

We now calculate

S_η(p, p) = kpk²₂− 1 b−a

Z b a

p(t)dt ²

− Z b

a

p(t)η₀(t)dt ² (2.42)

= (b−a)³ 48 .

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We also have

(2.43) C₃² =S_η(f⁰, f⁰).

From (2.41)-(2.43) we easily get (2.33).

Remark 2.4. It is not difficult to see that (2.33) is better than the first estimation in (1.7).

Finally, in [12] we can find the next inequality (2.44)

f(a) + 4f ^a+b₂

+f(b)

6 (b−a)−

Z b a

f(t)dt

≤ (b−a)²

12 (Γ−γ), wheref :I →R,(I ⊂Ris an open interval,a < b, a, b∈I) is a differentiable function, f⁰ is integrable and there exist constants γ,Γ ∈ R such that γ ≤ f⁰(t)≤Γ, t∈[a, b].

Inequality (2.44) is a variant of the Simpson’s inequality. On the other hand, we have

(2.45)

f(a) + 2f ^a+b₂

+f(b)

4 (b−a)−

Z b a

f(t)dt

≤ (b−a)² 8√

3 (Γ−γ).

Inequality (2.45) follows from (2.33), since (2.46) S_η(f⁰, f⁰)≤(b−a)

Γ−γ 2

2

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and (2.46) follows from (2.4) and (1.4).

Form (2.44) and (2.45) we see that the simple 3-point quadrature rule (2.32) has a better estimation of error than the well-known 3-point Simpson quadrature rule. Note that the estimations are expressed in terms of the first derivative f⁰ of integrand.

Finally, the following remark is valid.

Remark 2.5. The considered case n = 1 illustrates how to apply Lemma 2.1 to quadrature formulas. It is also shown that the derived results are better than some recently obtained results. We can use Lemma 2.1 to derive further improvements of the obtained results. However, in such a case we must require

Z b a

g(t)Ψ⁰_i(t)dt = 0,i= 0,1,2, ..., n.

Thus, the construction of such a finite sequence {Ψ⁰_i}ⁿ₀ can be complicated.

However, if we really need better error bounds, without taking into account possible complications, then we can apply the procedure described in this sec- tion.

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means and some numerical quadrature rules, Comput. Math. Appl., 33 (1997), 15–20.

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