A new inequality of Ostrowski–Grüss type and applications to some numerical quadrature rules

Computers and Mathematics with Applications 58 (2009) 589–596

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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa

A new inequality of Ostrowski–Grüss type and applications to some numerical quadrature rules Marek Niezgoda Department of Applied Mathematics and Computer Science, University of Life Sciences in Lublin, Akademicka 13, 20-950 Lublin, Poland

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Article history: Received 30 October 2008 Accepted 23 March 2009 Keywords: Grüss type inequality Ostrowski–Grüss type inequality Numerical quadrature rule

abstract In this paper we derive some Grüss and Ostrowski–Grüss type inequalities for functions in Lp -spaces. As applications, we provide some new estimates for the error in some numerical integration rules. In particular, we deal with the mid-point and trapezoid quadrature rules. © 2009 Elsevier Ltd. All rights reserved.

1. Motivation and summary p

We denote by L[a,b] (1 ≤ p < ∞) the space of p-power integrable functions on interval [a, b] equipped with the norm

1/p b kf kp = a |f (t )|p dt . The symbol L∞ [a,b] stands for the space of all essentially bounded functions on [a, b] with the norm kf k∞ = ess supx∈[a,b] |f (x)|. For two real functions f and g such that f , g , fg ∈ L1[a,b] , the Chebyshev functional is defined by Z b Z b Z b 1 1 1 T (f , g ) = f (x)g (x)dx − f (x)dx · g (x)dx. (1) b−a a b−a a b−a a R

The Grüss inequality [1] states that if f , g : [a, b] → R are two bounded integrable functions and α0 , β0 , γ0 and δ0 are numbers such that

α0 ≤ f (t ) ≤ β0 and γ0 ≤ g (t ) ≤ δ0 for all t ∈ [a, b],

(2)

then

|T (f , g )| ≤

1 4

(β0 − α0 )(δ0 − γ0 ).

(3)

See [2–8] for generalizations and extensions of (3). A related result is the following Ostrowski inequality [9, p. 468]. If f : [a, b] → R is a differentiable function with bounded derivative, then

# " Z b (x − a+2 b )2 1 f (x) − 1 f (t )dt ≤ + (b − a)kf 0 k∞ for x ∈ [a, b]. b−a a 4 ( b − a) 2

E-mail address: [email protected]. 0898-1221/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2009.03.089

(4)

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M. Niezgoda / Computers and Mathematics with Applications 58 (2009) 589–596

Dragomir and Wang [10] proved the Ostrowski–Grüss type inequality in the following form. If f : [a, b] → R is a differentiable function with bounded derivative and

α0 ≤ f 0 (t ) ≤ β0 for t ∈ [a, b], then

  Z b f (b) − f (a) a+b f (x) − 1 ≤ 1 (b − a)(β0 − α0 ) for x ∈ [a, b]. f (t )dt − x− 4 b−a a b−a 2

(5)

Some improvements of (5) are due to Matić, Pečarić and Ujević [11] and to Cheng [12]. In the last decade, Grüss and Ostrowski type inequalities have attracted much attention from researchers, because of their applications in numerical analysis (see [13,14,10,15–21]). In [2], Dragomir established a generalization of the Grüss inequality in inner product spaces. Developing Dragomir’s idea, Niezgoda [22] extended the above implication (2) ⇒ (3) by using some classes of bounding functions α(t ), β(t ), γ (t ), δ(t ) in place of bounding constants α0 , β0 , γ0 , δ0 , respectively. Recall that a function ϕ : [a, b] → R is said to be a constant function if there exists a constant c ∈ R such that ϕ(t ) = c for all t ∈ [a, b]. Lemma 1.1 ([22, Corollary 4.5]). Let f , g , α, β, γ , δ ∈ L2[a,b] be functions such that (a) α + β and γ + δ are constant functions, and (b) α(t ) ≤ f (t ) ≤ β(t ) and γ (t ) ≤ g (t ) ≤ δ(t ) for all t ∈ [a, b], or more generally b

Z

b

Z

(β(t ) − f (t ))(f (t ) − α(t ))dt ≥ 0 and a

(δ(t ) − g (t ))(g (t ) − γ (t ))dt ≥ 0. a

Then we have the inequality

|T (f , g )| ≤

1 4(b − a)

kβ − αk2 kδ − γ k2 .

(6)

It is worth emphasizing that for many functions f and g with appropriate choice of functions α , β , γ and δ , (6) provides a more precise estimate than (3) (see (16) and (17)). In the present paper, our aim is (1) to derive some Grüss and Ostrowski–Grüss like inequalities by using a generalization of Lemma 1.1 in Lp -spaces (p ≥ 1), (2) to provide some new estimates for the error in some numerical integration rules of functions which have first derivatives p in L[a,b] . The paper is organized as follows. Results are collected in Section 2. In Theorem 2.1 we present a Grüss like inequality for functions in Lp -spaces. Here we apply a method employing some bounding functions α(t ) and β(t ) (see conditions (a)– (b) of Lemma 1.1) in place of bounding constants α0 and β0 , respectively (see (2)). To prove Theorem 2.1, we use Lemma 2.2, which is an extension of Lemma 1.1 from p = 2 to 1 ≤ p ≤ ∞. In Lemma 2.3 we show that the method used in Lemma 2.2 and Theorem 2.1 can give a tighter estimate than the standard method based on bounding constants. Theorem 2.4 contains an Ostrowski–Grüss type inequality involving the above-mentioned bounding functions. In Section 3 we demonstrate some applications. We apply Theorem 2.4 to derive some new bounds for the error in some numerical integration rules. In particular, we provide special versions for the mid-point and trapezoid quadrature rules. Finally, by using the above results for the constant bounding functions, we recover some results of Dragomir and Wang [10] and Matić et al. [11]. 2. Ostrowski–Grüss type inequalities In this section we begin with a further extension of the Grüss inequality in Lp -spaces (cf. [23, Theorems 2 and 3], [24, p. 2]). For p = q = 2, the result below applied to constant functions reduces to [11, Lemma 1]. p

q

Theorem 2.1. Let f , α, β ∈ L[a,b] and g ∈ L[a,b] ( 1p +

1 q

= 1, 1 ≤ p ≤ ∞) be functions such that

(a) α + β is a constant function, and (b) α(t ) ≤ f (t ) ≤ β(t ) for all t ∈ [a, b]. Then we have the inequality

|T (f , g )| ≤

1 2(b − a)

kβ − αkp ·

g −

1 b−a

b

Z a



g (s)ds

.

(7)

q

The proof of Theorem 2.1 will be simplified if we first prove the following lemma which extends [2, Lemma 2.1] from inner product spaces to Lp -spaces.

M. Niezgoda / Computers and Mathematics with Applications 58 (2009) 589–596

591

p

Lemma 2.2. Let f , α, β ∈ L[a,b] (1 ≤ p ≤ ∞) be functions such that

α(t ) ≤ f (t ) ≤ β(t ) for all t ∈ [a, b].

(8)

Then we have the inequality



f − α + β ≤ 1 kβ − αkp .

2 p 2

(9)

Proof. It follows from (8) that

α(t ) −

α(t ) + β(t ) 2

≤ f (t ) −

α(t ) + β(t ) 2

α(t ) + β(t )

≤ β(t ) −

2

for all t ∈ [a, b],

or, equivalently,

α(t ) − β(t ) 2

≤ f (t ) −

α(t ) + β(t ) 2

≤

β(t ) − α(t ) 2

for all t ∈ [a, b],

which gives

f (t ) − α(t ) + β(t ) ≤ β(t ) − α(t ) 2 2

for all t ∈ [a, b].

Hence, for 1 ≤ p < ∞,

p 1/p Z b 1/p

Z b

β(t ) − α(t ) p

β − α



f − α + β = f (t ) − α(t ) + β(t ) dt



. ≤ =

dt 2 p 2 2 2 p a a The case p = ∞ can be proved similarly. This completes the proof of the lemma. 1 Proof of Theorem 2.1. Define I (g ) = b− a holds:

T (f , g ) =

b−a

a



g (s)ds. It is not hard to verify that for any constant c ∈ R the following identity

b

Z

1

Rb

(10)

(f (t ) − c ) (g (t ) − I (g )) dt . a

Using the Hölder inequality we get 1

|T (f , g )| ≤ Substituting c =

b−a α+β

|T (f , g )| ≤ as required.

kf − c kp · kg − I (g )kq for c ∈ R.

(11)

into (11) and applying Lemma 2.2, we obtain

2



1

f − α + β · kg − I (g )kq ≤ kβ − αkp · kg − I (g )kq ,

b−a 2 p 2(b − a) 1

(12)



In the case p = q = 2, Theorem 2.1 implies Lemma 1.1. In fact, we have the additional estimate

kg − I (g )k2 ≤ kg − c1 k2 for any constant c1 ∈ R, (13) Rb 1 because the function [a, b] 3 t → I (g ) = b−a a g (s)ds is the orthogonal projection of g into the subspace of constant functions in L2[a,b] . In consequence, (12) and (13) imply

|T (f , g )| ≤

1 2(b − a)

kβ − αk2 · kg − c1 k2 for all c1 ∈ R.

(14)

If, in addition, γ , δ ∈ L2[a,b] and γ + δ is a constant function, and γ (t ) ≤ g (t ) ≤ δ(t ) for t ∈ [a, b], then by (14) applied for the constant c1 =

|T (f , g )| ≤

γ +δ 2

and by Lemma 2.2 applied to functions g, γ and δ , we have

1

kβ − αk2 · kδ − γ k2 .

4(b − a) This gives Lemma 1.1. In particular, if α0 , β0 , γ0 , δ0 ∈ R are constants such that

(15)

α0 ≤ f (t ) ≤ β0 and γ0 ≤ g (t ) ≤ δ0 for all t ∈ [a, b], then (15) used for α(t ) = α0 , β(t ) = β0 , γ (t ) = γ0 and δ(t ) = δ0 reduces to (3), because the L2 -norm of a constant function ϕ(t ) = c0 ≥ 0, t ∈ [a, b], equals c0 (b − a)1/2 .

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M. Niezgoda / Computers and Mathematics with Applications 58 (2009) 589–596

It is easily seen that if f , g , α, β, γ , δ ∈ L2[a,b] are functions and α0 , β0 , γ0 , δ0 ∈ R are numbers satisfying

α0 ≤ α(t ) ≤ f (t ) ≤ β(t ) ≤ β0 and γ0 ≤ γ (t ) ≤ g (t ) ≤ δ(t ) ≤ δ0

for t ∈ [a, b],

(16)

then

kβ − αk2 ≤ (β0 − α0 )(b − a)1/2 and kδ − γ k2 ≤ (δ0 − γ0 )(b − a)1/2 .

(17)

This shows that (15) is a tighter estimate than (3) provided (16) holds. The following result is a complement to Lemma 2.2 and Theorem 2.1. p

Lemma 2.3. For given function f ∈ L[a,b] (1 ≤ p ≤ ∞) and constant c ∈ R, define



p

p

Bf ,c = (α, β) ∈ L[a,b] × L[a,b] :

α+β 2

 = c , α(t ) ≤ f (t ) ≤ β(t ) for t ∈ [a, b] ,

(18)

and

αf ,c (t ) = −|f (t ) − c | + c and βf ,c (t ) = |f (t ) − c | + c for t ∈ [a, b].

(19)

Then



βf ,c − αf ,c

β − α



. = kf − c kp = min

(α,β)∈Bf ,c 2 2 p

(20)

p

Proof. For any (α, β) ∈ Bf ,c , the inequality





α +β

≤ β − α kf − c kp = f −

2 2 p p

(21)

holds by Lemma 2.2. We shall show that the minimum is attained at (αf ,c , βf ,c ) ∈ Bf ,c defined by (19). As

−|f (t ) − c | ≤ f (t ) − c ≤ |f (t ) − c | for t ∈ [a, b], we have

αf ,c (t ) ≤ f (t ) ≤ βf ,c (t ) for t ∈ [a, b]. αf ,c +βf ,c

βf ,c −αf ,c

= |f − c |. In consequence, (αf ,c , βf ,c ) ∈ Bf ,c and 2

βf ,c − αf ,c

= k|f − c |kp = kf − c kp .

2

Furthermore,

2

= c and

(22)

p

Combining (21) and (22) yields (20).



By virtue of inequality (11) and Lemma 2.3, the assertion of Theorem 2.1 can be expressed as

!

β − α

· kg − I (g )kq kf − c kp · kg − I (g )kq = |T (f , g )| ≤ min b−a b − a (α,β)∈Bf ,c 2 p 1

1

(23)

for any c ∈ R. Now we proceed to present an improvement of the Ostrowski–Grüss inequality. This is a combination of Theorem 2.1 and a method given in [10]. For x, t ∈ [a, b], we define P (x, t ) =



t −a t −b

if t ∈ [a, x], if t ∈ (x, b].

(24) ◦

◦

Theorem 2.4. Let f : I → R, where I ⊂ R is an interval, be a function differentiable in the interior I of I, and let [a, b] ⊂ I . p Suppose that f 0 , α, β ∈ L[a,b] (1 ≤ p ≤ ∞) are functions such that (a) α + β is a constant function, and (b) α(t ) ≤ f 0 (t ) ≤ β(t ) for all t ∈ [a, b]. Then for x ∈ [a, b] we have the inequality

 1 (b − a)1/q    Z b  kβ − αkp 1 a + b f (b) − f (a) (q + 1)1/q f (x) − ≤ 4 f (t )dt − x − 1 b−a a 2 b−a  kβ − αk1 4

where

1 p

+

1 q

= 1.

if 1 ≤ q < ∞, if q = ∞,

(25)

M. Niezgoda / Computers and Mathematics with Applications 58 (2009) 589–596

593

Proof. In the first part of this proof, we follow the method presented in the proof of [10, Theorem 2.1]. By the Montgomery identity (see [7]) we get f ( x) =

b

Z

1 b−a

f (t )dt + a

Z

1 b−a

b

f 0 (t )P (x, t )dt .

(26)

a

It is easy to calculate that 1 b−a

f (b) − f (a)

b

Z

f 0 (t )dt =

(27)

b−a

a

and 1 b−a

b

Z

b−a

a

x

Z

1

P (x, t )dt =

(t − a)dt + a

b

Z

1 b−a

(t − b)dt = x −

a+b

x

2

.

(28)

According to Theorem 2.1 applied to f 0 and P (x, ·), we have the inequality

Z 1 b − a

b

f (t )P (x, t )dt −

b

Z

1

0

f (t )dt ·

1

b

Z

b−a

 

a+b 1

. P ( x , ·) − x − ≤ kβ − αkp ·

2(b − a) 2 q a

b−a

0

a

P (x, t )dt

a

(29)

Putting (26)–(28) into (29), we obtain



    Z b

a + b f (b) − f (a) 1

P (x, ·) − x − a + b . f (x) − 1 f ( t ) dt − x − ≤ kβ − αk p

b−a a 2 b−a 2(b − a) 2 q

(30)

Assume 1 ≤ q < ∞. It remains to prove that

  1+1/q

P (x, ·) − x − a + b = (b − a)

2 2(q + 1)1/q q

for 1 ≤ q < ∞.

(31)

Define c1 = x −

a+b 2

,

t∗ = a + c1

,

t∗ − x =

and

t ∗ = b + c1 .

Then x − t∗ =

b−a 2

b−a 2

,

t∗ − a = c1 ,

t ∗ − b = c1 .

(32)

Evidently, for x, t ∈ [a, b], we have P (x, t ) − c1 =



t − t∗ t − t∗

if t ∈ [a, x], if t ∈ (x, b].

(33)

b b . (In the case a+ ≤ x ≤ b, the proof is analogous.) Then Without lost of generality we can assume that a ≤ x ≤ a+ 2 2 c1 ≤ 0 and a ≤ x ≤ t ∗ ≤ b. Also,

P (x, t ) − c1 ≥ 0

for t ∈ [a, x) ∪ [t ∗ , b],

and

P (x, t ) − c1 ≤ 0

for t ∈ [x, t ∗ ]

(see [12, p. 111]). In consequence, we have

kP (x, ·) −

b

Z

c1 qq

|P (x, t ) − c1 |q dt

k = a x

Z

|t − t∗ |q dt +

= a

(t − t∗ )q dt + a

|t − t ∗ |q dt +

Z

b

|t − t ∗ |q dt t∗

x x

Z =

t∗

Z

t∗

Z

[−(t − t ∗ )]q dt + x

Z

b t∗

(t − t ∗ )q dt

(x − t∗ )q+1 (a − t∗ )q+1 [−(t ∗ − t ∗ )]q+1 [−(x − t ∗ )]q+1 (b − t ∗ )q+1 (t ∗ − t ∗ )q+1 − − + + − q+1 q+1 q+1 q+1 q+1 q+1   q +1 q +1 2 b−a (b − a) = = q . q+1 2 2 (q + 1) =

The last equality follows from (32). So, we conclude that (31) holds.

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M. Niezgoda / Computers and Mathematics with Applications 58 (2009) 589–596

On the other hand, if q = ∞ then b−a

kP (x, ·) − c1 k∞ = lim kP (x, ·) − c1 kr =

2

r →∞

,

because of (31) and limr →∞ (r + 1)1/r = 1. This completes the proof.

 ◦

◦

Corollary 2.5. Let f : I → R, where I ⊂ R is an interval, be a function differentiable in the interior I of I, and let [a, b] ⊂ I . Suppose that α0 , β0 ∈ R are numbers such that α0 ≤ f 0 (t ) ≤ β0 for all t ∈ [a, b]. Then for x ∈ [a, b] and 1 ≤ q ≤ ∞ we have the inequality

 1    Z b  (β0 − α0 )(b − a) a + b f (b) − f (a) 4 ( q + 1)1/q f (x) − 1 f ( t ) dt − x − ≤ 1 b−a a 2 b−a  (β − α )(b − a) 4

0

0

if 1 ≤ q < ∞, (34) if q = ∞.

p

Proof. Take 1 ≤ p ≤ ∞ satisfying 1p + 1q = 1. Clearly, f 0 ∈ L∞ [a,b] ⊂ L[a,b] . Define α(t ) = α0 and β(t ) = β0 for t ∈ [a, b]. Then conditions (a) and (b) in Theorem 2.4 are satisfied. Consequently, (25) holds. Observe that kβ − αkp = (β0 − α0 )(b − a)1/p . Therefore the right-hand side of (25) is equal to 1 4

kβ − αkp

1/q (b − a)1/q 1 1 ( b − a) 1/p (b − a) = (β − α )( b − a ) = (β0 − α0 ) 0 0 (q + 1)1/q 4 (q + 1)1/q 4 (q + 1)1/q

for 1 ≤ q < ∞, and 1 4

(β0 − α0 )(b − a) for q = ∞.

All of this gives (34).



Three special cases of Corollary 2.5 are well-known. Namely, if q = 2 then Corollary 2.5 becomes a result of Matić et al. [11, 1 Theorem 6] with the factor √ on the right-hand side of (34). 4 3

Also, the case q = 1 leads to a result of Cheng [12, Theorem 1.5] with the factor 18 on the right-hand side of (34). (See also a similar result of Liu [17, Theorem 3].) Finally, the aforementioned inequality (5) of Dragomir and Wang [10, Theorem 2.1] corresponds to (34) for q = ∞. 3. Applications to numerical quadrature rules In this section we apply (25) to obtain bounds for the error in some numerical integration rules for functions with first p derivatives in L[a,b] . Setting

1 q

1

= 0 and (q + 1) q = 1 for q = ∞, we have the following. ◦

◦

Theorem 3.1. Let f : I → R, where I ⊂ R is an interval, be a function differentiable in the interior I of I, and let [a, b] ⊂ I . p Suppose that f 0 , α, β ∈ L[a,b] (1 ≤ p ≤ ∞) are functions such that (a) α + β is a constant function, and (b) α(t ) ≤ f 0 (t ) ≤ β(t ) for all t ∈ [a, b].

Then for every partition In : a = x0 < x1 < · · · < xn−1 < xn = b of [a, b] and for any intermediate point vector ξ = (ξ0 , ξ1 , . . . , ξn−1 ) satisfying ξi ∈ [xi , xi+1 ] for i = 0, 1, . . . , n − 1, we have Z b n −1 X 1 1+1/q ≤ f ( t ) dt − A ( f , I , ξ ) kβ − αk(pi) hi , (35) G n 4(q + 1)1/q a

where

1 p

+

i =0

1 q

= 1, hi = xi+1 − xi , and Z x 1/p i+1   |β( t ) − α(t )|p dt kβ − αk(pi) = xi  ess sup |β(t ) − α(t )| t ∈[xi ,xi+1 ]

if 1 ≤ p < ∞, if p = ∞,

and AG denotes the generalized quadrature rules defined by AG (f , In , ξ ) =

n −1 X i=0

hi f (ξi ) −

n −1  X i =0

ξi −

xi + xi+1 2



(f (xi+1 ) − f (xi )).

(36)

M. Niezgoda / Computers and Mathematics with Applications 58 (2009) 589–596

595

Proof. By (25) with x = ξi ∈ [xi , xi+1 ] we have the inequality

Z

xi+1

f (t )dt − hi f (ξi ) +



xi + xi+1

ξi −



2

xi

1+1/q 1 h (f (xi+1 ) − f (xi )) ≤ kβ − αk(pi) i 1/q 4 (q + 1)

for i = 0, 1, . . . , n − 1. Summing the above and using the triangle inequality we get (35). Taking ξi =

xi +xi+1 2

(37)



for i = 0, 1, . . . , n − 1, we obtain the following result for the mid-point quadrature rule.

Corollary 3.2. Under the assumptions and notation of Theorem 3.1, we have

Z

b

a

where

1 p



f (t )dt − AM (f , In ) ≤

+

1 q

n −1 X

1 1/q

4(q + 1)

1+1/q

kβ − αk(pi) hi

,

(38)

i=0

= 1, and AM is the mid-point quadrature rule defined by

A M ( f , In ) =

n −1 X

 hi f

xi + xi+1



.

2

i =0

(39)

Applying Theorem 3.1 for ξi = xi and next for ξi = xi+1 leads to: Corollary 3.3. Under the assumptions and notation of Theorem 3.1, we have

Z

b

a

where

1 p



+

1 q

n −1 X

1

f (t )dt − AT (f , In ) ≤

1/q

4(q + 1)

1+1/q

kβ − αk(pi) hi

,

(40)

i=0

= 1, and AT is the trapezoid quadrature rule defined by

AT (f , In ) =

n−1 X

hi

f (xi ) + f (xi+1 ) 2

i=0

.

(41) 1/p

(i)

For constant functions α(t ) = α0 and β(t ) = β0 for t ∈ [a, b], we have kβ − αkp = (β0 − α0 )hi Consequently, the right-hand side of (35), (38) and (40) becomes 1 4(q + 1)

(β0 − α0 ) 1/q

n −1 X

h2i

for 1 ≤ p ≤ ∞.

for 1 ≤ q ≤ ∞.

(42)

i =0

Moreover, if α0 ≤ f 0 (t ) ≤ β0 for t ∈ [a, b], then f 0 ∈ L∞ [a,b] . So, Theorem 3.1 applied to p = ∞ and q = 1 implies the following result. ◦

◦

Corollary 3.4. Let f : I → R, where I ⊂ R is an interval, be a function differentiable in the interior I of I, and let [a, b] ⊂ I . Suppose that α0 , β0 ∈ R are numbers such that α0 ≤ f 0 (t ) ≤ β0 for all t ∈ [a, b]. Then for every partition In : a = x0 < x1 < · · · < xn−1 < xn = b of [a, b] and for any intermediate point vector ξ = (ξ0 , ξ1 , . . . , ξn−1 ) satisfying ξi ∈ [xi , xi+1 ] for i = 0, 1, . . . , n − 1, we have

Z

b

n−1 X 1 h2i , f (t )dt − AG (f , In , ξ ) ≤ (β0 − α0 ) 8

a

(43)

i=0

where hi = xi+1 − xi and AG is defined by (36). p

It is worth noting that the right-hand side of (43) can be replaced by (42) for any q ≥ 1, because L∞ [a,b] ⊂ L[a,b] for any p ≥ 1. For example, such a version of Corollary 3.4 for p = q = 2 gives a result of Matić et al. [11, Theorem 8] with the factor 1 √ in place of 18 (cf. [12, Theorem 1.5]). 4 3

Special versions of Corollary 3.4 are as follows (cf. [10, Corollaries 4.3 and 4.5] and [11, Corollaries 5 and 6]). Corollary 3.5. Under the assumptions and notation of Corollary 3.4, we have

Z

a

b



f (t )dt − AM (f , In ) ≤

where AM is defined by (39).

1 8

(β0 − α0 )

n−1 X i=0

h2i ,

(44)

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M. Niezgoda / Computers and Mathematics with Applications 58 (2009) 589–596

Corollary 3.6. Under the assumptions and notation of Corollary 3.4, we have

Z

b a



f (t )dt − AT (f , In ) ≤

1 8

(β0 − α0 )

n−1 X

h2i ,

(45)

i=0

where AT is defined by (41). References Rb

Rb

Rb

1 [1] G. Grüss, Über das maximum des absoluten betrages von b− f (x)g (x)dx − (b−1a)2 a f (x)dx a g (x)dx, Math. Z. 39 (1934) 215–226. a a [2] S.S. Dragomir, Some Grüss type inequalities in inner product spaces, J. Inequal. Pure Appl. Math. 4 (2) (2003) Article 42. [3] A.M. Fink, A treatise on Grüss’ inequality, in: T.M. Rassias, H.M. Srivastava (Eds.), Analytic and Geometric Inequalities and Applications, Kluwer Academic, 1999. [4] S. Izumino, J.E. Pečarić, B. Tepeš, A Grüss-type inequality and its applications, J. Inequal. Appl. 3 (2005) 277–288. [5] X. Li, R.N. Mohapatra, R.S. Rodriguez, Grüss-type inequalities, J. Math. Anal. Appl. 267 (2002) 434–443. [6] B.G. Pachpatte, On Grüss type inequality for double integrals, J. Math. Anal. Appl. 267 (2002) 454–459. [7] B.G. Pachpatte, On Čebyšev–Grüss type inequalities via Pečarić’s extension of the Montgomery identity, J. Inequal. Pure Appl. Math. 7 (1) (2006) Article 11. [8] N. Ujević, A new generalization of Grüss inequality in inner product spaces, J. Math. Anal. Appl. 250 (2000) 494–511. [9] D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1994. [10] S.S. Dragomir, S. Wang, An inequality of Ostrowski–Grüss’ type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl. 39 (11) (1997) 15–20. [11] M. Matić, J. Pečarić, N. Ujević, Improvement and further generalization of inequalities of Ostrowski–Grüss type, Comput. Math. Appl. 39 (2000) 161–175. [12] X.-L. Cheng, Improvement of some Ostrowski–Grüss type inequalities, Comput. Math. Appl. 42 (2001) 109–114. [13] P. Cerone, W.S. Cheung, S.S. Dragomir, On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation, Comput. Math. Appl. 54 (2007) 183–191. [14] S.S. Dragomir, A. Sofo, An inequality for monotonic functions generalizing Ostrowski and related results, Comput. Math. Appl. 51 (2006) 497–506. [15] S.S. Dragomir, S. Wang, Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and for some numerical quadrature rules, Appl. Math. Lett. 11 (1) (1998) 105–109. [16] I. Fedotov, S.S. Dragomir, An inequality of Ostrowski type and its applications for Simpson’s rule and special means, Math. Inequal. Appl. 2 (4) (1999) 491–499. [17] Z. Liu, Some Ostrowski–Grüss type inequalities and applications, Comput. Math. Appl. 53 (2007) 73–79. [18] Z. Liu, Some Ostrowski type inequalities, Math. Comput. Modelling 48 (2008) 949–960. [19] C.E.M. Pearce, J. Pečarić, N. Ujević, S. Varošanec, Generalizations of some inequalities of Ostrowski–Grüss type, Math. Inequal. Appl. 3 (1) (2000) 25–34. [20] K.-L. Tseng, S.R. Hwang, S.S. Dragomir, Generalizations of weighted Ostrowski type inequalities for mappings of bounded variation and their applications, Comput. Math. Appl. 55 (2008) 1785–1793. [21] N. Ujević, New bounds for the first inequality of Ostrowski–Grüss type and applications, Comput. Math. Appl. 46 (2003) 421–427. [22] M. Niezgoda, Translation-invariant maps and applications, J. Math. Anal. Appl. 354 (2009) 111–124. [23] P. Cerone, S.S. Dragomir, New bounds for the Čebyšev functional, Appl. Math. Lett. 18 (2005) 603–611. [24] S.S. Dragomir, Bounds for some perturbed Čebyšev functionals, J. Inequal. Pure Appl. Math. 9 (3) (2008) Article 64.