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volume 3, issue 5, article 68, 2002.

Received 6 May, 2002;

accepted 3 June, 2002.

Communicated by:P. Bullen

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Journal of Inequalities in Pure and Applied Mathematics

OSTROWSKI TYPE INEQUALITIES FOR ISOTONIC LINEAR FUNCTIONALS

S.S. DRAGOMIR

School of Communications and Informatics Victoria University of Technology

PO Box 14428 Melbourne City MC 8001 Victoria, Australia

EMail:sever@matilda.vu.edu.au

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html

c

2000Victoria University ISSN (electronic): 1443-5756 047-02

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Ostrowski Type Inequalities for Isotonic Linear Functionals

S.S. Dragomir

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J. Ineq. Pure and Appl. Math. 3(5) Art. 68, 2002

http://jipam.vu.edu.au

Abstract

Some inequalities of Ostrowski type for isotonic linear functionals defined on a linear class of functionL :={f : [a, b]→R}are established. Applications for integral and discrete inequalities are also given.

2000 Mathematics Subject Classification:Primary 26D15, 26D10.

Key words: Ostrowski Type Inequalities, Isotonic Linear Functionals.

1. Introduction

The following result is known in the literature as Ostrowski’s inequality [13].

Theorem 1.1. Letf : [a, b]→ Rbe a differentiable mapping on(a, b)with the property that|f⁰(t)| ≤M for allt∈(a, b). Then

(1.1)

f(x)− 1 b−a

Z b a

f(t)dt

≤

"

1

4+ x− ^a+b₂ 2

(b−a)²

#

(b−a)M for allx∈[a, b].

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

The following Ostrowski type result for absolutely continuous functions whose derivatives belong to the Lebesgue spacesL_p[a, b]also holds (see [9], [10] and [11]).

Theorem 1.2. Let f : [a, b] → Rbe absolutely continuous on [a, b]. Then, for allx∈[a, b], we have:

(1.2)

f(x)− 1 b−a

Z b a

f(t)dt

≤











1

4 +_x−a+b 2

b−a

²

(b−a)kf⁰k_∞ iff⁰ ∈L∞[a, b] ;

(b−a)¹^p (p+1)

1 p

h x−a b−a

p+1

+ ^b−x_b−ap+1i¹_p

kf⁰k_q iff⁰ ∈Lq[a, b], ¹_p +¹_q = 1, p >1;

h1 2 +

x−^a+b

2

b−a

ikf⁰k₁;

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wherek·k_r (r∈[1,∞]) are the usual Lebesgue norms onL_r[a, b], i.e., kgk_∞ :=ess sup

t∈[a,b]

|g(t)|

and

kgk_r :=

Z b a

|g(t)|^rdt ¹r

, r∈[1,∞).

The constants ¹₄, ¹

(p+1)

1

p and ¹₂ respectively are sharp in the sense presented in Theorem1.1.

The above inequalities can also be obtained from Fink’s result in [12] on choosingn= 1and performing some appropriate computations.

If one drops the condition of absolute continuity and assumes thatfis Hölder continuous, then one may state the result (see [7]):

Theorem 1.3. Letf : [a, b]→Rbe ofr−H−Hölder type, i.e., (1.3) |f(x)−f(y)| ≤H|x−y|^r, for all x, y ∈[a, b],

where r ∈ (0,1] and H > 0 are fixed. Then for allx ∈ [a, b] we have the inequality:

(1.4)

f(x)− 1 b−a

Z b a

f(t)dt

≤ H r+ 1

"

b−x b−a

r+1

+

x−a b−a

r+1#

(b−a)^r. The constant _r+1¹ is also sharp in the above sense.

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Note that ifr = 1, i.e.,f is Lipschitz continuous, then we get the following version of Ostrowski’s inequality for Lipschitzian functions (with Linstead of H) (see [3])

(1.5)

f(x)− 1 b−a

Z b a

f(t)dt

≤



 1

4+ x− ^a+b₂ b−a

!2

(b−a)L.

Here the constant ¹₄ is also best.

Moreover, if one drops the continuity condition of the function, and assumes that it is of bounded variation, then the following result may be stated (see [4]).

Theorem 1.4. Assume that f : [a, b] → Ris of bounded variation and denote byWb

a(f)its total variation. Then (1.6)

f(x)− 1 b−a

Z b a

f(t)dt

≤

"

1 2+

x−^a+b₂ b−a

# _b _

a

(f) for allx∈[a, b].

The constant ¹₂ is the best possible.

If we assume more about f, i.e., f is monotonically increasing, then the inequality (1.6) may be improved in the following manner [5] (see also [2]).

Theorem 1.5. Let f : [a, b] → R be monotonic nondecreasing. Then for all

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x∈[a, b], we have the inequality:

f(x)− 1 b−a

Z b a

f(t)dt (1.7)

≤ 1 b−a

[2x−(a+b)]f(x) + Z b

a

sgn(t−x)f(t)dt

≤ 1

b−a{(x−a) [f(x)−f(a)] + (b−x) [f(b)−f(x)]}

≤

"

1 2+

x− ^a+b₂ b−a

#

[f(b)−f(a)].

All the inequalities in (1.7) are sharp and the constant ¹₂ is the best possible.

The version of Ostrowski’s inequality for convex functions was obtained in [6] and is incorporated in the following theorem:

Theorem 1.6. Let f : [a, b] → Rbe a convex function on[a, b]. Then for any x∈(a, b)we have the inequality

1 2

(b−x)²f₊⁰ (x)−(x−a)²f_{_}⁰(x) (1.8)

≤ Z b

a

f(t)dt−(b−a)f(x)

≤ 1 2

(b−x)²f₋⁰ (b)−(x−a)²f₊⁰ (a) . In both parts of the inequality (1.8) the constant ¹₂ is sharp.

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For other Ostrowski type inequalities, see [8].

In this paper we extend Ostrowski’s inequality for arbitrary isotonic linear functionals A : L → R, where L is a linear class of absolutely continuous functions defined on[a, b].Some applications for particular instances of linear functionalsAare also provided.

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2. Preliminaries

LetLbe a linear class of real-valued functions,g :E →Rhaving the proper- ties

(L1) f, g∈Limply(αf +βg)∈Lfor allα, β ∈R; (L2) 1∈L,i.e., iff(t) = 1,t ∈E,thenf ∈L.

An isotonic linear functionalA:L→Ris a functional satisfying (A1) A(αf +βg) = αA(f) +βA(g)for allf, g∈Landα, β ∈R; (A2) Iff ∈Landf ≥0, thenA(f)≥0.

The mappingAis said to be normalised if (A3) A(1) = 1.

Usual examples of isotonic linear functional that are normalised are the following ones

A(f) := 1 µ(X)

Z

X

f(x)dµ(x), if µ(X)<∞ or

A_w(f) := 1 R

Xw(x)dµ(x) Z

X

w(x)f(x)dµ(x), where w(x) ≥ 0, R

Xw(x)dµ(x) > 0, X is a measurable space and µ is a positive measure onX.

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In particular, forx¯ = (x₁, . . . , x_n),w¯ := (w₁, . . . , w_n) ∈ Rⁿ withw_i ≥ 0, Wn:=Pn

i=1wi >0we have

A(¯x) := 1 n

n

X

i=1

xi

and

A_w_¯(¯x) := 1 W_n

n

X

i=1

w_ix_i,

are normalised isotonic linear functionals onRⁿ.

The following representation result for absolutely continuous functions holds.

Lemma 2.1. Letf : [a, b] → Rbe an absolutely continuous function on[a, b]

and define e(t) = t, t ∈ [a, b], g(t, x) = R1

0 f⁰[(1−λ)x+λt]dλ, t ∈ [a, b]

andx∈[a, b].IfA:L→Ris a normalised linear functional on a linear class Lof absolutely continuous functions defined on[a, b]and(x−e)·g(·, x)∈L, then we have the representation

(2.1) f(x) = A(f) +A[(x−e)·g(·, x)], forx∈[a, b].

Proof. For anyx, t ∈[a, b]witht6=x,one has f(x)−f(t)

x−t = Rx

t f⁰(u) x−t =

Z 1 0

f⁰[(1−λ)x+λt]dλ =g(t, x),

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giving the equality

(2.2) f(x) =f(t) + (x−t)g(t, x) for anyt, x∈[a, b].

Applying the functionalA,we get

A(f(x)·1) =A(f + (x−e)g(·, x)), for anyx∈[a, b].

Since

A(f(x)·1) =f(x)A(1) =f(x) and

A(f + (x−e)·g(·, x)) =A(f) +A((x−e)·g(·, x)), the equality (2.1) is obtained.

The following particular cases are of interest:

Corollary 2.2. Let f : [a, b] → R be an absolutely continuous function on [a, b].Then we have the representation:

(2.3) f(x) = 1 Rb

aw(t)dt Z b

a

w(t)f(t)dt

+ 1

Rb

a w(t)dt Z b

a

w(t) (x−t) Z 1

0

f⁰[(1−λ)x+λt]dλ

dt

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for anyx ∈[a, b],wherep : [a, b]→ Ris a Lebesgue integrable function with Rb

a w(t)dt6= 0.

In particular, we have

(2.4) f(x) = 1 b−a

Z b a

f(t)dt

+ 1

b−a Z b

a

(x−t) Z 1

0

f⁰[(1−λ)x+λt]dλ

dt for eachx∈[a, b].

The proof is obvious by Lemma2.1applied for the normalised linear functionals

A_w(f) := 1 Rb

a w(t)dt Z b

a

w(t)f(t)dt, A(f) := 1 b−a

Z b a

f(t)dt defined on

L:={f : [a, b]→R, f is absolutely continuous on [a, b]}. The following discrete case also holds.

Corollary 2.3. Let f : [a, b] → R be an absolutely continuous function on [a, b].Then we have the representation:

(2.5) f(x) = 1 W_n

n

X

i=1

w_if(x_i)

+ 1 W_n

n

X

i=1

w_i(x−x_i) Z 1

0

f⁰[(1−λ)x+λx_i]dλ

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for any x ∈ [a, b], where x_i ∈ [a, b], w_i ∈ R (i={1, . . . , n}) with W_n :=

Pn

i=1wi 6= 0.

In particular, we have

(2.6) f(x) = 1 n

n

X

i=1

f(x_i) + 1 n

n

X

i=1

(x−x_i) Z 1

0

f⁰[(1−λ)x+λx_i]dλ

for anyx∈[a, b].

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3. Ostrowski Type Inequalities

The following theorem holds.

Theorem 3.1. With the assumptions of Lemma2.1, and assuming thatA:L→ Ris isotonic, then we have the inequalities

(3.1) |f(x)−A(f)|

≤









 A

|x−e| kf⁰k_[x,·],∞

if |x−e| kf⁰k_[x,·],∞∈L, f⁰ ∈L_∞[a, b] ; A

|x−e|¹^q kf⁰k_[x,·],p

if |x−e|¹^q kf⁰k_[x,·],p ∈L, f⁰ ∈L_p[a, b], p > 1, ¹_p +¹_q = 1;

A

kf⁰k_[x,·],1

if kf⁰k_[x,·],1 ∈L, where

khk_[m,n],∞ :=ess sup

t∈[m,n]

(t∈[n,m])

|h(t)| and

khk_[m,n],p :=

Z n m

|h(t)|^pdt

1 p

, p≥1.

If we denote

M∞(x) := A

|x−e| kf⁰k_[x,·],∞

, M_p(x) := A

|x−e|¹^q kf⁰k_[x,·],p

, andM₁(x) := A

kf⁰k_[x,·],1 ,

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then we have the inequalities:

(3.2) M∞(x)

≤











kf⁰k_[a,b],∞A(|x−e|) if |x−e| ∈L, f⁰ ∈L_∞[a, b] ; h

A

kf⁰k^β_[x,·],∞i_β¹

[A(|x−e|^α)]^α¹

if kf⁰k^β_[x,·],∞,|x−e|^α ∈L, f⁰ ∈L∞[a, b], α >1, _α¹ + ¹_β = 1;

₁

2(b−a) +

x− ^a+b₂

A

kf⁰k_[x,·],∞

if kf⁰k_[x,·],∞∈L, f⁰ ∈L∞[a, b].

(3.3) M_p(x)

≤









 max

n

kf⁰k_[a,x],p,kf⁰k_[x,b],po A

|x−e|¹^q

if |x−e|¹^q ∈L, f⁰ ∈Lp[a, b] ; h

A

kf⁰k^β_[x,·],pi¹_β h A

|x−e|^α^qi_α¹

if kf⁰k^β_[x,·],p,|x−e|^α^q ∈L, f⁰ ∈L_p[a, b], α >1, _α¹ + ¹_β = 1;

₁

2(b−a) +

x− ^a+b₂

¹_q A

kf⁰k_[x,·],p

if kf⁰k_[x,·],p∈L, f⁰ ∈L_p[a, b]

and

(3.4) M₁(x)≤











1

2kf⁰k_[a,b],1+ ¹₂

kf⁰k_[a,x],1− kf⁰k_[x,b],1 , h

A

kf⁰k^β_[x,·],1i_β¹

, β >1.

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Proof. Using (2.1) and taking the modulus, we have

|f(x)−A(f)|=|A((x−e)·g(·, x))|

(3.5)

≤A(|(x−e)·g(·, x)|)

=A(|x−e| |g(·, x)|). Fort6=x(t, x∈[a, b])we may state

|g(t, x)| ≤ Z 1

0

|f⁰((1−λ)x+λt)|dλ

≤ess sup

λ∈[0,1]

|f⁰((1−λ)x+λt)|

=kf⁰k_[x,t],∞. Hölder’s inequality will produce

|g(t, x)| ≤ Z 1

0

≤ Z 1

0

|f⁰((1−λ)x+λt)|^pdλ ¹_p

= 1

x−t Z x

t

|f⁰(u)|^pdu ¹_p

=|x−t|⁻¹^p kf⁰k_[x,t],p, p > 1, 1 p +1

q = 1;

and finally

|g(t, x)| ≤ Z 1

0

|f⁰((1−λ)x+λt)|dλ = 1

t−xkf⁰k_[x,t],1.

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Consequently

(3.6) |(x−e)| |g(·, x)| ≤











|x−e| kf⁰k_[x,·],∞ if f⁰ ∈L∞[a, b] ;

|x−e|¹^q kf⁰k_[x,·],p if f⁰ ∈L_p[a, b], kf⁰k_[x,·],1

for anyx∈[a, b].

Applying the functionalAto (3.6) and using (3.5) we deduce the inequality (3.1).

We have

M∞(x)≤ sup

t∈[a,b]

nkf⁰k_[x,t],∞o

A(|x−e|)

= max n

kf⁰k_[a,x],∞,kf⁰k_[x,b],∞o

A(|x−e|)

=kf⁰k_[a,b],∞A(|x−e|) and the first inequality in (3.2) is proved.

Using Hölder’s inequality for the functionalA,i.e., (3.7) |A(hg)| ≤[A(|h|^α)]^α¹ h

A

|g|^βi_β¹

, α >1, 1 α + 1

β = 1, wherehg,|h|^α,|g|^β ∈L, we have

M_∞(x)≤[A(|x−e|^α)]^α¹ h A

kf⁰k^β_[x,·],∞i¹_β

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and the second part of (3.2) is proved.

In addition,

M∞(x)≤ sup

t∈[a,b]

|x−t|A

kf⁰k_[x,·],∞

= max{x−a, b−x}A

kf⁰k_[x,·],∞

= 1

2(b−a) +

x− a+b 2

A

kf⁰k_[x,·],∞

and the inequality (3.2) is completely proved.

We also have

M_p(x)≤ sup

t∈[a,b]

nkf⁰k_[x,t],po A

|x−e|¹^q

= maxn

kf⁰k_[a,x],p,kf⁰k_[x,b],po A

|x−e|¹^q . Using Hölder’s inequality (3.7) one has

M_p(x)≤h A

|x−e|^α^qi_α¹ h A

kf⁰k^β_[x,·],pi_β¹

, α >1, 1 α + 1

β = 1 and

M_p(x)≤ sup

t∈[a,b]

n|x−t|¹^qo A

kf⁰k_[x,·],p

= maxn

(x−a)¹^q ,(b−x)¹^qo A

kf⁰k_[x,·],p

= 1

2(b−a) +

x− a+b 2

¹_q A

kf⁰k_[x,·],p ,

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proving the inequality (3.3).

Finally, A

kf⁰k_[x,·],1

≤ sup

t∈[a,b]

nkf⁰k_[x,t],1o A(1)

= maxn

kf⁰k_[a,x],1,kf⁰k_[x,b],1o

= 1

2kf⁰k_[a,b],1+1 2

kf⁰k_[a,x],1− kf⁰k_[x,b],1 . By Hölder’s inequality, we have

A

kf⁰k_[x,·],1

≤h A

kf⁰k^β_[x,·],1i¹_β

, β >1, and the last part of (3.4) is also proved.

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4. The Case where |f

⁰

| is Convex

The following theorem also holds.

Theorem 4.1. Let f : [a, b] → R be an absolutely continuous function such thatf⁰ : (a, b) →Ris convex in absolute value, i.e.,|f⁰|is convex on (a, b).If A :L→Ris a normalised isotonic linear functional and|x−e|,|x−e| |f⁰| ∈ L,then

(4.1) |f(x)−A(f)| ≤ 1

2[|f⁰(x)|A(|x−e|) +A(|x−e| |f⁰|)]

≤











1 2

hkf⁰k_[a,b],∞+|f⁰(x)|i

A(|x−e|), if f⁰ ∈L∞[a, b] ;

1 2

|f⁰(x)|A(|x−e|) + [A(|x−e|^α)]^α¹ h A

|f⁰|^βi_β¹ if |x−e|^α,|f⁰|^β ∈L, α >1, _α¹ + ¹_β = 1;

1 2

|f⁰(x)|A(|x−e|) +₁

2(b−a) +

x− ^a+b₂

A(|f⁰|)

if |f⁰| ∈L.

Proof. Since|f⁰|is convex, we have

|g(t, x)| ≤ Z 1

0

=|f⁰(x)|

Z 1 0

(1−λ)dλ+|f⁰(t)|

Z 1 0

λdλ

= |f⁰(x)|+|f⁰(t)|

2 .

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Thus,

|f(x)−A(f)| ≤A

|x−e| · |f⁰(x)|+|f⁰(t)|

2

= 1

2[|f⁰(x)|A(|x−e|) +A(|x−e| |f⁰|)]

and the first part of (4.1) is proved.

We have

A(|x−e| |f⁰|)≤ess sup

t∈[a,b]

{|f⁰(t)|} ·A(|x−e|)

=kf⁰k_[a,b],∞A(|x−e|). By Hölder’s inequality for isotonic linear functionals, we have

A(|x−e| |f⁰|)≤[A(|x−e|^α)]^α¹ h

A

|f⁰|^βi_β¹

, α >1, 1 α + 1

β = 1 and finally,

A(|x−e| |f⁰|)≤ sup

t∈[a,b]

|x−t| ·A(|f⁰|)

= max (x−a, b−x)·A(|f⁰|)

= 1

2(b−a) +

x− a+b 2

A(|f⁰|). The theorem is thus proved.

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5. Some Integral Inequalities

If we consider the normalised isotonic linear functionalA(f) = _b−a¹ Rb

a f, then by Theorem3.1 forf : [a, b] → Ran absolutely continuous function, we may state the following integral inequalities

f(x)− 1 b−a

Z b a

f(t)dt (5.1)

≤ 1 b−a

Z b a

|x−t| kf⁰k_[x,t],∞dt

≤











kf⁰k_[a,b],∞

1 4 +

_x−a+b 2

b−a

2

(b−a) (Ostrowski’s inequality) providedf⁰ ∈L∞[a, b] ; h 1

b−a

Rb

a kf⁰k^β_[x,t],∞dti_β¹ h_(b−x)α+1+(x−a)^α+1 (α+1)(b−a)

i_α¹

iff⁰ ∈L_∞[a, b], kf⁰k_[x,·],∞ ∈L_β[a, b], α >1, _α¹ +_β¹ = 1;

1

2 +|^x−^a+b2 |

b−a

Rb

akf⁰k_[x,t],∞dt

iff⁰ ∈L∞[a, b], and if kf⁰k_[x,·],∞ ∈L1[a, b], for eachx∈[a, b] ;

(5.2)

f(x)− 1 b−a

Z b a

f(t)dt

≤ 1 b−a

Z b a

|x−t|¹^q kf⁰k_[x,t],pdt

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≤











qmaxn

kf⁰k_[a,x],p,kf⁰k_[x,b],po

(b−x)¹^q⁺¹+(x−a)¹^q⁺¹ (b−a)(q+1)

,

p >1, ¹_p +¹_q = 1 andf⁰ ∈Lp[a, b] ; q^α¹

1 b−a

Rb

akf⁰k^β_[x,t],pdt¹_β

(b−x)^α^q⁺¹+(x−a)^α^q⁺¹ (b−a)(q+α)

_α¹

iff⁰ ∈L_p[a, b], and kf⁰k_[x,·],p ∈Lβ[a, b], whereα >1, _α¹ +_β¹ = 1

1

2 + |^x−^a+b2 |

b−a

¹_q

1 b−a

Rb

akf⁰k_[x,t],pdt

iff⁰ ∈Lp[a, b], and kf⁰k_[x,·],p ∈L1[a, b], for eachx∈[a, b]and

f(x)− 1 b−a

Z b a

f(t)dt (5.3)

≤ 1 b−a

Z b a

kf⁰k_[x,t],1dt

≤











1

2 kf⁰k_[a,b],1+¹₂

kf⁰k_[a,x],1− kf⁰k_[x,b],1

iff⁰ ∈L₁[a, b] ; 1

b−a

Rb

a kf⁰k^β_[x,t],1dt ¹_β

iff⁰ ∈L₁[a, b], kf⁰k_[x,.],1 ∈L_β[a, b], whereβ >1, for eachx∈[a, b].

If we assume now thatf : [a, b]→Ris absolutely continuous and such that

|f⁰|is convex on (a, b),then by Theorem4.1 we obtain the following integral

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inequalities established in [1]

(5.4)

f(x)− 1 b−a

Z b a

f(t)dt

≤ 1 2



|f⁰(x)|



 1

4 + x− ^a+b₂ b−a

!2

(b−a) + 1 b−a

Z b a

|x−t| |f⁰(t)|dt





≤











1 2

hkf⁰k_[a,b],∞+|f⁰(x)|i

1

4 +_x−a+b 2

b−a

²

(b−a) iff⁰ ∈L∞[a, b] ;

1 2

|f⁰(x)|

1

4 +_x−a+b 2

b−a

2

(b−a) +h_(b−x)α+1+(x−a)^α+1

(α+1)(b−a)

i_α¹ h

1 b−a

Rb

a|f⁰(t)|^βdti_β¹ if f⁰ ∈L_β[a, b], α >1, _α¹ +_β¹ = 1;

1 2

|f⁰(x)|

1

4 +_x−a+b 2

b−a

2

(b−a) +

1

2 +|^x−^a+b₂ |

b−a

Rb

a|f⁰(t)|dt

iff⁰ ∈L₁[a, b], for eachx∈[a, b].

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6. Some Discrete Inequalities

For a given interval[a, b],consider the division

In:a=x0 < x1 <· · ·< xn−1 < xn =b

and the intermediate pointsξ_i ∈[x_i, x_i+1], i= 0, n−1.Ifh_i :=x_i+1−x_i >0 i= 0, n−1

we may define the following functionals

A(f;I_n, ξ) := 1 b−a

n−1

X

i=0

f(ξ_i)h_i (Riemann Rule)

A_T (f;I_n) := 1 b−a

n−1

X

i=0

f(x_i) +f(x_i+1)

2 ·h_i (Trapezoid Rule) A_M (f;I_n) := 1

b−a

n−1

X

i=0

f

x_i+x_i+1 2

·h_i (Mid-point Rule) AS(f;In) := 1

3AT (f;In) + 2

3AM (f;In). (Simpson Rule)

We observe that, all the above functionals are obviously linear, isotonic and normalised.

Consequently, all the inequalities obtained in Sections2 and 3may be applied for these functionals.

If, for example, we use the following inequality (see Theorem3.1) (6.1) |f(x)−A(f)| ≤ kf⁰k_[a,b]A(|x−e|), x∈[a, b],

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provided f : [a, b] → Ris absolutely continuous andf⁰ ∈ L∞[a, b],then we get the inequalities

(6.2)

f(x)− 1 b−a

n−1

X

i=0

f(ξ_i)h_i

≤ kf⁰k_[a,b],∞ 1 b−a

n−1

X

i=0

|x−ξ_i|h_i,

(6.3)

f(x)− 1 b−a

n−1

X

i=0

f(x_i) +f(x_i+1)

2 ·h_i

≤ kf⁰k_[a,b],∞· 1 b−a

n−1

X

i=0

|x−x_i|+|x−x_i+1|

2 hi,

(6.4)

f(x)− 1 b−a

n−1

X

i=0

f

x_i+x_i+1 2

·h_i

≤ kf⁰k_[a,b],∞ 1 b−a

n−1

X

i=0

x− x_i+x_i+1 2

h_i,

for eachx∈[a, b].

Similar results may be stated if one uses for example Theorem4.1. We omit the details.

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References

[1] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR, M.R. PINHEIRO AND

A. SOFO, Ostrowski type inequalities for functions whose modulus of derivatives are convex and applications, Res. Rep. Coll., 5(2) (2002), Arti- cle 1. [ONLINE:http://rgmia.vu.edu.au/v5n2.html]

[2] P. CERONEANDS.S. DRAGOMIR, Midpoint type rules from an inequal- ities point of view, in Analytic-Computational Methods in Applied Mathe- matics, G.A. Anastassiou (Ed), CRC Press, New York, 2000, 135-200.

[3] S.S. DRAGOMIR, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comp. and Math. with Appl., 38 (1999), 33- 37.

[4] S.S. DRAGOMIR, On the Ostrowski’s inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 33–40.

[5] S.S. DRAGOMIR, Ostrowski’s inequality for monotonous mappings and applications, J. KSIAM, 3(1) (1999), 127–135.

[6] S.S. DRAGOMIR, An Ostrowski type inequality for convex functions, Res. Rep. Coll., 5(1) (2002), Article 5. [ONLINE:

http://rgmia.vu.edu.au/v5n1.html]

[7] S.S. DRAGOMIR, P. CERONE, J. ROUMELIOTIS AND S. WANG, A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis, Bull. Math. Soc. Sci. Math. Roumanie, 42(90)(4) (1992), 301–314.

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[8] S.S. DRAGOMIRANDTh.M. RASSIAS (Eds.), Ostrowski Type Inequal- ities and Applications in Numerical Integration, Kluwer Academic Pub- lishers, Dordrecht/Boston/London, 2002.

[9] S.S. DRAGOMIRANDS. WANG, A new inequality of Ostrowski’s type in L1−norm and applications to some special means and to some numerical quadrature rules, Tamkang J. of Math., 28 (1997), 239–244.

[10] S.S. DRAGOMIR AND S. WANG, Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett., 11 (1998), 105–109.

[11] S.S. DRAGOMIRANDS. WANG, A new inequality of Ostrowski’s type in L_p−norm and applications to some special means and to some numerical quadrature rules, Indian J. of Math., 40(3) (1998), 245–304.

[12] A.M. FINK, Bounds on the deviation of a function from its averages, Czech. Math. J., 42(117) (1992), 289–310.

[13] A. OSTROWSKI, Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938), 226–227.