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volume 2, issue 3, article 31, 2001.

Received 10 January, 2001;

accepted 23 April, 2001.

Communicated by:B. Mond

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Journal of Inequalities in Pure and Applied Mathematics

IMPROVEMENT OF AN OSTROWSKI TYPE INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATION

FOR SOME SPECIAL MEANS

S.S. DRAGOMIR AND M.L. FANG

School of Communications and Informatics Victoria University of Technology

PO Box 14428, Melbourne City MC 8001 Victoria, AUSTRALIA.

EMail:[email protected]

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html Department of Mathematics,

Nanjing Normal University, Nanjing 210097, P. R. China EMail:[email protected]

c

2000School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

006-01

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Improvement of An Ostrowski Type Inequality for Monotonic Mappings and Its Application

for Some Special Means S.S. DragomirandM.L. Fang

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J. Ineq. Pure and Appl. Math. 2(3) Art. 31, 2001

http://jipam.vu.edu.au

Abstract

We first improve two Ostrowski type inequalities for monotonic functions, then provide its application for special means.

2000 Mathematics Subject Classification:26D15, 26D10.

Key words: Ostrowski’s inequality, Trapezoid inequality, Special means.

Project supported by the National Natural Science Foundation of China (Grant No.10071038)

1. Introduction

In [1], Dragomir established the following Ostrowski’s inequality for monotonic mappings.

Theorem 1.1. Let f : [a, b] → R be a monotonic nondecreasing mapping on [a, b]. Then for allx∈[a, b], we have the following inequality

f(x)− 1 b−a

Z b

a

f(t)dt

≤ 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)).

(1.1)

The constant ¹₂ is the best possible one.

In [2], Dragomir, Peˇcari´c and Wang generalized Theorem1.1and proved Theorem 1.2. Let f : [a, b] → R be a monotonic nondecreasing mapping on [a, b]andt₁,t₂,t₃ ∈(a, b)be such thatt₁ ≤t₂ ≤t₃. Then

Z b

a

f(x)dx−[(t1−a)f(a) + (t3−t1)f(t2) + (b−t3)f(b)]

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≤(b−t₃)f(b) + (2t₂−t₁ −t₃)f(t₂)−(t₁−a)f(a) +

Z b

a

T(x)f(x)dx

≤(b−t₃)(f(b)−f(t₃)) + (t₃−t₂)(f(t₃)−f(t₂))

+ (t₂−t₁)(f(t₂)−f(t₁)) + (t₁−a)(f(t₁)−f(a))

≤max{t₁−a, t₂−t₁, t₃ −t₂, b−t₃}(f(b)−f(a)), (1.2)

where T(x) = sgn(t₁ − x), for x ∈ [a, t₂], and T(x) = sgn(t₃ − x), for x∈[t₂, b].

In the present paper, we firstly improve the above results, and then provide its application for some special means.

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2. Main Result

We shall start with the following result.

Theorem 2.1. Let f : [a, b] → R be a monotonic nondecreasing mapping on [a, b]and lett1,t2,t3 ∈[a, b]be such thatt1 ≤t2 ≤t3. Then

Z b

a

f(x)dx−[(t1−a)f(a) + (t3−t1)f(t2) + (b−t3)f(b)]

≤max{(b−t₃)(f(b)−f(t₃)) + (t₂−t₁)(f(t₂)−f(t₁)), (t₃−t₂)(f(t₃)−f(t₂)) + (t₁−a)(f(t₁)−f(a))}

(2.1)

≤max{t₁−a, t₂−t₁, t₃ −t₂, b−t₃}(f(b)−f(a)).

(2.2)

Proof. Sincef(x)is a monotonic nondecreasing mapping on[a, b], we have

Z b

a

f(x)dx−[(t₁−a)f(a) + (t₃−t₁)f(t₂) + (b−t₃)f(b)]

=

Z t1

a

(f(x)−f(a))dx+ Z t3

t1

(f(x)−f(t₂))dx+ Z b

t3

(f(x)−f(b))dx

=

Z t1

a

(f(x)−f(a))dx+ Z t3

t2

(f(x)−f(t₂))dx

− Z t2

t1

(f(t₂)−f(x))dx+ Z b

t3

(f(b)−f(x))dx

≤max{(b−t₃)(f(b)−f(t₃)) + (t₂−t₁)(f(t₂)−f(t₁)), (t₃−t₂)(f(t₃)−f(t₂)) + (t₁−a)(f(t₁)−f(a))}

≤max{t₁−a, t₂−t₁, t₃−t₂, b−t₃}(f(b)−f(a)).

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Thus (2.1) and (2.2) is proved.

Fort₁ =t₂ =t₃ =x, Theorem2.1becomes the following corollary.

Corollary 2.2. Letf be defined as in Theorem2.1. Then

Z b

a

f(x)dx−[(x−a)f(a) + (b−x)f(b)]

≤max{(b−x)(f(b)−f(x)),(x−a)(f(x)−f(a))}

≤max{x−a, b−x} ·max{(f(x)−f(a)),(f(b)−f(x))}

≤ 1

2(b−a) +

x−a+b 2

(f(b)−f(a)).

Forx= ^a+b₂ , we get trapezoid inequality.

Corollary 2.3. Letf be defined as in Theorem2.1. Then

Z b

a

f(x)dx−f(a) +f(b)

2 (b−a)

≤ b−a 2 max

f

a+b 2

−f(a)

,

f(b)−f

a+b 2

(2.3)

≤ 1

2(b−a)(f(b)−f(a)).

Fort₁ =a,t₂ =x,t₃ =b, we get Theorem1.1.

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3. Application for Special Means

In this section, we shall give application of Corollary 2.3. Let us recall the following means.

1. The arithmetic mean:

A=A(a, b) := a+b

2 , a, b≥0.

2. The geometric mean:

G=G(a, b) := √

ab, a, b≥0.

3. The harmonic mean:

H =H(a, b) := 2

1/a+ 1/b, a, b≥0.

4. The logarithmic mean:

L=L(a, b) := b−a

lnb−lna, a, b≥0, a6=b; Ifa =b, thenL(a, b) =a.

5. The identric mean:

I =I(a, b) := 1 e

b^b a^a

_b−a¹

, a, b≥0, a6=b; Ifa=b, thenI(a, b) =a.

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6. Thep-logarithmic mean:

L_p =L_p(a, b) :=

b^p+1−a^p+1 (p+ 1)(b−a)

¹_p

, a6=b; Ifa=b, thenL_p(a, b) = a, wherep6=−1,0anda, b >0.

The following simple relationships are known in the literature H ≤G≤L≤I ≤A.

We are going to use inequality (2.3) in the following equivalent version:

1 b−a

Z b

a

f(t)dt− f(a) +f(b) 2

≤ 1 2max

f

a+b 2

−f(a)

,

f(b)−f

a+b 2

(3.1)

≤ 1

2(f(b)−f(a)),

wheref : [a, b]→Ris monotonic nondecreasing on[a, b].

3.1. Mapping f (x) = x

^p

Consider the mappingf : [a, b]⊂(0,∞)→R, f(x) = x^p, p >0. Then 1

b−a Z b

a

f(t)dt=L^p_p(a, b),

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f(a) +f(b)

2 =A(a^p, b^p), f(b)−f(a) = p(b−a)L^p−1_p−1. Then by (3.1), we get

L^p_p(a, b)−A(a^p, b^p) ≤ 1

2max

a+b 2

p

−a^p, b^p−

a+b 2

p

= 1 2

b^p−

a+b 2

p

= 1

2(b^p−a^p)− 1 2

a+b 2

p

−a^p

≤ 1

2p(b−a)L^p−1_p−1− p(b−a)a^p−1

4 .

(3.2)

Remark 3.1. The following result was proved in [2].

L^p_p(a, b)−A(a^p, b^p) ≤ 1

2p(b−a)L^p−1_p−1.

3.2. Mapping f (x) = −1/x

Consider the mappingf : [a, b]⊂(0,∞)→R, f(x) = −_x¹. Then 1

b−a Z b

a

f(t)dt=−L⁻¹(a, b),

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f(a) +f(b)

2 =− A(a, b) G²(a, b), f(b)−f(a) = b−a

G²(a, b). Then by (3.1), we get

A(a, b)

G²(a, b)−L⁻¹(a, b)

≤ 1 2max

1 a − 2

a+b, 2 a+b − 1

b

= 1

2· b−a a(a+b) = 1

2 · b−a ab − 1

2· b−a b(a+b)

≤ 1

2· b−a G²(a, b)− 1

2· b−a b(a+b). Thus we get

(3.3) 0≤AL−G² ≤ 1

2 b

a+b(b−a)L.

Remark 3.2. The following result was proved in [2].

0≤AG−G² ≤ 1

2(b−a)L.

3.3. Mapping f (x) = ln x

Consider the mappingf : [a, b]⊂(0,∞)→R, f(x) = lnx. Then 1

b−a Z b

a

f(t)dt= lnI(a, b),

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f(a) +f(b)

2 = lnG(a, b), f(b)−f(a) = b−a

L(a, b). Then by (3.1), we get

|lnI(a, b)−lnG(a, b)| ≤ 1 2max

lna+b

2 −lna,lnb−lna+b 2

= 1

2lna+b 2a = 1

2 b−a L(a, b)− 1

2ln 2b a+b. Thus we get

(3.4) 1≤ I

G ≤

ra+b

2b e¹²^·^L(a,b)^b−a . Remark 3.3. The following result was proved in [2].

1≤ I

G ≤e¹²^·^L(a,b)^b−a .

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References

[1] S.S. DRAGOMIR, Ostrowski’s inequality for monotonic mapping and ap- plications, J. KSIAM, 3(1) (1999), 129–135.

[2] S.S. DRAGOMIR, J. PE ˇCARI ´C ANDS. WANG, The unified treatment of trapezoid, Simpson, and Ostrowski type inequalities for monotonic map- pings and applications, Math. Comput. Modelling, 31 (2000), 61–70.

[3] S.S. DRAGOMIR AND 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 Computers Math. Applic., 33(11) (1997), 15–20.

[4] S.S. DRAGOMIRANDS. WANG, Applications of Ostrowski inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett., 11(1) (1998), 105–109.

[5] M. MATI Ć, J. PE ˇCARI ĆANDN. UJEVI Ć, Improvement and further gener- alization of inequalities of Ostrowski-Grüss type, Computers Math. Applic., 39(3/4) (2000), 161–175.

[6] D.S. MITRINOVI ´C, J. PE ˇCARI ´CANDA.M. FINK, Classical and New In- equalities in Analysis, Kluwer Academic, Dordrecht, 1993.

[7] D.S. MITRINOVI ´C, J. PE ˇCARI ´C AND A.M. FINK, Inequalities Involv- ing Functions and Their Integrals and Derivatives, Kluwer Academic, Dor- drecht, 1991.

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[8] G.S. YANG AND K.L. TSENG, On certain integral inequalities related to Hermite-Hadamard inequalities, J. Math. Anal. Appl., 239 (1999), 180–187.