(1)http://jipam.vu.edu.au/ Volume 2, Issue 3, Article 31, 2001 IMPROVEMENT OF AN OSTROWSKI TYPE INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATION FOR SOME SPECIAL MEANS S.S

http://jipam.vu.edu.au/

Volume 2, Issue 3, Article 31, 2001

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

S.S. DRAGOMIR AND M.L. FANG

SCHOOL OFCOMMUNICATIONS ANDINFORMATICS

VICTORIAUNIVERSITY OFTECHNOLOGY

PO BOX14428 MELBOURNECITYMC VICTORIA8001, AUSTRALIA. sever@matilda.vu.edu.au

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html DEPARTMENT OFMATHEMATICS,

NANJINGNORMALUNIVERSITY, NANJING210097, P. R. CHINA

mlfang@pine.njnu.edu.cn

Received 10 January, 2001; accepted 23 April, 2001 Communicated by B. Mond

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

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

2000 Mathematics Subject Classification. 26D15, 26D10.

1. INTRODUCTION

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

Theorem 1.1. Letf : [a, b]→Rbe a monotonic nondecreasing mapping on[a, b]. Then for all x∈[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)

ISSN (electronic): 1443-5756

c 2001 Victoria University. All rights reserved.

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

006-01

The constant ¹₂ is the best possible one.

In [2], Dragomir, Peˇcari´c and Wang generalized Theorem 1.1 and proved

Theorem 1.2. Let f : [a, b] → Rbe 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)]

≤(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)

whereT(x) = sgn(t₁−x), forx∈[a, t₂], andT(x) =sgn(t₃−x), forx∈[t₂, b].

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

2. MAINRESULT

We shall start with the following result.

Theorem 2.1. Let f : [a, b] → Rbe 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)).

Thus (2.1) and (2.2) is proved.

Fort₁ =t₂ =t₃ =x, Theorem 2.1 becomes the following corollary.

Corollary 2.2. Letf be defined as in Theorem 2.1. 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 Theorem 2.1. 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 Theorem 1.1.

3. APPLICATION FORSPECIALMEANS

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, a 6=b; Ifa =b, thenI(a, b) =a.

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

1 b−a

Z b

a

f(t)dt =−L⁻¹(a, b), 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. Mappingf(x) = lnx. Consider the mappingf : [a, b] ⊂ (0,∞) → R, f(x) = lnx.

Then

1 b−a

Z b

a

f(t)dt = lnI(a, b), 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 e^{12·L(a,b)b−a} . Remark 3.3. The following result was proved in [2].

1≤ I

G ≤e^{12·L(a,b)b−a} . REFERENCES

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