JJ II

(1)

volume 6, issue 4, article 128, 2005.

Received 17 August, 2005;

accepted 30 August, 2005.

Communicated by:I. Gavrea

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

ON OSTROWSKI-GRÜSS- ˇCEBYŠEV TYPE INEQUALITIES FOR FUNCTIONS WHOSE MODULUS OF DERIVATIVES ARE CONVEX

B.G. PACHPATTE

57 Shri Niketan Colony Near Abhinay Talkies Aurangabad 431 001 (Maharashtra) India.

EMail:bgpachpatte@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 245-05

(2)

On Ostrowski-Grüss- ˇCebyšev Type Inequalities for Functions Whose Modulus of Derivatives

are Convex B.G. Pachpatte

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of31

J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005

http://jipam.vu.edu.au

Abstract

The aim of the present paper is to establish some new Ostrowski-Grüss- ˇCebyšev type inequalities involving functions whose modulus of the derivatives are convex.

2000 Mathematics Subject Classification:26D15, 26D20.

Key words: Ostrowski-Grüss- ˇCebyšev type inequalities, Modulus of derivatives, Convex, Log-convex, Integral identities.

1. Introduction

In 1938, A.M. Ostrowski [5] proved the following classial inequality.

Letf : [a, b]→Rbe continuous on[a, b]and differentiable on(a, b)whose derivativef⁰ : (a, b)→Ris bounded on(a, b)i.e.,|f⁰(x)| ≤M < ∞.Then (1.1)

f(x)− 1 b−a

Z b a

f(t)dt

≤



 1

4+ x− ^a+b₂ b−a

!2

(b−a)M, for allx∈[a, b],whereM is a constant.

For two absolutely continuous functions f, g : [a, b] → R, consider the functional

(1.2) T(f, g) = 1 b−a

Z b a

f(x)g(x)dx

− 1

b−a Z b

a

f(x)dx 1 b−a

Z b a

g(x)dx

, provided the involved integrals exist. In 1882, P.L. ˇCebyšev [6] proved that, if f⁰, g⁰ ∈L∞[a, b], then

(1.3) |T (f, g)| ≤ 1

12(b−a)²kf⁰k_∞kg⁰k_∞. In 1934, G. Grüss [6] showed that

(1.4) |T (f, g)| ≤ 1

4(M −m) (N −n),

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of31

provided m, M, n, N are real numbers satisfying the condition −∞ < m ≤ f(x)≤M <∞,−∞< n≤g(x)≤N <∞,for allx∈[a, b].

During the past few years many researchers have given considerable atten- tion to the above inequalities and various generalizations, extensions and vari- ants of these inequalities have appeared in the literature, see [1] – [10] and the references cited therein. Motivated by the recent results given in [1] – [3], in the present paper, we establish some inequalities similar to those given by Os- trowski, Grüss and ˇCebyšev, involving functions whose modulus of derivatives are convex. The analysis used in the proofs is elementary and based on the use of integral identities proved in [1] and [2].

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of31

2. Statement of Results

Let I be a suitable interval of the real lineR. A function f : I → Ris called convex if

f(λx+ (1−λ)y)≤λf(x) + (1−λ)f(y),

for all x, y ∈ I andλ ∈ [0,1].A function f : I → (0,∞)is said to be log- convex, if

f(tx+ (1−t)y)≤[f(x)]^t[f(y)]^1−t,

for allx, y ∈Iandt∈[0,1](see [10]). We need the following identities proved in [1] and [2] respectively:

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]dt

dt,

f(x) = 1 b−a

Z b a

f(t)dt+ (x−a)² 1 b−a

Z 1 0

λf⁰[(1−λ)a+λx]dλ

−(b−x)² 1 b−a

Z 1 0

λf⁰[λx+ (1−λ)b]dλ, forx∈[a, b]wheref : [a, b]→Ris an absolutely continuous function on[a, b]

andλ∈[0,1].

We use the following notation to simplify the details of presentation:

S(f, g) =f(x)g(x)− 1 2 (b−a)

g(x)

Z b a

f(t)dt+f(x) Z b

a

g(t)dt

,

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of31

and definek·k_∞as the usual Lebesgue norm onL∞[a, b]i.e., khk_∞ := ess sup_t∈[a,b]|h(t)|

forh∈L∞[a, b].

The following theorems deal with Ostrowski type inequalities involving two functions.

Theorem 2.1. Letf, g: [a, b]→Rbe absolutely continuous functions on[a, b].

(a₁) If|f⁰|,|g⁰|are convex on[a, b], then

(2.1) |S(f, g)| ≤ 1 4



 1

4 + x− ^a+b₂ b−a

!2

(b−a)

× {|g(x)|[|f⁰(x)|+kf⁰k_∞] +|f(x)|[|g⁰(x)|+kg⁰k_∞]}, forx∈[a, b].

(a₂) If|f⁰|,|g⁰|are log-convex on[a, b], then (2.2) |S(f, g)| ≤ 1

2 (b−a)

|g(x)| |f⁰(x)|

Z b a

|x−t|

A−1 logA

dt +|f(x)| |g⁰(x)|

Z b a

|x−t|

B−1 logB

dt

,

forx∈[a, b], where

(2.3) A= |f⁰(t)|

|f⁰(x)|, B = |g⁰(t)|

|g⁰(x)|.

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of31

(b₁) If|f⁰|,|g⁰|are convex on[a, x]and[x, b], then (2.4) |S(f, g)| ≤ 1

2{|g(x)|F (x) +|f(x)|G(x)}, forx∈[a, b], where

(2.5) F (x) = 1 6

h|f⁰(a)|

x−a b−a

2

+|f⁰(b)|

b−x b−a

2

+







1 + 4 x− ^a+b₂ b−a

!2





|f⁰(x)|



(b−a),

(2.6) G(x) = 1 6 h

|g⁰(a)|

x−a b−a

2

+|g⁰(b)|

b−x b−a

2

+







1 + 4 x− ^a+b₂ b−a

!2





|g⁰(x)|



(b−a), forx∈[a, b].

(b2) If|f⁰|,|g⁰|are log-convex on[a, x]and[x, b], then

(2.7) |S(f, g)| ≤ |g(x)|H(x) +|f(x)|L(x),

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of31

forx∈[a, b], where

(2.8) H(x) = 1

2(b−a)

"

|f⁰(a)|

x−a b−a

2

A1logA1+ 1−A1

(logA₁)² +|f⁰(b)|

b−x b−a

2

B₁logB₁ + 1−B₁ (logB₁)²

# ,

(2.9) L(x) = 1

2(b−a)

"

|g⁰(a)|

x−a b−a

2

A₂logA₂ + 1−A₂ (logA2)² +|g⁰(b)|

b−x b−a

2

B₂logB₂ + 1−B₂ (logB₂)²

# ,

and

A₁ = |f⁰(x)|

|f⁰(a)|, B₁ = |f⁰(x)|

|f⁰(b)|, (2.10)

A₂ = |g⁰(x)|

|g⁰(a)|, B₂ = |g⁰(x)|

|g⁰(b)|, (2.11)

forx∈[a, b].

The Grüss type inequalities are embodied in the following theorems.

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of31

(c₁) If|f⁰|,|g⁰|are convex on[a, b], then

(2.12) |T (f, g)| ≤ 1 4 (b−a)²

Z b a

h|g(x)|[|f⁰(x)|+kf⁰k_∞] +|f(x)|[|g⁰(x)|+kg⁰k_∞]i

E(x)dx, where

(2.13) E(x) = (x−a)²+ (b−x)²

2 ,

forx∈[a, b].

(c₂) If|f⁰|,|g⁰|are log-convex on[a, b], then (2.14) |T (f, g)|

≤ 1

2 (b−a)² Z b

a

|g(x)|

Z b a

|x−t| |f⁰(x)|

A−1 logA

dt +|f(x)|

Z b a

|x−t| |g⁰(x)|

B−1 logB

dt

dx, whereA,Bare defined by (2.3).

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of31

(d₁) If|f⁰|,|g⁰|are convex on[a, b], then (2.15) |T (f, g)| ≤ 1

2 Z b

a

"

x−a b−a

2

|g(x)|

1

6|f⁰(a)|+ 1

3|f⁰(x)|

+|f(x)|

1

6|g⁰(a)|+ 1

3|g⁰(x)|

+

b−x b−a

2

|g(x)|

1

3|f⁰(x)|+ 1

6|f⁰(b)|

+|f(x)|

1

3|g⁰(x)|+1

6|g⁰(b)|

dx,

(d₂) If|f⁰|,|g⁰|are log-convex on[a, x]and[x, b], then (2.16) |T (f, g)|

≤ 1 2

Z b a

"

x−a b−a

2

{|g(x)| |f⁰(a)|A₁logA₁+ 1−A₁ (logA1)² + |f(x)| |g⁰(a)|A₂logA₂+ 1−A₂

(logA₂)²

+

b−x b−a

2

|g(x)| |f⁰(b)|B₁logB₁+ 1−B₁ (logB₁)² +|f(x)| |g⁰(b)|B₂logB₂+ 1−B₂

(logB₂)²

dx, whereA1, B1andA2, B2 are defined by (2.10) and (2.11).

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of31

The next theorem contains ˇCebyšev type inequalities.

(e₁) If|f⁰|,|g⁰|are convex on[a, b], then (2.17) |T (f, g)|

≤ 1

4 (b−a)³ Z b

a

[|f⁰(x)|+kf⁰k_∞] [|g⁰(x)|+kg⁰k_∞]E²(x)dx, whereE(x)is given by (2.13).

(e₂) If|f⁰|,|g⁰|are log-convex on[a, b], then

(2.18) |T (f, g)| ≤ 1 (b−a)³

Z b a

|x−t| |f⁰(x)|

A−1 logA

dt

× Z b

a

|x−t| |g⁰(x)|

B−1 logB

dt

dx, whereA, B are defined by (2.3).

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of31

3. Proofs of Theorems 2.1 and 2.2

Proof of Theorem2.1. From the hypotheses of Theorem2.1, the following identities hold:

(3.1) 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,

(3.2) g(x) = 1 b−a

Z b a

g(t)dt

+ 1

b−a Z b

a

(x−t) Z 1

0

g⁰[(1−λ)x+λt]dλ

dt, for x ∈ [a, b].Multiplying both sides of (3.1) and (3.2) by g(x)and f(x)respectively, adding the resulting identities and rewriting we have

(3.3) S(f, g) = 1 2 (b−a)

g(x)

Z b a

(x−t) Z 1

0

dt +f(x)

Z b a

(x−t) Z 1

0

dt

. (a₁)Since|f⁰|,|g⁰|are convex on[a, b], from (3.3) we observe that

|S(f, g)|

≤ 1

2 (b−a)

|g(x)|

Z b a

|x−t|

Z 1 0

|f⁰[(1−λ)x+λt]|dλ

dt

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of31

+|f(x)|

Z b a

|x−t|

Z 1 0

|g⁰[(1−λ)x+λt]|dλ

dt

≤ 1

2 (b−a)

|g(x)|

Z b a

|x−t|

Z 1 0

{(1−λ)|f⁰(x)|+λ|f⁰(t)|}dλ

dt +|f(x)|

Z b a

|x−t|

Z 1 0

{(1−λ)|g⁰(x)|+λ|g⁰(t)|}dλ

dt

= 1

2 (b−a)

|g(x)|

Z b a

|x−t|

|f⁰(x)|

Z 1 0

(1−λ)dλ+|f⁰(t)|

Z 1 0

λdλ

dt +|f(x)|

Z b a

|x−t|

|g⁰(x)|

Z 1 0

(1−λ)dλ+|g⁰(t)|

Z 1 0

λdλ

dt

= 1

2 (b−a)

|g(x)|

Z b a

|x−t|1

2[|f⁰(x)|+|f⁰(t)|]dt +|f(x)|

Z b a

|x−t|1

2[|g⁰(x)|+|g⁰(t)|]dt

≤ 1

4 (b−a)

|g(x)| ess.sup

t ∈[a, b] [|f⁰(x)|+|f⁰(t)|]

Z b a

|x−t|dt +|f(x)| ess.sup

t ∈[a, b] [|g⁰(x)|+|g⁰(t)|]

Z b a

|x−t|dt

= 1

4 (b−a){|g(x)|[|f⁰(x)|+kf⁰k_∞] +|f(x)|

|g⁰(x)|+kg⁰k_∞ Z b

a

|x−t|dt

(14)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of31

= 1 4

"

(x−a)²+ (b−x)² 2 (b−a)

#

× {|g(x)|[|f⁰(x)|+kf⁰k_∞] +|f(x)|

|g⁰(x)|+kg⁰k_∞

= 1 4



 1

4 + x− ^a+b₂ b−a

!2



×(b−a){|g(x)|[|f⁰(x)|+kf⁰k_∞] +|f(x)|

|g⁰(x)|+kg⁰k

∞ . This is the required inequality in (2.1).

(a₂)Since|f⁰|,|g⁰|are log-convex on[a, b], from (3.3) we observe that

|S(f, g)| ≤ 1 2 (b−a)

|g(x)|

Z b a

|x−t|

Z 1 0

dt +|f(x)|

Z b a

|x−t|

Z 1 0

dt

≤ 1

2 (b−a)

|g(x)|

Z b a

|x−t|

Z 1 0

[|f⁰(x)|]^1−λ[|f⁰(t)|]^λdλ

dt +|f(x)|

Z b a

|x−t|

Z 1 0

[|g⁰(x)|]^1−λ[|g⁰(t)|]^λdλ

dt

= 1

2 (b−a) (

|g(x)|

Z b a

|x−t|

"

|f⁰(x)|

Z 1 0

|f⁰(t)|

|f⁰(x)|

λ

dλ

# dt

+|f(x)|

Z b a

|x−t|

"

|g⁰(x)|

Z 1 0

|g⁰(t)|

|g⁰(x)|

λ

dλ

# dt

)

(15)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of31

= 1

2 (b−a)

|g(x)| |f⁰(x)|

Z b a

|x−t|

A−1 logA

dt +|f(x)| |g⁰(x)|

Z b a

|x−t|

B−1 logB

dt

.

This completes the proof of the inequality (2.2).

Proof of Theorem2.2. From the hypotheses of Theorem2.2, the following identities hold:

(3.4) f(x) = 1 b−a

Z b a

f(t)dt + (x−a)² 1 b−a

Z 1 0

−(b−x)² 1 b−a

Z 1 0

λf⁰[λx+ (1−λ)b]dλ,

(3.5) g(x) = 1 b−a

Z b a

g(t)dt + (x−a)² 1

b−a Z 1

0

λg⁰[(1−λ)a+λx]dλ

−(b−x)² 1 b−a

Z 1 0

λg⁰[λx+ (1−λ)b]dλ.

(16)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of31

Multiplying both sides of (3.4) and (3.5) byg(x)andf(x)respectively, adding the resulting identities and rewriting we have

(3.6) S(f, g) = 1 2

g(x)

(x−a)² 1 b−a

Z 1 0

−(b−x)² 1 b−a

Z 1 0

λf⁰[λx+ (1−λ)b]dλ

+f(x)

(x−a)² 1 b−a

Z 1 0

−(b−x)² 1 b−a

Z 1 0

λg⁰[λx+ (1−λ)b]dλ

.

(b₁)Since|f⁰|,|g⁰|are convex on[a, x]and[x, b], from (3.6) we observe that (3.7) |S(f, g)| ≤ 1

2{|g(x)|M(x) +|f(x)|N(x)}, where

(3.8) M(x) = (x−a)² 1 b−a

Z 1 0

λ|f⁰[(1−λ)a+λx]|dλ + (b−x)² 1

b−a Z 1

0

λ|f⁰[λx+ (1−λ)b]|dλ,

(3.9) N(x) = (x−a)² b−a

Z 1 0

λ|g⁰[(1−λ)a+λx]|dλ

(17)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of31

+(b−x)² b−a

Z 1 0

λ|g⁰[λx+ (1−λ)b]|dλ.

Next, we observe that Z 1

0

λ|f⁰[(1−λ)a+λx]|dλ (3.10)

≤ |f⁰(a)|

Z 1 0

λ(1−λ)dλ+|f⁰(x)|

Z 1 0

λ²dλ

= 1

6|f⁰(a)|+ 1

3|f⁰(x)|

and

Z 1 0

λ|f⁰[λx+ (1−λ)b]|dλ (3.11)

≤ |f⁰(x)|

Z 1 0

λ²dλ+|f⁰(b)|

Z 1 0

λ(1−λ)dλ

= 1

3|f⁰(x)|+1

6|f⁰(b)|. Similarly we have

(3.12)

Z 1 0

λ|g⁰[(1−λ)a+λx]|dλ≤ 1

6|g⁰(a)|+1

3|g⁰(x)|,

(3.13)

Z 1 0

λ|g⁰[λx+ (1−λ)b]|dλ≤ 1

3|g⁰(x)|+1

6|g⁰(b)|.

(18)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of31

From (3.8), (3.10) and (3.11) we observe that M(x) =

"

x−a b−a

2Z 1 0

λ|f⁰[(1−λ)a+λx]|dλ (3.14)

+

b−x b−a

2Z 1 0

λ|f⁰[λx+ (1−λ)b]|dλ

#

(b−a)

≤

"

x−a b−a

2 1

6|f⁰(a)|+1

3|f⁰(x)|

+

b−x b−a

2 1

3|f⁰(x)|+1

6|f⁰(b)|

#

(b−a)

= 1 6

"

x−a b−a

2

|f⁰(a)|+

b−x b−a

2

|f⁰(b)|

#

(b−a)

+1 3

"

x−a b−a

2

+

b−x b−a

2#

|f⁰(x)|(b−a)

= 1 6

"

x−a b−a

2

|f⁰(a)|+

b−x b−a

2

|f⁰(b)|

+ 2

"

x−a b−a

2

+

b−x b−a

2#

|f⁰(x)|

#

(b−a).

(19)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page19of31

It is easy to observe that 2

"

x−a b−a

2

+

b−x b−a

2#

= 4

b−a

"

(x−a)²+ (b−x)² 2 (b−a)

# (3.15)

= 4

b−a



 1

4 + x− ^a+b₂ b−a

!2

(b−a)

=



1 + 4 x− ^a+b₂ b−a

!2

.

Using (3.15) in (3.14) we get

(3.16) M(x)≤F (x).

Similarly, from (3.9), (3.12), (3.13) we get

(3.17) N(x)≤G(x).

Using (3.16), (3.17) in (3.7) we get the required inequality in (2.4).

(b₂)Since |f⁰|,|g⁰| are log-convex on [a, x]and [x, b] , from (3.6) we observe that

|S(f, g)|

(3.18)

≤ 1

2(b−a) (

|g(x)|

"

x−a b−a

2Z 1 0

λ|f⁰[(1−λ)a+λx]|dλ

(20)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page20of31

+

b−x b−a

2Z 1 0

λ|f⁰[λx+ (1−λ)b]|dλ

#

+|f(x)|

"

x−a b−a

2Z 1 0

+

b−x b−a

2Z 1 0

λ|g⁰[λx+ (1−λ)b]|dλ

#)

≤ 1

2(b−a) (

|g(x)|

"

x−a b−a

2Z 1 0

λ[|f⁰(a)|]^1−λ[|f⁰(x)|]^λdλ

+

b−x b−a

2Z 1 0

λ[|f⁰(x)|]^λ[|f⁰(b)|]^1−λ

#

+|f(x)|

"

x−a b−a

2Z 1 0

λ[|g⁰(a)|]^1−λ[|g⁰(x)|]^λdλ

+

b−x b−a

2Z 1 0

λ[|g⁰(x)|]^λ[|g⁰(b)|]^1−λdλ

#)

= 1

2(b−a) (

|g(x)|

"

x−a b−a

2

|f⁰(a)|

Z 1 0

λA^λ₁dλ

+

b−x b−a

2

|f⁰(b)|

Z 1 0

λB₁^λdλ

#

(21)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page21of31

+|f(x)|

"

x−a b−a

2

|g⁰(a)|

Z 1 0

λA^λ₂dλ

+

b−x b−a

2

|g⁰(b)|

Z 1 0

λB₂^λdλ

#) .

A simple calculation shows that for anyC > 0we have (see [2]) (3.19)

Z 1 0

λC^λdλ= ClogC+ 1−C (logC)² . Using this fact in (3.18) we get the required inequality in (2.7).

(22)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page22of31

4. Proofs of Theorems 2.3 and 2.4

Proof of Theorem2.3. From the hypotheses of Theorem2.3the identities (3.1) – (3.3) hold. Integrating both sides of (3.3) with respect toxfrom a tob and rewriting we have

(4.1) T(f, g)

= 1

2 (b−a)² Z b

a

g(x)

Z b a

(x−t) Z 1

0

dt

+f(x) Z b

a

(x−t) Z 1

0

dt

dx.

(c₁)Since|f⁰|,|g⁰|are convex on[a, b], from (4.1) we observe that

|T (f, g)|

≤ 1

2 (b−a)² Z b

a

|g(x)|

Z b a

|x−t|

Z 1 0

[(1−λ)|f⁰(x)|+λ|f⁰(t)|]dλ

dt

+|f(x)|

Z b a

|x−t|

Z 1 0

[(1−λ)|g⁰(x)|+λ|g⁰(t)|]dλ

dt

dx

= 1

2 (b−a)² Z b

a

|g(x)|

Z b a

|x−t|

|f⁰(x)|+|f⁰(t)|

2

dt +|f(x)|

Z b a

|x−t|

|g⁰(x)|+|g⁰(t)|

2

dt

dx

(23)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page23of31

≤ 1

2 (b−a)² Z b

a

|g(x)|

Z b a

|x−t| esssup t∈[a, b]

|f⁰(x)|+|f⁰(t)|

2

dt

+|f(x)|

Z b a

|x−t| ess sup t∈[a, b]

|g⁰(x)|+|g⁰(t)|

2

dt

dx

= 1

4 (b−a)² Z b

a

[|g(x)|[|f⁰(x)|+kf⁰k_∞]dt +|f(x)|[|g⁰(x)|+kg⁰k_∞]]

Z b a

|x−t|dt

dx

= 1

4 (b−a)² Z b

a

[|g(x)|[|f⁰(x)|+kf⁰k_∞]dt +|f(x)|[|g⁰(x)|+kg⁰k_∞]]E(x)dx.

This completes the proof of the inequality (2.14).

(c2)Since|f⁰|,|g⁰|are log-convex on[a, b]from (4.1) we observe that

|T (f, g)| ≤ 1 2 (b−a)²

Z b a

|g(x)|

Z b a

|x−t|

Z 1 0

[|f⁰(x)|]^1−λ[|f⁰(t)|]^λdλ

dt +|f(x)|

Z b a

|x−t|

Z 1 0

[|g⁰(x)|]^1−λ[|g⁰(t)|]^λdλ

dt

dx

= 1

2 (b−a)² Z b

a

"

|g(x)|

Z b a

|x−t|

"

|f⁰(x)|

Z 1 0

|f⁰(t)|

|f⁰(x)|

λ

dλ

# dt

(24)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page24of31

+|f(x)|

Z b a

|x−t|

"

|g⁰(x)|

Z 1 0

|g⁰(t)|

|g⁰(x)|

λ

dλ

# dt

# dx

= 1

2 (b−a)² Z b

a

|g(x)|

Z b a

|x−t| |f⁰(x)|

A−1 logA

dt

+|f(x)|

Z b a

|x−t| |g⁰(x)|

B−1 logB

dt

dx,

whereA, Bare defined by (2.3). This is the required inequality in (2.14).

Proof of Theorem2.4. From the hypotheses of Theorem2.4the identities (3.4) – (3.6) hold. Integrating both sides of (3.6) with respect toxfrom a tob and rewriting we have

(4.2) T(f, g) = 1 2

Z b a

"

x−a b−a

2 g(x)

Z 1 0

+f(x) Z 1

0

−

b−x b−a

2 g(x)

Z 1 0

λf⁰[λx+ (1−λ)b]dλ +f(x)

Z 1 0

λg⁰[λx+ (1−λ)b]dλ

dx.

(25)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page25of31

(d₁)Since|f⁰|,|g⁰|are convex on[a, x]and[x, b]from (4.2) we observe that

|T(f, g)| ≤ 1 2

Z b a

"

x−a b−a

2

|g(x)|

Z 1 0

λ|f⁰[(1−λ)a+λx]|dλ

+|f(x)|

Z 1 0

+

b−x b−a

2

|g(x)|

Z 1 0

λ|f⁰[λx+ (1−λ)b]|dλ +|f(x)|

Z 1 0

λ|g⁰[λx+ (1−λ)b]|dλ

dx

≤ 1 2

Z b a

"

x−a b−a

2

|g(x)|

Z 1 0

λ{(1−λ)|f⁰(a)|+λ|f⁰(x)|}dλ

+|f(x)|

Z 1 0

λ{(1−λ)|g⁰(a)|+λ|g⁰(x)|}dλ

+

b−x b−a

2

|g(x)|

Z 1 0

λ{λ|f⁰(x)|+ (1−λ)|f⁰(b)|}dλ +|f(x)|

Z 1 0

λ{λ|g⁰(x)|+ (1−λ)|g⁰(b)|}dλ

dx

= 1 2

Z b a

"

x−a b−a

2

|g(x)|

1

6|f⁰(a)|+1

3|f⁰(x)|

+|f(x)|

1

6|g⁰(a)|+1

3|g⁰(x)|

(26)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page26of31

+

b−x b−a

2

|g(x)|

1

3|f⁰(x)|+1

6|f⁰(b)|

+|f(x)|

1

3|g⁰(x)|+ 1

6|g⁰(b)|

dx.

This proves the inequality in (2.15).

(d₂) Since|f⁰|,|g⁰| are log-convex on [a, x]and [x, b], from (4.2) and the fact (3.19) we observe that

|T(f, g)|

≤ 1 2

Z b a

"

x−a b−a

2

|g(x)|

Z 1 0

λ[|f⁰(a)|]^1−λ[|f⁰(x)|]^λdλ

+|f(x)|

Z 1 0

λ[|g⁰(a)|]^1−λ[|g⁰(x)|]^λdλ

+

b−x b−a

2

|g(x)|

Z 1 0

λ[|f⁰(x)|]^λ[|f⁰(b)|]^1−λdλ +|f(x)|

Z 1 0

λ[|g⁰(x)|]^λ[|g⁰(b)|]^1−λdλ

dx

= 1 2

Z b a

"

x−a b−a

2

|g(x)| |f⁰(a)|

Z 1 0

λA^λ₁dλ+|f(x)| |g⁰(a)|

Z 1 0

λB₁^λdλ

+

b−x b−a

2

|g(x)| |f⁰(b)|

Z 1 0

λA^λ₂dλ+|f(x)| |g⁰(b)|

Z 1 0

λB₂^λdλ #

dx

(27)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page27of31

= 1 2

Z b a

"

x−a b−a

2

|g(x)| |f⁰(a)| A₁logA₁+ 1−A₁ (logA₁)² +|f(x)| |g⁰(a)|B1logB1+ 1−B1

(logB₁)²

+

b−x b−a

2

|g(x)| |f⁰(b)|A₂logA₂+ 1−A₂ (logA₂)² +|f(x)| |g⁰(b)|B₂logB₂+ 1−B₂

(logB₂)²

dx.

This is the desired inequality in (2.16).

(28)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page28of31

5. Proof of Theorem 2.5

From the hypotheses, the identities (3.1) and (3.2) hold. From (3.1) and (3.2) we observe that

f(x)− 1 b−a

Z b a

f(t)dt g(x)− 1 b−a

Z b a

g(t)dt

= 1

b−a Z b

a

(x−t) Z 1

0

dt

× 1

b−a Z b

a

(x−t) Z 1

0

dt

that is,

(5.1) f(x)g(x)−f(x) 1

b−a Z b

a

g(t)dt

−g(x) 1

b−a Z b

a

f(t)dt

+ 1

b−a Z b

a

f(t)dt 1 b−a

Z b a

g(t)dt

= 1

b−a Z b

a

(x−t) Z 1

0

dt

× 1

b−a Z b

a

(x−t) Z 1

0

dt

. Integrating both sides of (5.1) with respect to x from a tob and rewriting we have

(29)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page29of31

(5.2) T(f, g)

= 1

b−a Z b

a

1 b−a

Z b a

(x−t) Z 1

0

dt

× 1

b−a Z b

a

(x−t) Z 1

0

dt

dx.

(e1)Since|f⁰|,|g⁰|are convex on[a, b], from (5.2) we observe that

|T (f, g)| ≤ 1 b−a

Z b a

1 b−a

Z b a

|x−t|

Z 1 0

dt

× 1

b−a Z b

a

|x−t|

Z 1 0

dt

dx

≤ 1

(b−a)³ Z b

a

Z b a

|x−t|

Z 1 0

[(1−λ)|f⁰(x)|+λ|f⁰(t)|]dλ

dt

× Z b

a

|x−t|

Z 1 0

[(1−λ)|g⁰(x)|+λ|g⁰(t)|]dλ

dt

dx

= 1

(b−a)³ Z b

a

Z b a

|x−t|

|f⁰(x)|+|f⁰(t)|

2

dt

× Z b

a

|x−t|

|g⁰(x)|+|g⁰(t)|

2

dt

dx.

The rest of the proof of inequality (2.17) can be completed by closely looking at the proof of Theorem2.3, part(c1).

(e₂)The proof follows by closely looking at the proof of(e₁)given above and the proof of Theorem2.3, part(c₂). We omit the details.

(30)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page30of31

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, RGMIA Res. Rep. Coll., 5(2) (2002), 219–231. [ONLINE: http://rgmia.vu.edu.au/v5n2.

html]

[2] P.CERONE ANDS.S.DRAGOMIR, Ostrowski type inequalities for func- tions whose derivatives satisfy certain convexity assumptions, Demonstra- tio Math., 37(2) (2004), 299–308.

[3] S.S.DRAGOMIRANDA.SOFO, Ostrowski type inequalities for functions whose derivatives are convex, Proceedings of the 4th International Con- ference on Modelling and Simulation, November 11-13, 2002. Victoria University, Melbourne, Australia. RGMIA Res. Rep. Coll., 5(Supp) (2002), Art. 30. [ONLINE:http://rgmia.vu.edu.au/v5(E).html]

[4] S.S.DRAGOMIR AND Th.M. RASSIAS (Eds.), Ostrowski Type Inequal- ities and Applications in Numerical Integration, Kluwer Academic Pub- lishers, Dordrecht, 2002.

[5] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and Their Integrals and Derivatives, Kluwer Academic Pub- lishers, Dordrecht, 1991.

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

(31)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page31of31

[7] B.G. PACHPATTE, A note on integral inequalities involving two log- convex functions, Math. Inequal. Appl., 7(4) (2004), 511–515.

[8] B.G. PACHPATTE, A note on Hadamard type integral inequalities involv- ing several log-convex functions, Tamkang J. Math., 36(1) (2005), 43–47.

[9] B.G. PACHPATTE, Mathematical Inequalities, North-Holland Mathemat- ical Library, Vol. 67 Elsevier, 2005.

[10] J.E. PE ˇCARI ´C, F. PROSCHAN ANDY.L. TANG, Convex functions, Par- tial Orderings and Statistcal Applications, Academic Press, New York, 1991.