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volume 6, issue 1, article 19, 2005.

Received 12 January, 2005;

accepted 20 January, 2005.

Communicated by:R.P. Agarwal

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Journal of Inequalities in Pure and Applied Mathematics

NOTE ON DRAGOMIR-AGARWAL INEQUALITIES, THE GENERAL EULER TWO-POINT FORMULAE AND CONVEX FUNCTIONS

A. VUKELI ´C

Faculty of Food Technology and Biotechnology Mathematics Department

University of Zagreb Pierottijeva 6, 10000 Zagreb Croatia

EMail:avukelic@pbf.hr

c

2000Victoria University ISSN (electronic): 1443-5756 016-05

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Note on Dragomir-Agarwal Inequalities, the General Euler

Two-Point Formulae and Convex Functions

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J. Ineq. Pure and Appl. Math. 6(1) Art. 19, 2005

http://jipam.vu.edu.au

Abstract

The general Euler two-point formulae are used with functions possessing var- ious convexity and concavity properties to derive inequalities pertinent to numerical integration.

2000 Mathematics Subject Classification:26D15, 26D20, 26D99

Key words: Hadamard inequality, r-convexity, Integral inequalities, General Euler two-point formulae.

1. Introduction

One of the cornerstones of nonlinear analysis is the Hadamard inequality, which states that if [a, b] (a < b) is a real interval and f : [a, b] → R is a convex function, then

(1.1) f

a+b 2

≤ 1 b−a

Z b a

f(t)dt≤ f(a) +f(b)

2 .

Recently, S.S. Dragomir and R.P. Agarwal [3] considered the trapezoid formula for numerical integration of functionsfsuch that|f⁰|^qis a convex function for someq≥1. Their approach was based on estimating the difference between the two sides of the right-hand inequality in (1.1). Improvements of their results were obtained in [5]. In particular, the following tool was established.

Supposef :I⁰ ⊆R→Ris differentiable onI⁰and such that|f⁰|^qis convex on[a, b]for someq ≥1,wherea, b∈I⁰(a < b). Then

(1.2)

f(a) +f(b)

2 − 1

b−a Z b

a

f(t)dt

≤ b−a 4

|f⁰(a)|^q+|f⁰(b)|^q 2

¹_q . Some generalizations to higher-order convexity and applications of these results are given in [1]. Related results for Euler midpoint, Euler-Simpson, Euler two- point, dual Euler-Simpson, Euler-Simpson 3/8and Euler-Maclaurin formulae were considered in [7] and for Euler two-point formulae in [9] (see also [2] and [8]).

In the paper [4] Dah-Yan Hwang procured some new inequalities of this type and he applied the result to obtain a better estimate of the error in the trapezoidal formula.

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In this paper we consider some related results using the general Euler two- point formulae. We will use the interval[0,1]because of simplicity and since it involves no loss in generality.

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2. The General Euler Two-point Formulae

In the recent paper [6] the following identities, named the general Euler two- point formulae, have been proved. Let f ∈ Cⁿ([0,1],R)for some n ≥ 3 and letx∈[0,1/2]. Ifn= 2r−1, r ≥2, then

(2.1)

Z 1 0

f(t)dt = 1

2[f(x) +f(1−x)]−Tr−1(f)

+ 1

2(2r−1)!

Z 1 0

f^(2r−1)(t)F_2r−1^x (t)dt, while forn = 2r, r≥2we have

(2.2)

Z 1 0

f(t)dt = 1

2[f(x) +f(1−x)]−Tr−1(f)

+ 1

2(2r)!

Z 1 0

f^(2r)(t)F_2r^x(t)dt and

(2.3)

Z 1 0

f(t)dt = 1

2[f(x) +f(1−x)]−Tr(f)

+ 1

2(2r)!

Z 1 0

f^(2r)(t)G^x_2r(t)dt.

Here we defineT₀(f) = 0and for1≤m ≤ bn/2c T_m(f) =

m

X

k=1

B_2k(x) (2k)!

f^(2k−1)(1)−f^(2k−1)(0) ,

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G^x_n(t) =B_n^∗(x−t) +B_n^∗(1−x−t) and

F_n^x(t) = B_n^∗(x−t) +B_n^∗(1−x−t)−Bn(x)−Bn(1−x),

where B_k(·), k ≥ 0, is the k-th Bernoulli polynomial and B_k = B_k(0) = B_k(1)(k ≥ 0) the k-th Bernoulli number. By B_k^∗(·)(k ≥ 0) we denote the function of period one such thatB_k^∗(x) =B_k(x)for0≤x≤1.

It was proved in [6] thatF_n^x(1−t) = (−1)ⁿF_n^x(t), (−1)^r−1F_2r−1^x (t) ≥ 0, (−1)^rF_2r^x(t)≥0forx∈h

0,¹₂ −₂^√¹₃

andt∈[0,1/2], and(−1)^rF_2r−1^x (t)≥0, (−1)^r−1F_2r^x(t)≥0forx∈

1 2√

3,¹₂i

andt∈[0,1/2]. Also Z 1

0

F_2r−1^x (t)

dt= 2 r

B_2r 1

2 −x

−B_2r(x) ,

Z 1 0

|F_2r^x(t)|dt= 2|B2r(x)|

and

Z 1 0

|G^x_2r(t)|dt ≤4|B_2r(x)|.

With integration by parts, we have that the following identities hold:

C₁(x) = Z 1

0

F_2r−1^x y 2

dy (1)

=− Z 1

0

F_2r−1^x

1− y 2

dy = 2 r

B2r(x)−B2r

1 2−x

,

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C₂(x) = Z 1

0

yF_2r−1^x y 2

(2) dy

=− Z 1

0

yF_2r−1^x 1−y

2

dy

=−2 rB_2r

1 2 −x

,

C₃(x) = Z 1

0

(1−y)F_2r−1^x y 2

dy (3)

=− Z 1

0

(1−y)F_2r−1^x 1− y

2

dy

= 2

rB2r(x),

(4) C₄(x) = Z 1

0

F_2r^x y 2

dy=

Z 1 0

F_2r^x 1− y

2

dy=−2B_2r(x),

C₅(x) = Z 1

0

yF_2r^x y 2

dy (5)

= Z 1

0

yF_2r^x 1− y

2

dy

= 8

(2r+ 1)(2r+ 2)

B2r+2(x)−B2r+2

1 2−x

−B2r(x),

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C₆(x) = Z 1

0

(1−y)F_2r^x y 2

(6) dy

= Z 1

0

(1−y)F_2r^x 1−y

2

dy

= 8

(2r+ 1)(2r+ 2)

B_2r+2 1

2 −x

−B_2r+2(x)

−B_2r(x),

(7) C₇(x) = Z 1

0

G^x_2ry 2

dy=

Z 1 0

G^x_2r 1− y

2

dy= 0,

C₈(x) = Z 1

0

yG^x_2ry 2

dy (8)

= Z 1

0

yG^x_2r 1− y

2

dy

=− Z 1

0

(1−y)G^x_2ry 2

dy

= Z 1

0

(1−y)G^x_2r 1− y

2

dy

= 8

(2r+ 1)(2r+ 2)

B2r+2(x)−B2r+2

1 2−x

,

Theorem 2.1. Suppose f : [0,1] → R is n-times differentiable and x ∈ h

0,¹₂ − ¹

2√ 3

∪

1 2√

3,¹₂i .

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(a) If f⁽ⁿ⁾

qis convex for someq≥1,then forn= 2r−1, r ≥2, we have

(2.4)

Z 1 0

f(t)dt− 1

2[f(x) +f(1−x)] +Tr−1(f)

≤ 2 (2r)!

B_2r 1

2 −x

−B_2r(x)

1−¹

q

×

"

r 2C₃(x)

·

f^(2r−1)(0)

q+

f^(2r−1)(1)

q

2 +

r

2C₂(x) ·

f^(2r−1) 1

2

q¹_q .

Ifn = 2r, r≥2, then (2.5)

Z 1 0

f(t)dt− 1

2[f(x) +f(1−x)] +Tr−1(f)

≤ |B_2r(x)|¹⁻¹^q (2r)! ·

"

1 2C₆(x)

f^(2r)(0)

q+

f^(2r)(1)

q

2 +

1 2C₅(x)

f^(2r) 1

2

q¹_q

and we also have (2.6)

Z 1 0

f(t)dt− 1

2[f(x) +f(1−x)] +T_r(f)

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≤ 2|B_2r(x)|¹⁻¹^q (2r)!

1 8C₈(x)

f^(2r)(0)

q+ 2

f^(2r) 1

2

q

+

f^(2r)(1)

q¹_q .

(b) If f⁽ⁿ⁾

qis concave, then forn = 2r−1, r ≥2, we have

(2.7)

Z 1 0

f(t)dt− 1

2[f(x) +f(1−x)] +Tr−1(f)

≤ 1 (2r)!

r 2C₁(x)

·

f^(2r−1)

|C₂(x)|

2|C1(x)|

+

f^(2r−1)

C₃(x) + ¹₂C₂(x)

|C₁(x)|

!

# .

Ifn = 2r, r≥2, then (2.8)

Z 1 0

f(t)dt− 1

2[f(x) +f(1−x)] +Tr−1(f)

≤ |C₄(x)|

4(2r)!

f^(2r)

|C₅(x)|

2|C₄(x)|

+

f^(2r)

C₆(x) + ¹₂C₅(x)

|C₄(x)|

!

# .

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Proof. First, letn = 2r−1for somer≥2. Then by Hölder’s inequality

Z 1 0

f(t)dt− 1

2[f(x) +f(1−x)] +Tr−1(f)

≤ 1

2(2r−1)!

Z 1 0

F_2r−1^x (t) ·

f^(2r−1)(t) dt

≤ 1

2(2r−1)!

Z 1 0

F_2r−1^x (t) dt

¹⁻¹_q Z 1 0

F_2r−1^x (t) ·

f^(2r−1)(t)

qdt ¹_q

= 1

2(2r−1)!

2 r

B_2r 1

2−x

−B_2r(x)

1−¹

q

× Z 1

0

F_2r−1^x (t) ·

f^(2r−1)(t)

qdt ¹_q

.

Now, by the convexity of|f^(2r−1)|^qwe have Z 1

0

F_2r−1^x (t) ·

f^(2r−1)(t)

qdt

= Z ¹₂

0

F_2r−1^x (t) ·

f^(2r−1)(t)

qdt+ Z 1

1 2

F_2r−1^x (t) ·

f^(2r−1)(t)

qdt

= 1 2

Z 1 0

F_2r−1^x y 2

·

f^(2r−1)

(1−y)·0 +y· 1 2

q

dy +1

2 Z 1

0

F_2r−1^x 1− y

2

·

f^(2r−1)

(1−y)·1 +y· 1 2

q

dy

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≤ 1 2

Z 1 0

(1−y)F_2r−1^x y 2

dy

·

f^(2r−1)(0)

q

+

Z 1 0

yF_2r−1^x y 2

dy

·

f^(2r−1) 1

2

q

+

Z 1 0

(1−y)F_2r−1^x 1− y

2

dy

·

f^(2r−1)(1)

q

+

Z 1 0

yF_2r−1^x

1− y 2

dy

·

f^(2r−1) 1

2

q .

On the other hand, if

f^(2r−1)

qis concave, then

Z 1 0

f(t)dt− 1

2[f(x) +f(1−x)] +T_r−1(f)

≤ 1

2(2r−1)!

Z 1 0

F_2r−1^x (t) ·

f^(2r−1)(t) dt

= 1

2(2r−1)!

"

Z 1/2 0

F_2r−1^x (t) ·

f^(2r−1)(t) dt+

Z 1 1/2

F_2r−1^x (t) ·

f^(2r−1)(t) dt

#

= 1

2(2r−1)!

Z 1 0

F_2r−1^x y 2

·

f^(2r−1)

(1−y)·0 +y· 1 2

dy +

Z 1 0

F_2r−1^x 1− y

2

·

f^(2r−1)

(1−y)·1 +y· 1 2

dy

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≤ 1

4(2r−1)!

Z 1 0

F_2r−1^x y 2

dy

×

f^(2r−1)





R1

0 F_2r−1^x ^y₂

((1−y)·0 +y· ¹₂)dy

R1

0 F_2r−1^x ^y₂ dy



 +

Z 1 0

F_2r−1^x 1− y

2

dy

×

f^(2r−1)





R1

0 F_2r−1^x 1− ^y₂

((1−y)·1 +y· ¹₂)dy

R1

0 F_2r−1^x 1− ^y₂ dy







,

so the inequality (2.4) and (2.7) are completely proved.

The proofs of the inequalities (2.5), (2.8) and (2.6) are similar.

Remark 1. For (2.7) to be satisfied it is enough to suppose that |f^(2r−1)| is a concave function. For if |g|^q is concave and [0,1] for some q ≥ 1, then for x, y ∈[0,1]andλ ∈[0,1]

|g(λx+ (1−λ)y)|^q ≥λ|g(x)|^q+ (1−λ)|g(y)|^q≥(λ|g(x)|+ (1−λ)|g(y)|)^q, by the power-mean inequality. Therefore|g|is also concave on[0,1].

Remark 2. If in Theorem 2.1 we chose x = 0,1/2,1/3, we get generaliza- tions of the Dragomir-Agarwal inequality for Euler trapezoid (see [4]) , Euler midpoint and Euler two-point Newton-Cotes formulae respectively.

The resultant formulae in Theorem2.1whenr= 2are of special interest, so we isolate it as corollary.

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Corollary 2.2. Suppose f : [0,1] → R is 4-times differentiable and x ∈ h

0,¹₂ − ¹

2√ 3

∪

1 2√

3,¹₂i . (a) If

f⁽³⁾

qis convex for someq ≥1,then

Z 1 0

f(t)dt−1

2[f(x) +f(1−x)] + 1

12[f⁰(1)−f⁰(0)]

≤ 1 12

2x³−3

2x²+ 1 16

1−¹_q

×

"

x⁴−2x³+x²− 1 30

f⁽³⁾(0)

q+

f⁽³⁾(1)

q

2 +

−x⁴+x² 2 − 7

240

f⁽³⁾ 1

2

q¹_q

and if f⁽⁴⁾

qis convex for someq≥1,then

Z 1 0

f(t)dt−1

2[f(x) +f(1−x)] + 1

12[f⁰(1)−f⁰(0)]

≤ 1 24

x⁴−2x³+x²− 1 30

1−¹_q

×

"

2x⁵

5 −x⁴ +x³− 3x² 8 + 1

96

f⁽⁴⁾(0)

q+

f⁽⁴⁾(1)

q

2

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+

−2x⁵

5 +x³− 5x² 8 + 11

480

f⁽⁴⁾ 1

2

q¹_q .

(b) If f⁽³⁾

is concave, then

Z 1 0

f(t)dt−1

2[f(x) +f(1−x)] + 1

12[f⁰(1)−f⁰(0)]

≤ 1 24

2x³− 3

2x²+ 1 16





f⁽³⁾





−x⁴+ ^x₂² − ₂₄₀⁷ −4x³+ 3x²− ¹₈





+

f⁽³⁾





x⁴

2 −2x³ +^5x₄² − ₄₈₀²³ −2x³+ ^3x₂² − ₁₆¹









and if f⁽⁴⁾

is concave, then

Z 1 0

f(t)dt−1

2[f(x) +f(1−x)] + 1

12[f⁰(1)−f⁰(0)]

≤ 1 48

x⁴−2x³+x²− 1 30





f⁽⁴⁾





−^4x₅⁵ + 2x³− ^5x₄² +₂₄₀¹¹ −4x⁴+ 8x³ −4x²+ ₁₅²





+

f⁽⁴⁾





2x⁵

5 −2x⁴+ 3x³− ^11x₈² +₁₆₀⁷ −2x⁴+ 4x³−2x² +₁₅¹







.

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Now, we will give some results of the same type in the case whenr= 1.

Theorem 2.3. Supposef : [0,1]→Ris2-times differentiable.

(a) If|f⁰|^qis convex for someq≥1, then forx∈[0,1/2]we have

Z 1 0

f(t)dt−1

2[f(x) +f(1−x)]

≤ |8x²−4x+ 1|¹⁻¹^q

4 ·

2x² −2x+ 2 3

|f⁰(0)|^q+|f⁰(1)|^q 2

+

−2x²+ 2x+1 3

f⁰ 1

2

q¹_q .

If|f⁰⁰|^q is convex for someq≥1andx∈[0,1/4], then

Z 1 0

f(t)dt−1

2[f(x) +f(1−x)]

≤

−6x²+6x−1

3 +²₃(1−4x)^3/2

1−¹_q

4

−x²+x− 1 8

|f⁰⁰(0)|^q+|f⁰⁰(1)|^q 2

+

−2x²+ 2x− 5 24

f⁰⁰ 1

2

q¹_q ,

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while forx∈[1/4,1/2]we have

Z 1 0

f(t)dt−1

2[f(x) +f(1−x)]

≤

−6x²+6x−1 3

1−¹_q

4

−x²+x−1 8

|f⁰⁰(0)|^q+|f⁰⁰(1)|^q 2

+

−2x²+ 2x− 5 24

f⁰⁰ 1

2

q¹_q .

(b) If|f⁰|is concave for someq≥1, then forx∈[0,1/2]we have

Z 1 0

f(t)dt−1

2[f(x) +f(1−x)]

≤ 1 8

f⁰

−x²+x+ 1 6

+

f⁰

x²−x+5 6

. If|f⁰⁰|is concave for someq ≥1andx∈[0,1/2], then

Z 1 0

f(t)dt−1

2[f(x) +f(1−x)]

≤ 1 8

−3x²+ 3x− 1 3

"

f⁰⁰

−2x²+ 2x− ₂₄⁵ −6x²+ 6x− ²₃

! +

f⁰⁰

−2x² + 2x− ¹¹₄₈ −3x²+ 3x− ¹₃

!

# .

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Proof. It was proved in [6] that forx∈[0,1/2]

Z 1 0

|F₁^x(t)|dt = 8x²−4x+ 1

2 ,

forx∈[0,1/4]

Z 1 0

|F₂^x(t)|dt = −6x²+ 6x−1

3 +2

3(1−4x)^3/2, and forx∈[1/4,1/2]

Z 1 0

|F₂^x(t)|dt= −6x²+ 6x−1

3 .

So, using identities (2.1) and (2.2) with calculation of C₁(x), C₂(x), C₃(x), C4(x), C5(x)andC6(x)similar to that in Theorem2.1we get the inequalities in (a) and (b).

Remark 3. Forx= 0in the above theorem we have the trapezoid formula and for|f⁰⁰|^q a convex function and|f⁰⁰|a concave function we get the results from [4].

If|f⁰|^qis convex for someq ≥1, then

Z 1 0

f(t)dt−1

2[f(0) +f(1)]

≤ 1 4

"

|f⁰(0)|^q+|f⁰ ¹₂

|^q+|f⁰(1)|^q 3

#¹_q

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and if|f⁰|is concave, then

Z 1 0

f(t)dt−1

2[f(0) +f(1)]

≤ 1 8

f⁰ 1

6

+

f⁰ 5

6

.

Forx= 1/4we get two-point Maclaurin formula and then if|f⁰|^qis convex for someq ≥1, then

Z 1 0

f(t)dt− 1 2

f

1 4

+f

3 4

≤ 1 8

"

7|f⁰(0)|^q+ 34|f⁰ ¹₂

|^q+ 7|f⁰(1)|^q 24

#¹_q

and if|f⁰⁰|^q is convex for someq ≥1, then

Z 1 0

f(t)dt− 1 2

f

1 4

+f

3 4

≤ 1 96

"

3|f⁰⁰(0)|^q+ 16|f⁰⁰ ¹₂

|^q+ 3|f⁰⁰(1)|^q 4

#¹_q .

If|f⁰|is concave, then

Z 1 0

f(t)dt−1 2

f

1 4

+f

3 4

≤ 1 8

f⁰ 17

48

+

f⁰ 31

48

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and if|f⁰⁰|is concave for someq ≥1, then

Z 1 0

f(t)dt− 1 2

f

1 4

+f

3 4

≤ 11 384

f⁰⁰ 4

11

+

f⁰⁰ 7

11

. For x = 1/3 we get two-point Newton-Cotes formula and then if |f⁰|^q is convex for someq ≥1, then

Z 1 0

f(t)dt− 1 2

f

1 3

+f

2 3

≤ 5 36

"

|f⁰(0)|^q+ 7|f⁰ ¹₂

|^q+|f⁰(1)|^q 5

#¹_q

Z 1 0

f(t)dt− 1 2

f

1 3

+f

2 3

≤ 1 36

"

7|f⁰⁰(0)|^q+ 34|f⁰⁰ ¹₂

|^q+ 7|f⁰⁰(1)|^q 16

#¹_q .

Z 1 0

f(t)dt−1 2

f

1 3

+f

2 3

≤ 1 8

f⁰ 7

18

+

f⁰ 11

18

Z 1 0

f(t)dt−1 2

f

1 3

+f

2 3

≤ 1 24

f⁰⁰ 17

48

+

f⁰⁰ 31

48

.

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For x = 1/2 we get midpoint formula and then if|f⁰|^q is convex for some q ≥1, then

Z 1 0

f(t)dt−f 1

2

≤ 1 4

"

|f⁰(0)|^q+ 10|f⁰ ¹₂

|^q+|f⁰(1)|^q 12

#¹_q

Z 1 0

f(t)dt−f 1

2

≤ 1 24

"

3|f⁰⁰(0)|^q+ 14|f⁰⁰ ¹₂

|^q+ 3|f⁰⁰(1)|^q 8

#¹_q .

Z 1 0

f(t)dt−f 1

2

≤ 1 8

f⁰ 5

12

+

f⁰ 7

12

Z 1 0

f(t)dt−f 1

2

≤ 5 96

f⁰⁰ 7

20

+

f⁰⁰ 13

20

.

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References

[1] Lj. DEDI ´C, C.E.M. PEARCEANDJ. PE ˇCARI ´C, Hadamard and Dragomir- Agarwal inequalities, higher-order convexity and the Euler formula, J. Ko- rean Math. Soc., 38 (2001), 1235–1243.

[2] Lj. DEDI ´C, C.E.M. PEARCEAND J. PE ˇCARI ´C, The Euler formulae and convex functions, Math. Inequal. Appl., 3(2) (2000), 211–221.

[3] S.S. DRAGOMIR AND R.P. AGARWAL, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Mat. Lett., 11(5) (1998), 91–95.

[4] D.Y. HWANG, Improvements of Euler-trapezodial type inequalities with higher-order convexity and applications, J. of Inequal. in Pure and Appl.

Math., 5(3) (2004), Art. 66. [ONLINE:http://jipam.vu.edu.au/

article.php?sid=412]

[5] C.E.M. PEARCE AND J. PE ˇCARI ´C, Inequalities for differentiable map- pings and applications to special means and quadrature formulas, Appl.

Math. Lett., 13 (2000), 51–55.

[6] J. PE ˇCARI Ć, I. PERI Ć AND A. VUKELI Ć, On general Euler two-point formulae, to appear in ANZIAM J.

[7] J. PE ˇCARI ´C AND A. VUKELI ´C, Hadamard and Dragomir-Agarwal in- equalities, the Euler formulae and convex functions, Functional Equations, Inequalities and Applications, Th.M. Rassias (ed.)., Dordrecht, Kluwer Academic Publishers, 2003.

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[8] J. PE ˇCARI ´CANDA. VUKELI ´C, On generalizations of Dragomir-Agarwal inequality via some Euler-type identities, Bulletin de la Société des Mathé- maticiens de R. Macédoine, 26 (LII) (2002), 17–42.

[9] J. PE ˇCARI ´C AND A. VUKELI ´C, Hadamard and Dragomir-Agrawal inequalities, the general Euler two-point formulae and convex function, to appear in Rad HAZU, Matematiˇcke Znanosti.