(1)http://jipam.vu.edu.au/ Volume 6, Issue 1, Article 19, 2005 NOTE ON DRAGOMIR-AGARWAL INEQUALITIES, THE GENERAL EULER TWO-POINT FORMULAE AND CONVEX FUNCTIONS A

(1)

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

Volume 6, Issue 1, Article 19, 2005

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

A. VUKELI ´C

FACULTY OFFOODTECHNOLOGY ANDBIOTECHNOLOGY

MATHEMATICSDEPARTMENT

UNIVERSITY OFZAGREB

PIEROTTIJEVA6, 10000 ZAGREB

CROATIA

avukelic@pbf.hr

Received 12 January, 2005; accepted 20 January, 2005 Communicated by R.P. Agarwal

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

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

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

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 andf : [a, b]→Ris 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 functionsf such that|f⁰|^qis a convex function for some q ≥ 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 → R is differentiable onI⁰ and such that|f⁰|^q is 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,

ISSN (electronic): 1443-5756

016-05

(2)

Euler-Simpson 3/8 and 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.

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.

2. THEGENERALEULERTWO-POINTFORMULAE

In the recent paper [6] the following identities, named the general Euler two-point formulae, have been proved. Letf ∈Cⁿ([0,1],R)for somen≥3and 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)]−T_r−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) , 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)−B_n(x)−B_n(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 that B_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) ≥ 0 for x ∈ h

0,¹₂ − ¹

2√ 3

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

2√ 3,¹₂i

andt∈[0,1/2]. Also Z 1

0

F_2r−1^x (t)

dt= 2 r

B2r

1 2−x

−B2r(x) , Z 1

0

|F_2r^x(t)|dt= 2|B_2r(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:

(1) C₁(x) = Z 1

0

F_2r−1^x y 2

dy=− Z 1

0

F_2r−1^x 1−y

2

dy= 2 r

B_2r(x)−B_2r 1

2−x

,

(3)

(2) C₂(x) = Z 1

0

yF_2r−1^x y 2

dy=− Z 1

0

yF_2r−1^x 1−y

2

dy=−2 rB_2r

1 2−x

,

(3) C₃(x) = Z 1

0

(1−y)F_2r−1^x y 2

dy=− Z 1

0

(1−y)F_2r−1^x 1− y

2

dy= 2

rB_2r(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)

B_2r+2(x)−B_2r+2 1

2 −x

−B_2r(x),

C₆(x) = Z 1

0

(1−y)F_2r^x y 2

dy (6)

= 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_2r y

2

dy= Z 1

0

G^x_2r

1− y 2

dy= 0,

C₈(x) = Z 1

0

yG^x_2ry 2

(8) dy

= 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)

B_2r+2(x)−B_2r+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

.

(4)

(a) Iff is 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)!

B2r

1 2 −x

−B2r(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

q#¹_q

and we also have (2.6)

Z 1 0

f(t)dt− 1

2[f(x) +f(1−x)] +T_r(f)

≤ 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)|

!

# . 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

.

(5)

Now, by the convexity of|f^(2r−1)|^q we 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

≤ 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)] +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 1/2 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

≤ 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 2.2. For (2.7) to be satisfied it is enough to suppose that|f^(2r−1)|is a concave function.

For if|g|^qis concave and[0,1]for someq≥1, then forx, 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,

(6)

by the power-mean inequality. Therefore|g|is also concave on[0,1].

Remark 2.3. If in Theorem 2.1 we chose x = 0,1/2,1/3, we get generalizations of the Dragomir-Agarwal inequality for Euler trapezoid (see [4]) , Euler midpoint and Euler two-point Newton-Cotes formulae respectively.

The resultant formulae in Theorem 2.1 whenr= 2are of special interest, so we isolate it as corollary.

Corollary 2.4. Suppose f : [0,1] → R is 4-times differentiable and x ∈ h

0,¹₂ − ¹

2√ 3

∪ 1

2√ 3,¹₂

i . (a) If

f⁽³⁾

q is 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 +

−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)]

(7)

≤ 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







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

Theorem 2.5. Supposef : [0,1]→Ris2-times differentiable.

(a) If|f⁰|^q is 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|^qis 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 ,

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

.

(8)

If|f |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

!

# . 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)and C₆(x)similar to that in Theorem 2.1 we get the inequalities in (a) and (b).

Remark 2.6. For x = 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

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|^qis 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

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

.

(9)

Forx = 1/3 we get two-point Newton-Cotes formula and then if |f⁰|^q is convex for some q≥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

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

. Forx= 1/2we get midpoint formula and then if|f⁰|^qis convex for someq ≥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

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

. REFERENCES

[1] Lj. DEDI ´C, C.E.M. PEARCE AND J. PE ˇCARI ´C, Hadamard and Dragomir-Agarwal inequalities, higher-order convexity and the Euler formula, J. Korean 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. DRAGOMIRANDR.P. AGARWAL, Two inequalities for differentiable mappings and applica- tions 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]

(10)

[5] C.E.M. PEARCE AND J. PE ˇCARI ´C, Inequalities for differentiable mappings 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 ´CANDA. VUKELI ´C, Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions, Functional Equations, Inequalities and Applications, Th.M. Rassias (ed.)., Dordrecht, Kluwer Academic Publishers, 2003.

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

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