Journal of Inequalities in Pure and
Applied Mathematics
NOTE ON DRAGOMIR-AGARWAL INEQUALITIES, THE GENERAL
EULER TWO-POINT FORMULAE AND CONVEX FUNCTIONS
volume 6, issue 1, article 19,
2005.
A. VUKELIĆ
Faculty of Food Technology and Biotechnology
Mathematics Department
University of Zagreb
Pierottijeva 6, 10000 Zagreb
Croatia
EMail: avukelic@pbf.hr
Received 12 January, 2005;
accepted 20 January, 2005.
Communicated by: R.P. Agarwal
Abstract
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2000
Victoria University
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Abstract
The general Euler two-point formulae are used with functions possessing various 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.
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The General Euler Two-point Formulae . . . . . . . . . . . . . . . . . .
References
A. Vukelić
3
5
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J. Ineq. Pure and Appl. Math. 6(1) Art. 19, 2005
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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
Z b
a+b
1
f (a) + f (b)
(1.1)
f
≤
f (t)dt ≤
.
2
b−a a
2
Recently, S.S. Dragomir and R.P. Agarwal [3] considered the trapezoid formula for numerical integration of functions f such that |f 0 |q is 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.
Suppose f : I 0 ⊆ R → R is differentiable on I 0 and such that |f 0 |q is convex
on [a, b] for some q ≥ 1, where a, b ∈ I 0 (a < b). Then
1
Z b
f (a) + f (b)
b − a |f 0 (a)|q + |f 0 (b)|q q
1
(1.2) −
f (t)dt ≤
.
2
b−a a
4
2
Some generalizations to higher-order convexity and applications of these results
are given in [1]. Related results for Euler midpoint, Euler-Simpson, Euler twopoint, dual Euler-Simpson, 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.
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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In this paper we consider some related results using the general Euler twopoint formulae. We will use the interval [0, 1] because of simplicity and since it
involves no loss in generality.
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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2.
The General Euler Two-point Formulae
In the recent paper [6] the following identities, named the general Euler twopoint formulae, have been proved. Let f ∈ C n ([0, 1], R) for some n ≥ 3 and
let x ∈ [0, 1/2]. If n = 2r − 1, r ≥ 2, then
Z 1
1
(2.1)
f (t)dt = [f (x) + f (1 − x)] − Tr−1 (f )
2
0
Z 1
1
x
+
f (2r−1) (t)F2r−1
(t)dt,
2(2r − 1)! 0
while for n = 2r, r ≥ 2 we have
Z 1
1
(2.2)
f (t)dt = [f (x) + f (1 − x)] − Tr−1 (f )
2
0
Z 1
1
x
+
f (2r) (t)F2r
(t)dt
2(2r)! 0
and
Z
1
f (t)dt =
(2.3)
0
1
[f (x) + f (1 − x)] − Tr (f )
2
Z 1
1
+
f (2r) (t)Gx2r (t)dt.
2(2r)! 0
Here we define T0 (f ) = 0 and for 1 ≤ m ≤ bn/2c
m
X
B2k (x) (2k−1)
Tm (f ) =
f
(1) − f (2k−1) (0) ,
(2k)!
k=1
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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Gxn (t) = Bn∗ (x − t) + Bn∗ (1 − x − t)
and
Fnx (t) = Bn∗ (x − t) + Bn∗ (1 − x − t) − Bn (x) − Bn (1 − x),
where Bk (·), k ≥ 0, is the k-th Bernoulli polynomial and Bk = Bk (0) =
Bk (1)(k ≥ 0) the k-th Bernoulli number. By Bk∗ (·)(k ≥ 0) we denote the
function of period one such that Bk∗ (x) = Bk (x) for 0 ≤ x ≤ 1.
x
n x
r−1 x
It was proved in [6] that
h Fn (1 − t) = (−1) Fn (t), (−1) F2r−1 (t) ≥ 0,
x
x
(t) ≥ 0,
(−1)r F2r
(t) ≥ 0 for x ∈ 0, 21 − 2√1 3 and t ∈ [0, 1/2], and (−1)r F2r−1
i
x
(−1)r−1 F2r
(t) ≥ 0 for x ∈ 2√1 3 , 12 and t ∈ [0, 1/2]. Also
Z
0
1
x
F2r−1 (t) dt = 2 B2r 1 − x − B2r (x) ,
r
2
Z 1
x
|F2r
(t)| dt = 2|B2r (x)|
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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0
and
Z
1
|Gx2r (t)| dt ≤ 4|B2r (x)|.
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0
With integration by parts, we have that the following identities hold:
Z 1
y x
F2r−1
(1) C1 (x) =
dy
2
0
Z 1
y
2
1
x
=−
F2r−1 1 −
dy =
B2r (x) − B2r
−x ,
2
r
2
0
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1
Z
x
yF2r−1
C2 (x) =
(2)
y 0
Z
1
2
dy
y
x
yF2r−1
1−
dy
2
0
2
1
= − B2r
−x ,
r
2
=−
1
Z
(1 −
C3 (x) =
(3)
0
Z
=−
0
1
x
y)F2r−1
y dy
2
y
x
(1 − y)F2r−1
1−
dy
2
2
= B2r (x),
r
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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1
Z
(4)
x
F2r
C4 (x) =
y 0
Z
(5)
1
C5 (x) =
Z0 1
x
yF2r
2
y 2
Z
dy =
0
1
x
F2r
y
1−
dy = −2B2r (x),
2
dy
y
=
1−
dy
2
0
8
1
=
B2r+2 (x) − B2r+2
− x − B2r (x),
(2r + 1)(2r + 2)
2
x
yF2r
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Z
(6)
1
x
(1 − y)F2r
C6 (x) =
y 2
0
Z
dy
1
y
x
(1 − y)F2r
1−
dy
2
0
8
1
=
B2r+2
− x − B2r+2 (x) − B2r (x),
(2r + 1)(2r + 2)
2
=
Z
(7)
C7 (x) =
1
Gx2r
y 2
0
Z
(8)
1
yGx2r
C8 (x) =
0
Z
y 2
Z
dy =
0
1
y
Gx2r 1 −
dy = 0,
2
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
dy
1
y
yGx2r 1 −
dy
2
0
Z 1
y x
=−
(1 − y)G2r
dy
2
0
Z 1
y
=
(1 − y)Gx2r 1 −
dy
2
0
8
1
=
B2r+2 (x) − B2r+2
−x ,
(2r + 1)(2r + 2)
2
=
Theorem
2.1.
Suppose
i f : [0, 1] → R is n-times differentiable and x ∈
h
1
1
1
1
0, 2 − 2√3 ∪ 2√3 , 2 .
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q
(a) If f (n) is convex for some q ≥ 1, then for n = 2r − 1, r ≥ 2, we have
(2.4)
Z
0
1
1
f (t)dt − [f (x) + f (1 − x)] + Tr−1 (f )
2
1− 1
q
1
2 ≤
B
−
x
−
B
(x)
2r
2r
(2r)! 2
"
f (2r−1) (0)q + f (2r−1) (1)q
r
× C3 (x) ·
2
2
q 1
r
(2r−1) 1 q
.
+ C2 (x) · f
2
2 If n = 2r, r ≥ 2, then
Z 1
1
(2.5) f (t)dt − [f (x) + f (1 − x)] + Tr−1 (f )
2
0
"
(2r) q (2r) q
1
f (0) + f (1)
1
|B2r (x)|1− q
≤
· C6 (x)
(2r)!
2
2
1
1
(2r) 1 q q
+ C5 (x) f
2
2 and we also have
Z 1
1
(2.6) f (t)dt − [f (x) + f (1 − x)] + Tr (f )
2
0
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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1 1
(2r) q
(2r) 1 q (2r) q q
2|B2r (x)|1− q 1
≤
+ f (1)
.
8 C8 (x) f (0) + 2 f
(2r)!
2 q
(b) If f (n) is concave, then for n = 2r − 1, r ≥ 2, we have
(2.7)
Z
0
1
1
f (t)dt − [f (x) + f (1 − x)] + Tr−1 (f )
2
1 r
(2r−1) |C2 (x)| ≤
C1 (x) · f
(2r)! 2
2 |C1 (x)| !#
1
C
(x)
+
C
(x)
3
2 2
+ f (2r−1)
.
|C1 (x)|
If n = 2r, r ≥ 2, then
(2.8)
1
1
f (t)dt − [f (x) + f (1 − x)] + Tr−1 (f )
2
0
|C4 (x)| (2r) |C5 (x)| ≤
f
4(2r)! 2|C4 (x)| !#
1
(2r) C6 (x) + 2 C5 (x) + f
.
|C4 (x)|
Z
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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Proof. First, let n = 2r − 1 for some r ≥ 2. Then by Hölder’s inequality
Z 1
1
f
(t)dt
−
[f
(x)
+
f
(1
−
x)]
+
T
(f
)
r−1
2
0
Z 1
x
1
F2r−1 (t) · f (2r−1) (t) dt
≤
2(2r − 1)! 0
1− 1q Z 1
Z 1
1q
x
x
(2r−1) q
1
F2r−1 (t) dt
F2r−1 (t) · f
≤
(t) dt
2(2r − 1)!
0
0
1− 1
q
2 1
1
B2r
− x − B2r (x)
=
2(2r − 1)! r
2
Z 1
1q
x
(2r−1) q
F2r−1 (t) · f
×
(t) dt .
0
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0
1
2
x
F2r−1 (t) · f (2r−1) (t)q dt +
=
0
1
=
2
Z
0
1
A. Vukelić
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Now, by the convexity of |f (2r−1) |q we have
Z 1
x
F2r−1 (t) · f (2r−1) (t)q dt
Z
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
1
Z
1
2
x
F2r−1 (t) · f (2r−1) (t)q dt
q
y 1 x
(2r−1)
(1 − y) · 0 + y ·
dy
F2r−1
· f
2
2 q
Z
1 1 x y (2r−1)
1 +
(1 − y) · 1 + y ·
dy
F2r−1 1 −
· f
2 0
2
2 II
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Z
y q
1 1
x
≤
(1 − y)F2r−1
dy · f (2r−1) (0)
2
2
0
Z 1
q
y
1 x
(2r−1)
+ dy · f
yF2r−1
2
2 Z0 1
y (2r−1) q
x
+ (1 − y)F2r−1
1−
dy · f
(1)
2
0
Z 1
q y
1 x
(2r−1)
+ yF2r−1 1 −
dy · f
.
2
2 0
q
On the other hand, if f (2r−1) is concave, then
A. Vukelić
1
1
f (t)dt − [f (x) + f (1 − x)] + Tr−1 (f )
2
0
Z 1
x
1
F2r−1 (t) · f (2r−1) (t) dt
≤
2(2r − 1)! 0
"Z
#
Z 1
1/2 1
x
x
F2r−1
F2r−1
=
(t) · f (2r−1) (t) dt +
(t) · f (2r−1) (t) dt
2(2r − 1)! 0
1/2
Z 1 y 1
1 x
(2r−1)
=
(1 − y) · 0 + y ·
dy
F2r−1
· f
2(2r − 1)! 0
2
2 Z 1
y (2r−1)
1 x
+
(1 − y) · 1 + y ·
dy
F2r−1 1 −
· f
2
2 Z
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Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
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Z 1
y 1
x
≤
F
dy 4(2r − 1)! 0 2r−1 2

 R 1
y
1
x
F
((1
−
y)
·
0
+
y
·
)dy
(2r−1) 0 2r−1 2
2


R
× f
1 x
y
0 F2r−1 2 dy Z 1
y x
+
F2r−1 1 −
dy 2
0
 
 R 1
y
1
x
F
1
−
((1
−
y)
·
1
+
y
·
)dy
(2r−1) 0 2r−1
2
2

  ,
R
× f
1 x
y
0 F2r−1 1 − 2 dy Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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] and λ ∈ [0, 1]
q
q
q
q
|g(λx + (1 − λ)y)| ≥ λ|g(x)| + (1 − λ)|g(y)| ≥ (λ|g(x)| + (1 − λ)|g(y)|) ,
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 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 when r = 2 are of special interest, so
we isolate it as corollary.
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hCorollary 2.2.
Supposei f : [0, 1] → R is 4-times differentiable and x ∈
1
1
0, 2 − 2√3 ∪ 2√1 3 , 21 .
q
(a) If f (3) is convex for some q ≥ 1, then
Z
1
1
1 0
0
f (t)dt − [f (x) + f (1 − x)] +
[f (1) − f (0)]
2
12
0
1− 1
1 q
1 3 3 2
≤
2x
−
x
+
12 2
16 "
(3) q (3) q
f (0) + f (1)
4
1
× x − 2x3 + x2 − 30
2
q 1
4 x2
7 (3) 1 q
+ −x +
−
f
2
240 2 q
and if f (4) is convex for some q ≥ 1, then
Z
0
1
1 0
1
0
[f (1) − f (0)]
f (t)dt − [f (x) + f (1 − x)] +
2
12
1− 1
1 q
1 4
3
2
≤
x
−
2x
+
x
−
24 30 "
(4) q (4) q
2
2x5
f (0) + f (1)
3x
1
4
3
× −x +x −
+ 5
8
96
2
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Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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q 1
2x5
5x2
11 (4) 1 q
3
+ −
f
+x −
+
.
5
8
480 2 (b) If f (3) is concave, then
Z
1
0
1 0
1
f (t)dt − [f (x) + f (1 − x)] +
[f (1) − f 0 (0)]
2
12


 x2
7 4
−x
+
−
2
240 1 3 3 2
1  (3) 

≤
2x − x + f
1
−4x3 + 3x2 − 24
2
16 8
 
 4
5x2
23 x
3
−
2x
+
−
2
4
480 
+ f (3)  −2x3 + 3x2 − 1 2
16
and if f (4) is concave, then
1
1
1 0
0
[f (1) − f (0)]
f (t)dt − [f (x) + f (1 − x)] +
2
12
0


 4x5
5x2
11 3
−
+
2x
−
+
5
4
240 1 4
1  (4) 
3
2

≤
x
−
2x
+
x
−
f
−4x4 + 8x3 − 4x2 + 2 48 30 15
 
 5
2x
11x2
7 4
3
−
2x
+
3x
−
+
5
8
160  .
+ f (4)  1 −2x4 + 4x3 − 2x2 + 15
Z
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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Now, we will give some results of the same type in the case when r = 1.
Theorem 2.3. Suppose f : [0, 1] → R is 2-times differentiable.
(a) If |f 0 |q is convex for some q ≥ 1, then for x ∈ [0, 1/2] we have
Z
0
1
1
f (t)dt − [f (x) + f (1 − x)]
2
0
1
2
|f (0)|q + |f 0 (1)|q
|8x2 − 4x + 1|1− q
2
≤
· 2x − 2x + 4
3
2
q 1
1 q
1 .
+ −2x2 + 2x + f 0
3
2 If |f 00 |q is convex for some q ≥ 1 and x ∈ [0, 1/4], then
Z 1
1
f (t)dt − [f (x) + f (1 − x)]
2
0
1− 1
q
−6x2 +6x−1 2
00
3/2 +
(1
−
4x)
2
|f (0)|q + |f 00 (1)|q
3
3
1
≤
−x + x − 8 4
2
q 1
5 1 q
+ −2x2 + 2x − f 00
,
24
2 Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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while for x ∈ [1/4, 1/2] we have
Z 1
1
f (t)dt − [f (x) + f (1 − x)]
2
0
1
−6x2 +6x−1 1− q 00
2
|f (0)|q + |f 00 (1)|q
3
1
≤
−x + x − 8 4
2
q 1
5
1 q
.
+ −2x2 + 2x − f 00
24
2 (b) If |f 0 | is concave for some q ≥ 1, then for x ∈ [0, 1/2] we have
Z 1
1
f
(t)dt
−
[f
(x)
+
f
(1
−
x)]
2
0
1 0 2
1 0 2
≤
f −x + x + + f x − x +
8 6
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
5 .
6 If |f 00 | is concave for some q ≥ 1 and x ∈ [0, 1/2], then
Z 1
1
f (t)dt − [f (x) + f (1 − x)]
2
0
!
"
5 2
−2x
+
2x
−
1
1 24
≤ −3x2 + 3x − f 00 2
2
8
3
−6x + 6x − 3
!#
11 2
−2x
+
2x
−
48
+ f 00 .
1
2
−3x + 3x − 3
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Proof. It was proved in [6] that for x ∈ [0, 1/2]
Z 1
8x2 − 4x + 1
x
|F1 (t)|dt =
,
2
0
for x ∈ [0, 1/4]
Z
1
|F2x (t)|dt =
0
−6x2 + 6x − 1 2
+ (1 − 4x)3/2 ,
3
3
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
and for x ∈ [1/4, 1/2]
Z
0
1
|F2x (t)|dt =
−6x2 + 6x − 1
.
3
A. Vukelić
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 3. For x = 0 in the above theorem we have the trapezoid formula and
for |f 00 |q a convex function and |f 00 | a concave function we get the results from
[4].
If |f 0 |q is convex for some q ≥ 1, then
Z
0
1
"
1 |f 0 (0)|q + |f 0 2 |q + |f 0 (1)|q
1
f (t)dt − [f (0) + f (1)] ≤
2
4
3
1
# 1q
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and if |f 0 | is concave, then
Z 1
1 0 1 0 5 1
f
+ f
.
f (t)dt − [f (0) + f (1)] ≤
2
8 6 6 0
For x = 1/4 we get two-point Maclaurin formula and then if |f 0 |q is convex
for some q ≥ 1, then
Z
1
0
1
1
3 f (t)dt −
f
+f
2
4
4 # 1q
"
1 7|f 0 (0)|q + 34|f 0 12 |q + 7|f 0 (1)|q
≤
24
8
and if |f 00 |q is convex for some q ≥ 1, then
Z
0
1
1
1
3 f (t)dt −
f
+f
2
4
4 "
#1
q
00
q
00 1
00
q q
3|f
(0)|
+
16|f
|
+
3|f
(1)|
1
2
≤
.
96
4
If |f 0 | is concave, then
Z 1
1 0 17 0 31 1
1
3
≤
f
+ f
f (t)dt −
f
+f
2
4
4 8 48 48 0
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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and if |f 00 | is concave for some q ≥ 1, then
Z 1
1
3 1
11 00 4 00 7 f
+f
≤
f
+ f
.
f (t)dt −
2
4
4 384 11 11 0
For x = 1/3 we get two-point Newton-Cotes formula and then if |f 0 |q is
convex for some q ≥ 1, then
#1
"
q
Z 1
0
q q
0
q
0 1
|f
(0)|
+
7|f
|
+
|f
(1)|
1
1
2
5
2
≤
f (t)dt −
f
+f
36
5
2
3
3
0
and if |f 00 |q is convex for some q ≥ 1, then
Z 1
1
1
2 f
(t)dt
−
f
+
f
2
3
3 0
"
# 1q
1 7|f 00 (0)|q + 34|f 00 12 |q + 7|f 00 (1)|q
≤
.
36
16
If |f 0 | is concave, then
Z 1
1 0 7 0 11 1
1
2
≤
f
+ f
f
+f
f (t)dt −
2
3
3 8 18 18 0
and if |f 00 | is concave for some q ≥ 1, then
Z 1
00 17 00 31 1
1
2
1
≤
f
+ f
.
f
+
f
f
(t)dt
−
2
3
3 24 48 48 0
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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For x = 1/2 we get midpoint formula and then if |f 0 |q is convex for some
q ≥ 1, then
Z
0
1
"
#1
q
0
q
0 1
0
q q
|f
(0)|
+
10|f
|
+
|f
(1)|
1 1
2
≤
f (t)dt − f
2 4
12
and if |f 00 |q is convex for some q ≥ 1, then
Z
0
1
# 1q
"
1 1 3|f 00 (0)|q + 14|f 00 12 |q + 3|f 00 (1)|q
.
f (t)dt − f
≤
8
2 24
If |f 0 | is concave, then
Z 1
1 1 0 5 0 7 f (t)dt − f
≤
f
+ f
2 8 12 12 0
00
and if |f | is concave for some q ≥ 1, then
Z 1
00 7 00 13 1
5
≤
f
+ f
.
f
(t)dt
−
f
2 96 20 20 0
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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References
[1] Lj. DEDIĆ, C.E.M. PEARCE AND J. PEČARIĆ, Hadamard and DragomirAgarwal inequalities, higher-order convexity and the Euler formula, J. Korean Math. Soc., 38 (2001), 1235–1243.
[2] Lj. DEDIĆ, C.E.M. PEARCE AND J. PEČARIĆ, 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]
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
Title Page
Contents
[5] C.E.M. PEARCE AND J. PEČARIĆ, Inequalities for differentiable mappings and applications to special means and quadrature formulas, Appl.
Math. Lett., 13 (2000), 51–55.
[6] J. PEČARIĆ, I. PERIĆ AND A. VUKELIĆ, On general Euler two-point
formulae, to appear in ANZIAM J.
[7] J. PEČARIĆ AND A. VUKELIĆ, 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.
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[8] J. PEČARIĆ AND A. VUKELIĆ, 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. PEČARIĆ AND A. VUKELIĆ, Hadamard and Dragomir-Agrawal inequalities, the general Euler two-point formulae and convex function, to
appear in Rad HAZU, Matematičke Znanosti.
Note on Dragomir-Agarwal
Inequalities, the General Euler
Two-Point Formulae and
Convex Functions
A. Vukelić
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