Journal of Inequalities in Pure and
Applied Mathematics
IMPROVEMENTS OF EULER-TRAPEZOIDAL TYPE INEQUALITIES
WITH HIGHER-ORDER CONVEXITY AND APPLICATIONS
volume 5, issue 3, article 66,
2004.
DAH-YAN HWANG
Department of General Education
Kuang Wu Institute of Technology
Peito, Taipei, Taiwan 11271, R.O.C.
EMail: dyhuang@mail.apol.com.tw
Received 28 May, 2003;
accepted 12 April, 2004.
Communicated by: C.E.M. Pearce
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
072-03
Quit
Abstract
We improve a bounds relating to Euler’s formula for the case of a function with
higher-order convexity properties. These are used to improve estimates of the
error involved in the use of the trapezoidal formula for integrating such a function.
2000 Mathematics Subject Classification: 26D15, 26D20.
Key words: Hadamard inequality, Euler formula, Convex functions, Integral inequalities, Numerical integration.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
The Euler-Trapezoidal Formula and Some Identities . . . . . . . 5
3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
1.
Introduction
Let f : I ⊆ R → R be a convex mapping defined on the interval I of real
numbers and a, b ∈ I with a < b. The following inequality:
Z b
a+b
1
f (a) + f (b)
(1.1)
f
≤
f (x)dx ≤
2
b−a a
2
is known in the literature as Hadamard’s inequality for convex mappings. Note
that some of the classical inequalities for means can be derived from (1.1) for
appropriate particular selections of the mappings f . Over the last decade this
pair of inequalities (1.1) have been improved and extended in a number of ways,
including the derivation of estimates of the differences between the two sides of
each inequality.
In [4], Dragomir and Agarwal have made use of the latter to derive bounds
for the error term in the trapezoidal formula for the numerical integration of an
integrable function f such that |f 0 |q is convex for some q ≥ 1. Some improvements of their result have been derived in [6] and [7].
Recently, Lj Dedic et al. [3, Theorem 2 for r = 1] establish the following
basic result which was obtained for the difference between the two side of the
right-hand Hadamard inequality.
Suppose f : [a, b] → R is a real-valued twice differentiable function. If |f 00 |q
is convex for some q ≥ 1, then
Z b
1
(b − a)3 |f 00 (a)|q + |f 00 (b)|q q
b
−
a
(1.2) f (t)dt −
[f (a) + f (b)] ≤
.
2
12
2
a
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
If |f 00 |q is concave, then
Z b
(b − a)3
b
−
a
≤
(1.3)
f
(t)dt
−
[f
(a)
+
f
(b)]
2
12
a
00 a + b f
.
2
In this paper, using Euler-trapezoidal formula, we shall generalize and improve the inequalities (1.2) and (1.3). Also, we apply the result to obtain a better
estimates of the error in the trapezoidal formula.
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
2.
The Euler-Trapezoidal Formula and Some
Identities
In what follows, let Bn (t), n ≥ 0 be the Bernoulli polynomials and Bn =
Bn (0), n ≥ 0, the Bernoulli numbers. The first few Bernoulli polynomials are
1
B1 (t) = t − ,
2
1
B4 (t) = t4 − 2t3 + t2 −
30
B0 (t) = 1,
1
B2 (t) = t2 − t + ,
6
3
1
B3 (t) = t3 − t2 + t,
2
2
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
and the first few Bernoulli numbers are
B0 = 1,
1
B1 = − ,
2
1
B2 = ,
6
B3 = 0,
1
B4 = − ,
30
Dah-Yan Hwang
B5 = 0.
For some details on the Bernoulli polynomials and the Bernoulli numbers, see
for example [1, 5].
The relevant key properties of the Bernoulli polynomials are
Bn0 (t) = nBn−1 (t), (n ≥ 1)
Bn (1 + t) − Bn (t) = ntn−1 , (n ≥ 0)
(See for example, [1, Chapter 23]). Let P2n (t) = [Bn (t) − Bn ]/n!. We note that
P2n (t), P2n ( 2t ) and P2n (1 − 2t ) do not change sign on (0.1).
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
By [2, p. 274], we have the following Euler-trapezoidal formula:
Z
b
b−a
[f (a) + f (b)]
2
r−1
X
(b − a)2k B2k (2k−1)
−
f
(b) − f (2k−1) (a)
(2k)!
k=1
Z 1
2r+1
+ (b − a)
P2r (t)f (2r) (a + t(b − a))dt.
f (x)dx =
(2.1)
a
0
and by [2, p. 275], we have
Z
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
a+nh
f (x)dx = T (f ; h)
(2.2)
a
−
r−1
X
B2k h2k
k=1
(2k)!
Title Page
[f
(2k−1)
(a + nh) − f
2r+1
+h
n−1 Z
X
k=0
where
(2k−1)
(a)]
Contents
1
P2r (t)f
(2r)
(a + h(t + k))dt,
0
JJ
J
II
I
Go Back
"
#
n−1
X
1
1
f (a + nh) + f (a + nh)
T (f ; h) = h f (a) +
2
2
k=1
R a+h
is estimates of integral a f (x)dx.
Close
Quit
Page 6 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
In this article, we adopt the terminology that f is (j + 2)-convex if f (j)
is convex, so ordinary convexity is two-convexity. A corresponding definition
applies for (j + 2)-concavity.
For our results, we need the following identities. By B2j+1 21 = 0 for
j ≥ 0; Bj = Bj (1) for j ≥ 0 and integrating by parts, we have that the
following identities hold:
Z 1
t
2B2r+1
B2r
(I)
aI (r) =
P2r
dt = −
−
,
2
(2r + 1)! (2r)!
0
Z 1
4 B2r+2 21 − B2r+2
t
B2r
(II)
aII (r) =
tP2r
−
,
dt = −
(2r + 2)!
2(2r)!
2
0
Z 1
t
(III) aIII (r) =
dt
(1 − t)P2r
2
0
4 B2r+2 21 − B2r+2
2B2r+1
B2r
=
−
−
,
(2r + 2)!
(2r + 1)! 2(2r)!
Z 1
2B2r+1
B2r
t
(IV) aIV (r) =
P2r 1 −
dt =
−
,
2
(2r + 1)! (2r)!
0
Z 1
4 B2r+2 12 − B2r+2
t
B2r
(V)
aV (r) =
tP2r 1 −
dt = −
−
,
2
(2r + 2)!
2(2r)!
0
Z 1
t
(VI) aVI (r) =
(1 − t)P2r 1 −
dt
2
0
4 B2r+2 21 − B2r+2
2B2r+1
B2r
+
−
.
=−
(2r + 2)!
(2r + 1)! 2(2r)!
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
3.
Results
In the remainder of the paper we shall use the notation
Z b
(b − a)
r
Ir = (−1)
f (x)dx −
[f (a) + f (b)]
2
a
)
r−1
X
B2k (b − a)2k (2k−1)
[f
(b) − f (2k−1) (a)] ,
+
(2k)!
k=1
As above, the empty sum for r = 1 is interpreted as zero.
Theorem 3.1. Suppose f : [a, b] → R is a real-valued (2r)-times differentiable
function.
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
(a) If |f (2r) |q is convex for q ≥ 1, then
1− 1q (
1q |B2r |
1
2r+1
|aIII (r)| · |f (2r) (a)|q
(3.1) |Ir | ≤ (b − a)
(2r)!
2
Title Page
Contents
1
q
)
(2r) a + b q
(2r)
q
+ |aVI (r)| · |f (b)|
+ (|aII (r)| + |aV (r)|) · f
.
2
(b) If |f (2r) |q is concave for q ≥ 1, then
(b − a)2r+1
(3.2) |Ir | ≤
( 2 × |aI (r)| · f (2r)
JJ
J
II
I
Go Back
Close
Quit
!
|aIII (r)| · a + |aII (r)| · ( a+b
)
2
|aI (r)|
Page 8 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
+|aIV (r)| · f (2r)
)
|aVI (r)| · b + |aV (r)| · ( a+b
2
|aIV (r)|
!)
.
Proof. From (2.1) and the definition of Ir , we have
Z 1
2r+1
|Ir | ≤ (b − a)
|P2r (t)f (2r) (a + t(b − a))|dt
0
Z b
x
−
a
2r
P2r
· |f (2r) (x)|dx.
= (b − a)
b
−
a
a
It follows from Hölder’s inequality that
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
Z b 1− 1q
x
−
a
P2r
dx
(3.3) |Ir | ≤
b−a a
Z b 1q
x − a (2r)
q
×
P2r b − a · |f (x)| dx .
a
Now,
(3.4)
Z
Z b
x
−
a
P2r
dx = (b − a) b−a a
0
=
1
P2r (t)dt
(b − a)|B2r |
,
(2r)!
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
and, by the convexity of |f (2r) |q , we have
Z b
x − a (2r)
q
(3.5)
P2r b − a · |f (x)| dx
a
Z a+b 2 x − a (2r)
q
=
P2r b − a · |f (x)|
a
Z b x
−
a
P2r
· |f (2r) (x)|q dx
+
a+b
b−a Z 21 q
(2r)
b−a
t
a
+
b
P2r
· f
dt
=
(1
−
t)a
+
t
2
2 2
0
q Z 1
P2r 1 − t · f (2r) (1 − t)b + t a + b
dt
+
2
2
0
Z
b − a 1
t
≤
(1 − t)P2r
dt · |f (2r) (a)|q
2
2
Z 01
(2r) a + b q
t
+ tP2r
dt · f
2
2
0
Z 1
t
+ (1 − t)P2r 1 −
dt · |f (2r) (b)|q
2
Z 0 1
(2r) a+b q
t
+ P2r 1 −
dt · f
2
2 0
Thus, by (3.3), (3.4) and (3.5) and the identities (II), (III), (V) and (VI), we have
the inequality (3.1).
On the other hand, since |f (2r) |q is concave implies that |f (2r) | is concave
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
and by Jensen’s integral inequality, we have
"Z a+b 2 x
−
a
2r
P2r
· |f (2r) (x)|dx
|Ir | ≤ (b − a)
b−a a
#
Z b x
−
a
P2r
· |f (2r) (x)|dx
+
a+b
b
−
a
2
Z 1 a+b
t (2r)
2r+1
dt
= (b − a)
·
f
(1
−
t)a
+
t
P
2r
2 2
0
Z 1
P2r 1 − t · f (2r) (1 − t)b + t a + b
+
2
2
0
Z
(b − a)2r+1 1
t
≤
P
dt
2r
2
2
0
 R 1

a+b
t
0 P2r 2 ((1 − t)a + t 2 )dt 
R
× f (2r) 
1
t
0 P2r 2 dt
Z 1
t
+ P2r 1 −
dt
2
0
 R 1

t
a+b
dt 0 P2r 1 − 2 (1 − t)b + t 2
 .
R
× f (2r) 
1
t
0 P2r 1 − 2 dt
Thus, by identities I, II, III, IV, V and VI, we have the inequality (3.2).
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 16
For r = 1 in Theorem 3.1, we have the following corollary.
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
Corollary 3.2. Under the assumptions of Theorem 3.1. If f is a 4-convex function, we have
Z b
b−a
(3.6) f (x)dx −
[f (a) + f (b)]
2
a
"
#1
q
q
+ 3|f 00 (b)|q q
(b − a)3 3 |f 00 (a)| + 10 f 00 a+b
2
≤
12
16
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
and if f is a 4-concave function, we have
Z b
b
−
a
(3.7) f (x)dx −
[f (a) + f (b)]
2
a
"
(b − a) f 00
≤
12
3
Dah-Yan Hwang
11a+5b
16
00
+ f
5a+11b
16
2
Remark 3.1. Using the convexity of |f 00 |q , we have
00 a + b q |f 00 (a)|q + |f 00 (b)|q
f
≤
,
2
2
Hence inequality (3.6) is an improvement of inequality (1.2).
Since |f 00 |q is concave implies that |f 00 | is concave, we have
00 a + b 1 00 11a + 5b 00 5a + 11b .
f
+ f
≤ f
2 16
16
2
#
.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 16
Thus inequality (3.7) is an improvement of inequality (1.3).
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
4.
Application
To obtain estimates of the error in the trapezoidal formula, we apply the results
of the previous section on each interval from the subdivision
[a, a + b], [a + h, a + 2h], . . . , [a + (n − 1)h, a + nh].
We define
Z a+nh
r−1
X
B2k h2k (2k−1)
Jr =
f (x)dx − T (f ; h) +
[f
(a + nh) − f (2k−1) (a)].
(2k)!
a
k=1
Theorem 4.1. If f : [a, a + nh] → R is a (2r)-time differentiable function
(r ≥ 1).
(a) If |f (2r) |q is convex for some q ≥ 1, then
Dah-Yan Hwang
Title Page
Contents
1− 1q
1q |B2r |
1
2
(2r)!
(
n
X
×
|aIII (r)| · |f (2r) (a + (m − 1)h)|q
2r+1
|J2r| ≤ h
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
JJ
J
II
I
Go Back
m=1
q
(2r)
1
+ (|aII (r)| + |aV (r)|) · f
a+ m−
h 2
) 1q
+ |aVI (r)| · |f (2r) (a + mh)|q
.
Close
Quit
Page 13 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
(b) If |f (2r) |q is concave for some q ≥ 1, then
n h2r+1 X
|J2r | ≤
|aI (r)|
2 m=1
(2r) |aIII (r)|(a + (m − 1 · h) + |aII (r)|(a + (m − 12 ) · h) × f
|aI (r)|
(2r) |aVI (r)|(a + mh) + |aII (r)| · (a + (m − 12 )h) .
+|aIV (r)| · f
|aIV (r)|
Proof. Since
|Jr | ≤
n Z
X
m=1
a+mh
h
f (x)dx − [f (a + (m − 1)h) + f (a + mh)]
2
a+(m−1)h
+
r−1
X
B2k h2k
k=1
(2k)!
Dah-Yan Hwang
)
[f (2k−1) (a + mh) − f (2k−1) (a + (m − 1)h)] ,
we have
1q 1− 1q (
1
|B
|
2r
|aIII (r)| · |f (2r) (a + (m − 1)h)|q
|Jr | ≤
h2r+1
2
(2r)!
m=1
q
(2r)
1
+ (|aII (r)| + |aV (r)|) · f
h a+ m−
2
1q
(2r)
q
+ |aII (r)| · |f (a + mh)|
,
n
X
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
by Theorem 3.1(a) applied to each interval [a + (m − 1)h, a + mh]. Obviously,
the proof (a) is complete.
The proof (b) is similar.
Remark 4.1. As the same discussion in the Section 3, Theorem 4.1 for r = 1 is
an improvement of Theorem 4 for r = 1 in [3].
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au
References
[1] M. ABRAMOWITZ AND I.A. STEGUN, Handbook of Mathematical Functions, Dover, New York, 1965.
[2] I.S. BEREZIN AND N.P. ZHIDKOV, Computing Methods, Vol. I, Pergamon
Press, Oxford, 1965.
[3] LJ. DEDIC, C.E.M. PEARCE AND J. PEČARIĆ, Hadamard and DragomirAgarwal inequalities, Higher-order convexity and the Euler formula, J. Korean Math., 38(6) (2001), 1235–1243.
[4] 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. Math. Lett., 11(5)(1998), 91–95.
[5] V.I. KRYLOV, Approximate Calculation of Integrals, Macmillan, New
York, 1962.
[6] C.E.M. PEARCE AND J. PEČARIĆ, Inequalities for differentiable mapping with application to special means and quadrature, Appl. Math. Lett.,
13 (2000), 51–55.
[7] GAU-SHENG YANG, DAH-YAN HWANG AND KUEI-LIN TSENG,
Some inequalities for differentiable convex and concave mappings, Computer and Mathematics with Applications, 47 (2004), 207–216.
Improvements of
Euler-Trapezoidal Type
Inequalities with Higher-Order
Convexity and Applications
Dah-Yan Hwang
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 16
J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004
http://jipam.vu.edu.au