Miskolc Mathematical Notes
Vol. 13 (2012), No. 2, pp. 233–248
HU e-ISSN 1787-2413
ON COMPANION OF OSTROWSKI INEQUALITY FOR
MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE
ARE CONVEX WITH APPLICATIONS
MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMAC
Received 21 February, 2012
Abstract. Several inequalities for a companion of Ostrowski inequality for absolutely continuous mappings whose first derivatives absolute value are convex (resp. concave) are established.
Applications to a composite quadrature rule, to p.d.f.’s, and to special means are provided.
2000 Mathematics Subject Classification: 26D10; 26A15; 26A16; 26A51
Keywords: Convex mappings, Hermite-Hadamard inequality, Ostrowski inequality
1. I NTRODUCTION
In 1938, Ostrowski established a very interesting inequality for differentiable mappings with bounded derivatives, as follows [8]:
Theorem 1. Let f W I R ! R be a differentiable mapping on I ı ; interior of the
interval I; such that f 0 2 LŒa; b, where a; b 2 I with a < b. If jf 0 .x/j M , then
the following inequality,
2
2 3
ˇ
ˇ
aCb
Z b
ˇ
ˇ
x
2
1
7
ˇ
ˇ
61
f .u/ duˇ M .b a/ 4 C
ˇf .x/
5
2
ˇ
ˇ
b a a
4
.b a/
holds for all x 2 Œa; b. The constant
be replaced by a smaller constant.
1
4
is the best possible in the sense that it cannot
For recent results concerning Ostrowski inequality see [1], [2] and [3]. Also, the
reader may be refer to the monograph [8] where various inequalities of Ostrowski
type are discussed.
In [9], Guessab and Schmeisser have proved among others, the following companion of Ostrowski inequality:
c 2012 Miskolc University Press
234
MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMAC
Theorem 2. Let f W Œa; b ! R satisfy the Lipschitz condition, i.e., jf .t / f .s/j M jt sj. Then for each x 2 Œa; aCb
2 , we have the inequality,
"
3aCb 2 #
ˇ
ˇ
R
x
ˇ
ˇ f .x/Cf .aCb x/
b
1
4
.b a/ M:
f .t / dt ˇ 81 C 2
(1.1)
ˇ
2
b a a
b a
The constant 1=8 is the best possible in the sense that it cannot be replaced by a
smaller constant.
We may also note that the best inequality in (1.1) is obtained for x D 3aCb
4 , giving
the trapezoid type inequality,
ˇ ˇ
ˇ f 3aCb C f aC3b
ˇ
Z b
ˇ
ˇ .b a/
4
4
1
ˇ
f .t/ dt ˇˇ M
(1.2)
ˇ
2
b a a
8
ˇ
ˇ
The constant 1=8 is sharp in (1.2) in the sense mentioned above.
Companions of Ostrowski integral inequality for absolutely continuous functions
was considered by Dragomir in [6], pp.228, as follows :
Theorem 3. Let f W I R ! R be an absolutely continuous function on Œa; b.
Then we have the inequalities,
ˇ
ˇ
Z b
ˇ f .x/ C f .a C b x/
ˇ
1
ˇ
ˇ
f .t/ dt ˇ
(1.3)
ˇ
ˇ
ˇ
2
b a a
8 "
3aCb 2 #
ˆ
x
1
ˆ
4
ˆ
.b a/ kf 0 k1 ; f 0 2 L1 Œa; b
ˆ 8 C2
b a
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
"
ˆ
aCb qC1 #1=q
ˆ
1=q
ˆ
ˆ
x
qC1
2
x a
2
<
.b a/1=q kf 0 kŒa;b;p ;
C
qC1
b a
b a
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
p > 1; p1 C q1 D 1; and f 0 2 Lp Œa; b
ˆ
ˆ
ˆ
ˆ
ˆ
ˇ
ˇ
ˆ
ˆ
ˇ 3aCb ˇ
ˆ
ˆ
: 1 C ˇ x 4 ˇ kf 0 kŒa;b;1
ˇ b a ˇ
4
for all x 2 Œa; aCb
2 .
In [7], the following theorem which was obtained by Dragomir and Agarwal contains the Hermite-Hadamard type integral inequality:
ON COMPANION OF OSTROWSKI INEQUALITY
235
Theorem 4. Let f W I ı R ! R be a differentiable mapping on I ı , a; b 2 I ı with
a < b: If jf 0 j is convex on Œa; b, then the following inequality holds:
ˇ
ˇ
Z b
ˇ f .a/ C f .b/
ˇ .b a/ .jf 0 .a/j C jf 0 .b/j/
1
ˇ
ˇ
f .u/duˇ :
(1.4)
ˇ
ˇ
ˇ
2
b a a
8
In [5], S.S. Dragomir established some inequalities for this companion for mappings of bounded variation. Also, Z. Liu introduced some companions of an Ostrowski type integral inequality for functions whose derivatives are absolutely continuous in [10]. Recently, N.S. Barnett et al. have proved some companions for the
Ostrowski inequality and the generalized trapezoid inequality in [4].
The aim of this paper is to study a companion of Ostrowski inequality Theorem
2 for the class of functions whose derivatives in absolute value are convex (concave)
functions.
2. R ESULTS
In order to prove our results, we need the following lemma (see [6]):
Lemma 1. Let f W I R ! R be an absolutely continuous mapping on I ı ; where
a; b 2 I with a < b, such that f 0 2 L1 Œa; b. Then, the following equality holds
Z b
Z b
1
1
f .x/ C f .a C b x/
f .t/ dt D
p .x; t / f 0 .t/ dt ;
2
b a a
b a a
where
8
t
ˆ
ˆ
ˆ
ˆ
<
p .x; t / D t
ˆ
ˆ
ˆ
ˆ
:
t
a;
aCb
2 ;
b;
t 2 Œa; x
t 2 .x; a C b
t 2 .a C b
x ;
x; b
for all x 2 Œa; aCb
2 .
A simple proof of the equality can be done by performing integration by parts.
The details are left to the interested reader (see [6]).
Let us begin with the following result:
Theorem 5. Let f W I R ! R be an absolutely continuous mapping on I ı ,
where a; b 2 I with a < b, such that f 0 2 L1 Œa; b. If jf 0 j is convex on Œa; b, then
we have the following inequality:
ˇ
ˇ
Z b
ˇ
ˇ f .x/ C f .a C b x/
1
ˇ
ˇ
(2.1)
f .t / dt ˇ
ˇ
ˇ
ˇ
2
b a a
236
MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMAC
.x a/2 ˇˇ 0 ˇˇ ˇˇ 0 ˇˇ
f .a/ C f .b/
6 .b a/
C
8 .x
a/2 C 3 .a C b
24 .b a/
2x/2 ˇˇ 0 ˇˇ ˇˇ 0
f .x/ C f .a C b
ˇ
x/ˇ
for all x 2 Œa; aCb
2 .
Proof. Using Lemma 1 and the modulus, we have
ˇ
ˇ
Z b
ˇ f .x/ C f .a C b x/
ˇ
1
ˇ
ˇ
f .t/ dt ˇ
ˇ
ˇ
ˇ
2
b a a
Z b
ˇ
ˇ
1
jp .x; t /j ˇf 0 .t/ˇ dt
b a a
"Z
Z aCb x
x
ˇ 0 ˇ
ˇ
ˇ
1
ˇ
ˇ
D
jp .x; t /j f .t / dt C
jp .x; t /j ˇf 0 .t /ˇ dt
b a a
x
#
Z b
ˇ
ˇ
C
jp .x; t /j ˇf 0 .t /ˇ dt
aCb x
Since jf
have
0j
is convex on Œa; b D Œa; x [ .x; a C b
x [ .a C b
x; b, therefore we
ˇ 0 ˇ t aˇ 0 ˇ x t ˇ 0 ˇ
ˇf .x/ˇ C
ˇf .a/ˇ ; t 2 Œa; xI
ˇf .t /ˇ x a
x a
ˇ
ˇ
ˇ 0 ˇ
ˇ
ˇ
ˇf .t /ˇ t x ˇf 0 .a C b x/ˇ C a C b x t ˇf 0 .x/ˇ ; t 2 .x; a C b xI
a C b 2x
a C b 2x
and
ˇ
ˇ 0 ˇ t a b Cx ˇ 0 ˇ b t ˇ 0
ˇf .b/ˇ C
ˇf .a C b x/ˇ ; t 2 .a C b x; bI
ˇf .t /ˇ x a
x a
which follows that,
ˇ
ˇ
Z b
ˇ f .x/ C f .a C b x/
ˇ
1
ˇ
ˇ
f .t/ dt ˇ
ˇ
ˇ
ˇ
2
b a a
Z x
1
t a ˇˇ 0 ˇˇ x t ˇˇ 0 ˇˇ
f .x/ C
f .a/ dt
jt aj
b a a
x a
x a
ˇ
Z aCb x ˇ
ˇ
ˇ aCbˇ
1
t x ˇˇ 0
ˇ
ˇt
f .a C b x/ˇ
C
ˇ
ˇ
b a x
2
a C b 2x
a C b x t ˇˇ 0 ˇˇ
C
f .x/ dt
a C b 2x
Z b
ˇ
1
t a b C x ˇˇ 0 ˇˇ b t ˇˇ 0
ˇ
C
f .b/ C
f .a C b x/ dt
jt bj
b a aCb x
x a
x a
ON COMPANION OF OSTROWSKI INEQUALITY
D
.b
C
C
1
a/ .x
"
a/
ˇ 0 ˇ .x
ˇf .x/ˇ
a/3
3
ˇ
ˇ .x
C ˇf 0 .a/ˇ
a/3
237
#
6
3
ˇ ˇ
ˇ
.a C b 2x/ ˇˇ 0
f .a C b x/ˇ C ˇf 0 .x/ˇ
2x/
8
"
#
ˇ 0 ˇ .x a/3 ˇ 0
ˇ .x a/3
ˇf .b/ˇ
C ˇf .a C b x/ˇ
a/
6
3
.b
1
a/ .a C b
.b
1
a/ .x
ˇ
ˇ
ˇ
ˇ ˇ
ˇ
.x a/2 ˇˇ 0 ˇˇ
f .a/ C 2 ˇf 0 .x/ˇ C 2 ˇf 0 .a C b x/ˇ C ˇf 0 .b/ˇ
6 .b a/
ˇ ˇ
ˇ
.a C b 2x/2 ˇˇ 0
C
f .a C b x/ˇ C ˇf 0 .x/ˇ
8 .b a/
.x a/2 ˇˇ 0 ˇˇ ˇˇ 0 ˇˇ
f .a/ C f .b/
D
6 .b a/
ˇ
8 .x a/2 C 3 .a C b 2x/2 ˇˇ 0 ˇˇ ˇˇ 0
f .x/ C f .a C b x/ˇ ;
C
24 .b a/
where
ˇ
ˇ
ˇ
ˇ
R aCb x
R aCb x
ˇ
ˇ
aCb ˇ
aCb ˇ
.a
/
.t
C
b
x
t
t
dt
D
x/
t
ˇ
ˇ
ˇ
x
x
2
2 ˇ dt D
D
.aCb 2x/3
;
8
which completes the proof.
An Ostrowski type inequality may be deduced as follows:
Corollary 1. Let f as in Theorem 5. Additionally, if f is symmetric about the
x-axis, i.e., f .a C b x/ D f .x/, we have
ˇ
ˇ
Z b
ˇ
ˇ
1
ˇ
ˇ
f .t / dt ˇ
ˇf .x/
ˇ
ˇ
b a a
.x a/2 ˇˇ 0 ˇˇ ˇˇ 0 ˇˇ
f .a/ C f .b/
6 .b a/
C
8 .x
a/2 C 3 .a C b
24 .b a/
2x/2 ˇˇ 0 ˇˇ ˇˇ 0
f .x/ C f .a C b
ˇ
x/ˇ
for all x 2 Œa; aCb
2 .
Remark 1. In Theorem 5, if we choose x D a, then we get
ˇ
ˇ
Z b
ˇ b a ˇ
ˇ f .a/ C f .b/
ˇ ˇ
ˇ
1
ˇ
ˇ
ˇf 0 .a/ˇ C ˇf 0 .b/ˇ :
f .t / dt ˇ ˇ
ˇ
ˇ
2
b a a
8
which is the inequality in (1.4).
238
MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMAC
Corollary 2. In Theorem 5, if we choose
(1) x D 3aCb
4 , then we get
ˇ ˇ
Z b
ˇ1
ˇ
3a C b
a C 3b
1
ˇ
ˇ
Cf
f .t/ dt ˇ
(2.2)
ˇ f
ˇ2
ˇ
4
4
b a a
ˇ ˇ ˇ
ˇ
ˇ 0 3a C b ˇ
ˇ 0 a C 3b ˇ ˇ 0 ˇ
.b a/ ˇˇ 0 ˇˇ
ˇ C 5 ˇf
ˇ C ˇf .b/ˇ
f .a/ C 5 ˇˇf
ˇ
ˇ
ˇ
96
4
4
(2) x D aCb
2 , then we get
ˇ
ˇ Rb
ˇ
ˇ
1
.t
/
f
dt
ˇ
ˇf aCb
2
b a a
.b a/
24
ˇ h
ˇ
i
ˇ
ˇ
0 .b/j : (2.3)
C
jf 0 .a/j C 4 ˇf 0 aCb
jf
ˇ
2
Another result may be considered using the Hölder inequality, as follows:
Theorem 6. Let f W I R ! R be an absolutely continuous mapping on I ı ;
where a; b 2 I with a < b, such that f 0 2 L1 Œa; b. If jf 0 jq , q > 1 is convex on Œa; b,
then we have the following inequality:
ˇ
ˇ
Z b
ˇ
ˇ f .x/ C f .a C b x/
1
ˇ
ˇ
f .t/ dt ˇ
(2.4)
ˇ
ˇ
ˇ
2
b a a
n
hˇ
ˇq ˇ 0 ˇq i 1=q
1
2 ˇ 0
ˇ C ˇf .x/ˇ
.x
.a/
a/
f
21=q .b a/ .p C 1/1=p
ˇq i1=q
.a C b 2x/2 hˇˇ 0 ˇˇq ˇˇ 0
C
f .x/ C f .a C b x/ˇ
2 h
ˇ
ˇq ˇ
ˇq i o
C .x a/2 ˇf 0 .a C b x/ˇ C ˇf 0 .b/ˇ 1=q ;
for all x 2 Œa; aCb
2 , where
1
p
C q1 D 1.
Proof. Using Lemma 1 and the Hölder inequality, we have
ˇ
ˇ
Z b
ˇ f .x/ C f .a C b x/
ˇ
1
ˇ
ˇ
f .t / dt ˇ
ˇ
ˇ
ˇ
2
b a a
Z b
ˇ
ˇ
1
jp .x; t /j ˇf 0 .t/ˇ dt
b a a
"Z
ˇ
Z aCb x ˇ
x
ˇ a Cb ˇˇ 0 ˇ
ˇ 0 ˇ
1
ˇ
ˇ
ˇ
ˇ ˇf .t/ˇ dt
D
jt aj f .t/ dt C
ˇt
b a a
2 ˇ
x
#
Z b
ˇ 0 ˇ
C
jt bj ˇf .t/ˇ dt
aCb x
(2.5)
ON COMPANION OF OSTROWSKI INEQUALITY
"Z
1
b
a
a
x
Z
C
jt
p
1=p Z
b
aCb x
xˇ
aj dt
a
ˇ
ˇf 0 .t/ˇq dt
!1=p Z
ˇp
ˇ
ˇ
ˇt a C b ˇ dt
ˇ
2 ˇ
!1=p Z
p
jt bj dt
aCb x ˇ
Z
C
x
aCb x ˇ
x
b
aCb x
239
1=q
ˇ
ˇf .t /ˇq dt
!1=q
0
ˇ 0 ˇq
ˇf .t/ˇ dt
!1=q 3
5
Since jf 0 jq is convex on Œa; b D Œa; x [ .x; a C b x [ .a C b x; b, therefore we
have
ˇ 0 ˇq t a ˇ 0 ˇq x t ˇ 0 ˇq
ˇf .x/ˇ C
ˇf .a/ˇ ; t 2 Œa; xI
ˇf .t /ˇ x a
x a
ˇ
ˇ
ˇ
ˇ
ˇ 0 ˇq
ˇf .t /ˇ t x ˇf 0 .a C b x/ˇq C a C b x t ˇf 0 .x/ˇq ; t 2 .x; a Cb xI
a C b 2x
a C b 2x
and
ˇ 0 ˇq t a b C x ˇ 0 ˇq b t ˇ 0
ˇ
ˇf .t/ˇ ˇf .b/ˇ C
ˇf .a C b x/ˇq ; t 2 .a C b x; bI
x a
x a
which follows that,
ˇ
ˇ
Z b
ˇ f .x/ C f .a C b x/
ˇ
1
ˇ
ˇ
f .t / dt ˇ
ˇ
ˇ
ˇ
2
b a a
"Z
1=p Z x 1=q
x
1
t a ˇˇ 0 ˇˇq x t ˇˇ 0 ˇˇq
p
f .x/ C
f .a/ dt
jt aj dt
b a
x a
x a
a
a
!1=p
ˇ
Z aCb x ˇ
ˇ a C b ˇp
ˇ dt
ˇt
C
ˇ
2 ˇ
x
!1=q
Z aCb x ˇq a C b x t ˇ 0 ˇq
t x ˇˇ 0
ˇf .x/ˇ dt
f .a C b x/ˇ C
a C b 2x
a C b 2x
x
!1=p
Z
b
C
bjp dt
a b C x ˇˇ 0 ˇˇq b t ˇˇ 0
f .b/ C
f .a C b
x a
x a
aCb x
Z
D
jt
b
aCb x
1
b
2
t
pC1
4 .x a/
.p C 1/
a
!1=p
x
2
3
!1=q
ˇq
5
x/ˇ dt
a 1=q ˇˇ 0 ˇˇq ˇˇ 0 ˇˇq 1=q
f .a/ C f .x/
240
MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMAC
2
C
.p C 1/
aCb
2
pC1 !1=p
x
1=q aCb
2
.x a/pC1
C
.p C 1/
!1=p
x
x
ˇ 0 ˇq ˇ 0
ˇf .x/ˇ C ˇf .a C b
a 1=q ˇˇ 0
f .a C b
ˇq 1=q
x/ˇ
3
ˇq 1=q
ˇq ˇ
5
x/ˇ C ˇf 0 .b/ˇ
2
ˇ
ˇq ˇ 0 ˇq 1=q
1
2 ˇ 0
ˇ C ˇf .x/ˇ
.a/
.x
D
f
a/
21=q .b a/ .p C 1/1=p
ˇq 1=q
.a C b 2x/2 ˇˇ 0 ˇˇq ˇˇ 0
C
f .x/ C f .a C b x/ˇ
2
ˇ
ˇq 1=q
ˇq ˇ
;
C .x a/2 ˇf 0 .a C b x/ˇ C ˇf 0 .b/ˇ
since
1
p
C q1 D 1, q > 1, which completes the proof.
Corollary 3. Let f as in Theorem 6. Additionally, if f is symmetric about the
x-axis, i.e., f .a C b x/ D f .x/, we have
ˇ
ˇ
Z b
ˇ
ˇ
1
1
ˇ
ˇ
f .t / dt ˇ ˇf .x/
ˇ
ˇ 21=q .b a/ .p C 1/1=p
b a a
n
hˇ
ˇq ˇ
ˇq i
.x a/2 ˇf 0 .a/ˇ C ˇf 0 .x/ˇ 1=q
ˇq i1=q
.a C b 2x/2 hˇˇ 0 ˇˇq ˇˇ 0
f .x/ C f .a C b x/ˇ
2
hˇ
ˇq ˇ 0 ˇq i1=q
2 ˇ 0
ˇ
ˇ
ˇ
C .x a/ f .a C b x/ C f .b/
C
for all x 2 Œa; aCb
2 .
Corollary 4. In Theorem 6, if we choose
(1) x D a, then we get
ˇ
ˇ
Rb
ˇ f .a/Cf .b/
ˇ
1
.t/
f
dt
ˇ
ˇ
2
b a a
(2) x D
.b a/
1
1C q
2
.1Cp/
1=p
jf 0 .a/jq C jf 0 .b/jq
1=q
: (2.6)
3aCb
4 ,
then we get
ˇ ˇ1
3a C b
a C 3b
ˇ
Cf
ˇ f
ˇ2
4
4
1
b
Z
a
a
b
ˇ
ˇ
ˇ
f .t/ dt ˇ
ˇ
(2.7)
ON COMPANION OF OSTROWSKI INEQUALITY
241
ˇ ˇ ˇ 0 ˇq ˇ 0 3a C b ˇq 1=q
ˇ
ˇ
ˇ
ˇ
f .a/ C ˇf
ˇ
4C q1
1=p
4
.1 C p/
2
ˇ ˇ
ˇ ˇ ˇ 0 3a C b ˇq ˇ 0 a C 3b ˇq 1=q
ˇ C ˇf
ˇ
C2 ˇˇf
ˇ
ˇ
ˇ
4
4
)
ˇ ˇ
ˇ 0 a C 3b ˇq ˇ 0 ˇq 1=q
ˇ
ˇ
ˇ
ˇ
C ˇf
:
ˇ C f .b/
4
.b
(3) x D
a/
(
aCb
2 ,
then we get
ˇ ˇ
Z b
ˇ
ˇ
aCb
1
ˇ
ˇ
f .t/ dt ˇ
ˇf
ˇ
ˇ
2
b a a
"
ˇ ˇ ˇ 0 ˇq ˇ 0 a C b ˇq 1=q
.b a/
ˇf .a/ˇ C ˇf
ˇ
1
ˇ
ˇ
2
22C q .1 C p/1=p
ˇ ˇ
#
ˇ 0 a C b ˇq ˇ 0 ˇq 1=q
ˇ C ˇf .b/ˇ
C ˇˇf
:
ˇ
2
The following result holds for concave mappings.
Theorem 7. Let f W I R ! R be an absolutely continuous mapping on I ı ;
where a; b 2 I with a < b, such that f 0 2 L1 Œa; b. If jf 0 jq , q > 1 is concave on
Œa; b, then we have the following inequality:
ˇ
ˇ
Z b
ˇ
ˇ f .x/ C f .a C b x/
1
ˇ
ˇ
f .t / dt ˇ
(2.8)
ˇ
ˇ
ˇ
2
b a a
1
a/ .1 C p/1=p
ˇ ˇ ˇ ˇ
ˇ ˇ 0 2b C a
2 ˇ 0 aCx ˇ
ˇ
.x a/ ˇf
ˇ C ˇf
2
2
.b
"
for all x 2 Œa; aCb
2 , where
1
p
ˇ ˇ
ˇ#
.a C b 2x/2 ˇˇ 0 a C b ˇˇ
x ˇˇ
ˇ C
ˇf
ˇ
2
2
C q1 D 1.
Proof. From Lemma 1, and by (2.5), we have
ˇ
ˇ
Z b
ˇ
ˇ f .x/ C f .a C b x/
1
ˇ
ˇ
f .t / dt ˇ
ˇ
ˇ
ˇ
2
b a a
"Z
Z
1=q
1=p
x
xˇ
ˇq
1
p
0
ˇ
ˇ
f .t/ dt
jt aj dt
b a
a
a
242
MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMAC
C
x
Z
C
Now, let us write,
Z x
a
Z
!1=q
ˇp !1=p Z aCb x
ˇ
ˇ
ˇ 0 ˇq
a
C
b
ˇ dt
ˇt
ˇf .t/ˇ dt
ˇ
2 ˇ
x
!1=q 3
!1=p Z
b
ˇ
ˇ
ˇf 0 .t /ˇq dt
5:
jt bjp dt
aCb x ˇ
Z
b
aCb x
aCb x
aCb x ˇ
x
1ˇ
Z
ˇ 0 ˇq
ˇf .t/ˇ dt D .x
a/
0
1ˇ
Z
ˇ
ˇf 0 .t /ˇq dt D .a C b
2x/
ˇ 0 ˇq
ˇf .t/ˇ dt D .x
Z
ˇf 0 .x C .1
0
ˇf 0 . .a C b
ˇq
/ a/ˇ d ;
x/ C .1
ˇq
/ x/ˇ d ;
and
Z
b
aCb x
a/
0
1ˇ
ˇf 0 .b C .1
/ .a C b
ˇq
x//ˇ d :
Since jf 0 jq , q > 1 is concave on Œa; b D Œa; x [ .x; a C b x [ .a C b x; b, we
can use the Jensen integral inequality to obtain
Z 1
ˇ 0
ˇ
ˇf .x C .1 / a/ˇq d .x a/
0
Z 1
ˇ
ˇq
D .x a/
0 ˇf 0 .x C .1 / a/ˇ d 0
!ˇq
Z 1
ˇˇ
Z 1
ˇ
1
ˇ
ˇ 0
0
.x C .1 / a/ d ˇ
.x a/
d ˇf R 1
0
ˇ
ˇ
0
0 d 0
ˇ ˇq
ˇ
x C a ˇˇ
D .x a/ ˇˇf 0
ˇ
2
and analogously
Z 1
ˇ 0
ˇf . .a C b
.a C b 2x/
0
Z
.x
a/
0
1ˇ
ˇf 0 .b C .1
x/ C .1
/ .a C b
ˇq
/ x/ˇ d .a C b
ˇq
x//ˇ d .x
Combining all above inequalities, we get
ˇ
ˇ
Z b
ˇ f .x/ C f .a C b x/
ˇ
1
ˇ
ˇ
f .t/ dt ˇ
ˇ
ˇ
ˇ
2
b a a
ˇ ˇ
ˇ 0 a C b ˇq
ˇ ;
ˇ
2x/ ˇf
ˇ
2
ˇ ˇ
2b C a
a/ ˇˇf 0
2
ˇ
x ˇˇq
ˇ :
ON COMPANION OF OSTROWSKI INEQUALITY
2
1
b
a
pC1
4 .x a/
.p C 1/
2
C
.p C 1/
!1=p
.x
aCb
x
2
!1=p
.x a/pC1
C
.p C 1/
3
ˇ ˇ
ˇ
ˇ
x C a ˇ5
a/1=q ˇˇf 0
ˇ
2
pC1 !1=p
2x/
ˇ ˇ 0 2b C a
ˇf
2
ˇ
x ˇˇ
ˇ
1=q ˇ
a/
ˇ ˇ
ˇ 0 aCb ˇ
ˇ
ˇ
ˇf
2
1=q ˇ
.a C b
.x
243
1
D
a/ .1 C p/1=p
ˇ ˇ ˇ ˇ ˇ 0 2b C a
ˇ
2 ˇ 0 aCx ˇ
ˇ
.x a/ ˇf
ˇ C ˇf
2
2
.b
"
for all x 2 Œa; aCb
2 , where
1
p
ˇ ˇ
ˇ#
.a C b 2x/2 ˇˇ 0 a C b ˇˇ
x ˇˇ
ˇ C
ˇf
ˇ ;
2
2
C q1 D 1, q > 1, which is required.
Therefore, we may state the following Ostrowski type inequality:
Corollary 5. Let f as in Theorem 7. Additionally, if f is symmetric about the
x-axis, i.e., f .a C b x/ D f .x/, we have
ˇ
ˇ
Z b
ˇ
ˇ
1
ˇ
ˇ
f .t / dt ˇ
ˇf .x/
ˇ
ˇ
b a a
1
a/ .1 C p/1=p
ˇ ˇ ˇ ˇ
ˇ ˇ 0 2b C a
2 ˇ 0 aCx ˇ
ˇ
.x a/ ˇf
ˇ C ˇf
2
2
.b
"
ˇ ˇ
ˇ#
.a C b 2x/2 ˇˇ 0 a C b ˇˇ
x ˇˇ
ˇ C
ˇf
ˇ ;
2
2
for all x 2 Œa; aCb
2 .
Corollary 6. In Theorem 7, if we choose
(1) x D a, then we get
ˇ
ˇ
ˇ ˇ
Z b
ˇ f .a/ C f .b/
ˇ
1
.b a/ ˇˇ 0 a C b ˇˇ
ˇ
ˇ
f .t / dt ˇ f
ˇ
ˇ:
ˇ
ˇ 2 .1 C p/1=p ˇ
2
b a a
2
(2) x D
3aCb
4 ,
then we get
ˇ ˇ1
3a C b
a C 3b
ˇ
Cf
ˇ f
ˇ2
4
4
1
b
Z
a
a
b
ˇ
ˇ
ˇ
f .t/ dt ˇ
ˇ
(2.9)
244
MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMAC
ˇ ˇ
ˇ ˇ ˇ
ˇ ˇ 0 a C b ˇ ˇ 0 a C 7b ˇ
ˇ 0 7a C b ˇ
ˇ
ˇ
ˇ
ˇ
ˇ :
ˇ
ˇ C 2 ˇf
ˇ C ˇf
ˇ
ˇf
8
2
8
16 .1 C p/1=p
.b
a/
(3) x D aCb
2 , then we get
ˇ ˇ
Rb
ˇ
ˇ
1
.t
/
f
dt
ˇf aCb
ˇ
2
b a a
.b a/
4.1Cp/1=p
(2.10)
hˇ ˇ ˇ ˇi
ˇ 0 3aCb ˇ ˇ 0 aC3b ˇ
C
f
ˇf
ˇ
ˇ
ˇ :
4
4
3. A COMPOSITE QUADRATURE FORMULA
Let In W a D x0 < x1 < < xn D b be a division of the interval Œa; b and
hi D xi C1 xi , (i D 0; 1; 2; ; n 1).
Consider the general quadrature formula
n 1 1X
3xi C xi C1
xi C 3xi C1
f
Qn .In ; f / WD
Cf
hi :
2
4
4
(3.1)
i D0
The following result holds.
Theorem 8. Let f W I R ! R be an absolutely continuous mapping on I ı ;
where a; b 2 I with a < b, such that f 0 2 L1 Œa; b. If jf 0 j is convex on Œa; b. Then,
we have
Z
b
a
f .t / dt D Qn .In ; f / C Rn .In ; f / :
(3.2)
where, Qn .In ; f / is defined by formula (3.1), and the remainder term Rn .In ; f /
satisfies the error estimates
ˇ ˇ
n 1
ˇ
ˇ 0 3xi C xi C1 ˇ
1 X 2 ˇˇ 0
ˇ
ˇ
ˇ
hi f .xi / C 5 ˇf
jRn .In ; f /j ˇ
96
4
i D0
ˇ ˇ
ˇ
ˇ 0 xi C 3xi C1 ˇ ˇ 0
ˇ C ˇf .xi C1 /ˇ :
C5 ˇˇf
ˇ
4
Proof. Applying inequality (3.1) and (3.2) on the intervals Œxi ; xi C1 , we may state
that
Z xiC1
1
3xi C xi C1
xi C 3xi C1
Ri .Ii ; f / D
f .t / dt
f
Cf
hi :
2
4
4
xi
Summing the above inequality over i from 0 to n 1, we get
n
n 1 X1 Z xi C1
3xi C xi C1
xi C 3xi C1
1X
Rn .In ; f / D
f .t/ dt
f
Cf
hi
2
4
4
i D0
i D0 xi
Z b
n 1 3xi C xi C1
xi C 3xi C1
1X
f .t/ dt
f
Cf
hi ;
D
2
4
4
a
i D0
ON COMPANION OF OSTROWSKI INEQUALITY
245
which follows from (2.2), that
ˇZ
ˇ b
ˇ
f .t / dt
jRn .In ; f /j D ˇ
ˇ a
ˇˇ
n 1 1X
3xi C xi C1
xi C 3xi C1
ˇ
f
Cf
hi ˇ
ˇ
2
4
4
i D0
ˇ ˇ
n 1
ˇ
ˇ 0 3xi C xi C1 ˇ
1 X 2 ˇˇ 0
ˇ
hi f .xi /ˇ C 5 ˇˇf
ˇ
96
4
i D0
ˇ ˇ
ˇ 0 xi C 3xi C1 ˇ ˇ 0
ˇ
ˇ
ˇ
ˇ
ˇ
C5 ˇf
ˇ C f .xi C1 / :
4
which completes the proof.
Remark 2. One may state more inequalities, using (2.7) and (2.10). We shall omit
the details.
4. A PPLICATIONS FOR P.D.F.’ S
Let X be a random variable taking values in the finite interval Œa; b, with the probability density function f W Œa; b ! Œ0; 1 with the cumulative distribution function
Rb
F .x/ D P r.X x/ D a f .t /dt.
Theorem 9. With the assumptions of Theorem 5, we have the inequality
ˇ
ˇ
ˇ
ˇ1
ˇ ŒF .x/ C F .a C b x/ b E .X/ ˇ
ˇ2
b a ˇ
.x a/2 ˇˇ 0 ˇˇ ˇˇ 0 ˇˇ
F .a/ C F .b/
6 .b a/
C
8 .x
a/2 C 3 .a C b
24 .b a/
2x/2 ˇˇ 0 ˇˇ ˇˇ 0
F .x/ C F .a C b
ˇ
x/ˇ
for all x 2 Œa; aCb
2 , where E.X / is the expectation of X.
Proof. In the proof of Theorem 5 , let f D F , and taking into account that
Z b
Z b
E .X/ D
t dF .t/ D b
F .t/ dt :
a
a
We left the details to the interested reader.
Corollary 7. In Theorem 9, if we choose x D 3aCb
4 , then we get
ˇ ˇ
ˇ
ˇ1
.X/
3a
C
b
a
C
3b
b
E
ˇ
ˇ F
CF
ˇ2
4
4
b a ˇ
ˇ ˇ ˇ
ˇ
ˇ 0 3a C b ˇ
ˇ 0 a C 3b ˇ ˇ 0 ˇ
.b a/ ˇˇ 0 ˇˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
F .a/ C 5 ˇF
ˇ C 5 ˇF
ˇ C F .b/ :
96
4
4
246
MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMAC
Corollary 8. In Theorem 9, if F is symmetric about the x-axis, i.e., F .a C b
F .x/, we have
ˇ
ˇ
ˇ
ˇ
ˇF .x/ b E .X/ ˇ
ˇ
b a ˇ
x/ D
.x a/2 ˇˇ 0 ˇˇ ˇˇ 0 ˇˇ
F .a/ C F .b/
6 .b a/
C
8 .x
a/2 C 3 .a C b
24 .b a/
2x/2 ˇˇ 0 ˇˇ ˇˇ 0
F .x/ C F .a C b
ˇ
x/ˇ
for all x 2 Œa; aCb
2 .
Remark 3. One may state more inequalities, using Theorem 6 and Theorem 7. We
shall omit the details.
5. A PPLICATIONS FOR SPECIAL MEANS
Recall the following means which could be considered extensions of arithmetic,
logarithmic and generalized logarithmic for positive real numbers.
(1) The arithmetic mean:
aCb
A D A .a; b/ D
I a; b 2 RC
2
(2) The logarithmic mean:
b a
L .a; b/ D
I jaj ¤ jbj ; a; b 2 RC
ln jbj ln jaj
(3) The generalized logarithmic mean:
1
nC1
b
anC1 n
I n 2 Zn f 1; 0g ; a; b 2 RC ; a ¤ b
Ln .a; b/ D
.b a/ .n C 1/
(4) The identric mean:
8 1
ˆ
< 1 b ba b a ; a ¤ b
e a
I .a; b/ D
a; b 2 RC :
ˆ
:
a
;a D b
Now using our results, we give some applications to special means for positive
real numbers.
Proposition 1. Let a; b 2 RC ; a < b. Then, we have
ˇ ˇ
h 2 2
ˇ
ˇ
4
4
; aC3b
L 1 .a; b/ˇ .b96a/ aa2Cb
C
80
ˇA 3aCb
2
b
1
.3aCb/2
C
1
.aC3b/2
i
:
Proof. The assertion was obtained by the inequality in (2.2) applied to the convex
mapping f W Œa; b ! R; f .x/ D x1 :
ON COMPANION OF OSTROWSKI INEQUALITY
247
Proposition 2. Let a; b 2 RC ; a < b; and p 2 Z; jpj 2:Then,
ˇ p
ˇA .a; b/
ˇ p .b a/ p
Lpp .a; b/ˇ a
24
1
C 4Ap
1
.a; b/ C b p
1
:
Proof. The assertion was obtained by the inequality in (2.3) applied to the convex
mapping f W Œa; b ! R; f .x/ D x p :
Proposition 3. Let a; b 2 RC ; a < b. Then, we have
1
.b a/
1 q
1
:
C q
jln I A .ln a; ln b/j 1
1
q
b
21C q .1 C p/ p a
Proof. The assertion was obtained by the inequality in (2.6) applied to the convex
mapping f W Œa; b ! Œ0; 1/; f .x/ D ln x:
R EFERENCES
[1] M. Alomari and M. Darus, “Some Ostrowski type inequalities for quasi-convex functions with
applications to special means,” RGMIA Preprint, vol. 13, no. 2, p. article No. 3., 2010.
[2] M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone, “Ostrowski type inequalities for functions
whose derivatives are s-convex in the second sense,” Appl. Math. Lett., vol. 23, no. 9, pp. 1071–
1076, 2010.
[3] M. W. Alomari, “A companion of Ostrowski ’s inequality with applications,” Transylv. J. Math.
Mech., vol. 3, no. 1, pp. 9–14, 2011.
[4] N. S. Barnett, S. S. Dragomir, and I. Gomm, “A companion for the Ostrowski and the generalised
trapezoid inequalities,” Math. Comput. Modelling, vol. 50, no. 1-2, pp. 179–187, 2009.
[5] S. S. Dragomir, “A companion of Ostrowski’s inequality for functions of bounded variation and
applications,” RGMIA Preprint, vol. 5, p. article No. 28, 2002.
[6] S. S. Dragomir, “Some companions of Ostrowski’s inequality for absolutely continuous functions
and applications,” Bull. Korean Math. Soc., vol. 42, no. 2, pp. 213–230, 2005.
[7] 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., vol. 11, no. 5, pp.
91–95, 1998.
[8] S. S. Dragomir and T. M. Rassias, Eds., Ostrowski type inequalities and applications in numerical
integration. Dordrecht: Kluwer Academic Publishers, 2002.
[9] A. Guessab and G. Schmeisser, “Sharp integral inequalities of the Hermite-Hadamard type,” J.
Approximation Theory, vol. 115, no. 2, pp. 260–288, 2002.
[10] Z. Liu, “Some companions of an Ostrowski type inequality and applications,” JIPAM, J. Inequal.
Pure Appl. Math., vol. 10, no. 2, p. 12, 2009.
Authors’ addresses
Mohammad W. Alomari
Department of Mathematics, Faculty of Science, Jerash University, Jerash, Jordan
E-mail address: [email protected]
248
MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMAC
M. Emin Özdemir
Ağrı İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, 04100,
Ağrı, Turkey
E-mail address: [email protected]
Havva Kavurmac
Ağrı İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, 04100,
Ağrı, Turkey
E-mail address: [email protected]