Vol. 20 (2019), No. 2, pp. 1101–1118 DOI: 10.18514/MMN.2019.2785
A WEIGHTED COMPANION OF OSTROWSKI’S INEQUALITY USING THREE STEP WEIGHTED KERNEL
S. OBEIDAT, M. A. LATIF, AND A. QAYYUM Received 10 December, 2018
Abstract. A weighted version of Ostrowski type integral inequalities is established . We use a newly developed special type of three steps kernel. Our findings give some new error bounds for various quadrature rules. We apply our results to cumulative distributive functions.
2010Mathematics Subject Classification: 26D15; 26D20; 26D99
Keywords: Ostrowski inequality, weight function, composite quadrature rule, cumulative dis- tributive function
1. INTRODUCTION
The importance of mathematical inequalities is due to its applications in several branches of Mathematics such as numerical integration, optimization theory, and in- tegral operator theory. During the past few decades, many researchers worked on inequalities and their applications (see for instance [4]-[5], [7]-[8]).
In 1938, Ostrowski [6] introduced an interesting integral inequality which meas- ures the deviation between a function and its integral mean. It is stated as follows:
Theorem 1. Let fWŒc; d !R be continuous on Œc; d and differentiable on .c; d / ;whose derivativef0W.c; d /!Ris bounded on .c; d / ;i.e.
f0
1D sup
t2Œc;d
ˇˇf0.t /ˇ ˇ<1 then for allx2Œc; d
ˇ ˇ ˇ ˇ ˇ ˇ
f .x/ 1
d c
d
Z
c
f .t /dt ˇ ˇ ˇ ˇ ˇ ˇ
2 4 1
4C x cC2d
d c
!23
5.d c/
f0
1: (1.1) Another important inequality is so-called Gr¨uss inequality [5] which links the in- tegral mean of a product of two functions with the product of their integral means. It
c 2019 Miskolc University Press
is stated as follows:
ˇ ˇ ˇ ˇ ˇ ˇ
1
d c
d
Z
c
f .x/g.x/dx 1
d c
d
Z
c
f .x/dx: 1
d c
d
Z
c
g.x/dx ˇ ˇ ˇ ˇ ˇ ˇ
(1.2) 1
4.˛2 ˛1/.2 1/;
where
˛1f .x/˛2 and1g .x/2, for allx2Œc; d :The constant 14 is sharp in (1.2) .
In [3], Dragomir and Wang combined Ostrowski and Gr¨uss inequality and ob- tained a new inequality known in the literature as Ostrowski-Gr¨uss type inequalities.
Dragomir [2] established some companions of Ostrowski type integral inequalities.
In [1], Alomari proved the following companion inequality of Ostrowski’s type using Gr¨uss inequality.
Theorem 2. LetfWŒc; d !R be a differentiable function on.c; d / ; 1< c <
d <1:Iff 02L1Œc; d andB1f 0.t /B2for allt 2Œc; d , whereB1; B2 are constants, then the inequality
ˇ ˇ ˇ ˇ ˇ ˇ
f .x/Cf .cCd x/
2
1
d c
d
Z
c
f .t /dt ˇ ˇ ˇ ˇ ˇ ˇ
1
8.d c/ .B2 B1/ (1.3) holds for allx2h
c;cC2di :
In the proof of Theorem2, Alomari defined the following mapping p .x; t /D
8
<
:
t c t2Œc; x
t aC2b t2.x; cCd x
t b t2.cCd x; d for allx2h
c;cC2di
:The upper bound of Inequality (1.3) is obtained using the fact that
x cCd
2 p .x; t /x c (1.4)
for each t 2Œc; d and each x2h c;cC2d
i
:However, this fact is not correct. For example, ifxDcC2d d c100 andtD cC2dCd c50 , then
p .x; t /Dt d D 12 .d c/
25 ;
and
x cCd
2 D .d c/
100 > p .x; t / ;
which contradics Inequality (1.4). The correct property ofp .x; t /is that min
x cCd 2 ; c x
p .x; t /max cCd
2 x; x c
: (1.5)
for eacht2Œc; d and eachx2h
c;cC2di
:Using this property ofp .x; t /, the correc- tion of Inequality (1.3) becomes
ˇ ˇ ˇ ˇ ˇ ˇ
f .x/Cf .cCd x/
2
1
d c
d
Z
c
f .t /dt ˇ ˇ ˇ ˇ ˇ ˇ
1
4.d c/ .B2 B1/ : (1.6) In this paper, we will present a weighted version of Inequality (1.6) using a new three step kernel, and discuss some applications of our results. Throughout the present paper, a weight function (or density function) over some intervalŒc; d ;where
1< c < d <1;is a functionwWŒc; d !Œ0;1/with0 <
d
R
c
w.t /dt <1. 2. MAIN RESULTS
Definition 1. Let 1< c < d <1:Letwbe a weight function overŒc; d :The 3-step linear kernel with respect towis denoted byGw, and is defined as follows:
Gw.x; t /D 8 ˆˆ ˆˆ ˆˆ ˆ<
ˆˆ ˆˆ ˆˆ ˆ:
t
R
c
w .u/ du; t2Œc; x
t
R
c
w .u/ du 12
d
R
c
w .u/ du; t2.x; cCd x
t
R
d
w .u/ du; t2.cCd x; d
; (2.1)
forx2h
c;cC2di
andt 2Œ0; 1 :
The following lemma will be used repeatedly throughout the present paper.
Lemma 1. Let 1< c < d <1:For a weight functionwoverŒc; d ;the identity
d
Z
c
Gw.x; t /f 0.t /dt (2.2)
D1 2
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
d
Z
c
w .t / f .t / dt;
holds for allx2h c;cC2d
i
andt2Œ0; 1 : Proof. Using Definition1, we have
d
Z
c
Gw.x; t /f 0.t /dtD
x
Z
c
2 4
t
Z
c
w .u/ du 3
5f 0.t /dt
C
cCd x
Z
x
2 4
t
Z
c
w .u/ du 1 2
d
Z
c
w .u/ du 3
5f 0.t /dt
C
d
Z
cCd x
2 4
t
Z
b
w .u/ du 3
5f 0.t /dt:
Applying integration by parts on each integral, we get
x
Z
c
2 4
t
Z
c
w .u/ du 3
5f 0.t /dtD 2 4
x
Z
c
w .u/ du 3 5f .x/
x
Z
c
w .t / f .t / dt;
cCd x
Z
x
2 4
t
Z
c
w .u/ du 1 2
d
Z
c
w .u/ du 3
5f 0.t /dt
D 0
@
cCd x
Z
c
w .u/ du 1 2
d
Z
c
w .u/ du 1
Af .cCd x/
0
@
x
Z
c
w .u/ du 1 2
d
Z
c
w .u/ du 1 Af .x/
cCd x
Z
x
w .t / f .t / dt;
and
d
Z
cCd x
2 4
t
Z
d
w .u/ du 3
5f 0.t /dtD 2 6 4
d
Z
cCd x
w .u/ du 3 7
5f .cCd x/
d
Z
cCd x
w .t / f .t / dt:
Thus,
d
Z
c
Gw.x; t /f 0.t /dt
D1 2
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
d
Z
c
w .t / f .t / dt:
Remark1. If we setf .t /Dtin Identity (2.2), we have
d
Z
c
Gw.x; t /dtDcCd 2
2 4
d
Z
c
w .t / dt 3 5
d
Z
c
w .t / dt; (2.3)
forx2h
c;cC2di :
Lemma 2. Let 1< c < d <1:Ifw is a weight function overŒc; d , andwis symmetric about cC2d, then
d
Z
c
w .t / dtDcCd 2
2 4
d
Z
c
w .t / dt 3 5;
and hence
d
Z
c
Gw.x; t /dtD0;
forx2h
c;cC2di :
Proof. Sincewis symmetric about cC2d;
d
Z
c
t w .t / dt D
d
Z
c
.cCd t / w .t / dt;
which implies that
d
Z
c
t w .t / dtD cCd 2
d
Z
c
w .t / dt:
Using Identity (2.3), we get that
d
Z
c
Gw.x; t /dtD0:
Theorem 3. Let 1< c < d <1andfWŒc; d !R be a differentiable function on .c; d /. Suppose that w is a weight function over Œc; d . If f 02L1Œc; d and B1f 0.t /B2, for allt2Œc; d ;whereB1; B2are constants, then the inequality
ˇ ˇ ˇ ˇ ˇ ˇ
2 4 1 2
0
@
d
Z
c
w .t / dt 1
A.f .x/Cf .cCd x// I .c; d; x/H .c; d / 3 5
d
Z
c
w .t / f .t / dt ˇ ˇ ˇ ˇ ˇ ˇ 1
4V .c; d; x/ .B2 B1/ .d c/ (2.4)
holds for allx2h
c;cC2di , where
I .c; d; x/D
d
Z
c
Gw.x; t /dt; H .c; d /Df .d / f .c/
d c ;
and
V .c; d; x/Dmax 8
<
:
cCd x
Z
x
w .t / dt;
d
Z
c
w .t / dt
cCd x
Z
x
w .t / dt 9
=
; Proof. Note that fort2Œ0; 1andx2h
c;cC2di
, we have
L .c; d; x/Gw.x; t /U .c; d; x/ ; (2.5) where
U .c; d; x/Dmax 8
<
:
x
Z
c
w .t / dt;
cCd x
Z
c
w .t / dt 1 2
d
Z
c
w .t / dt 9
=
;
; and
L .c; d; x/Dmin 8
<
:
x
Z
c
w .t / dt 1 2
d
Z
c
w .t / dt;
cCd x
Z
d
w .t / dt 9
=
; :
By Lemma1, we have 1
d c
d
Z
c
Gw.x; t /f 0.t /dt 1 .d c/2
d
Z
c
Gw.x; t /dt
d
Z
c
f 0.t /dt
D 1 2 .d c/
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
f .d / f .c/
.d c/2
d
Z
c
Gw.x; t /dt 1
d c
d
Z
c
w .t / f .t / dt:
Applying Gr¨uss Inequality and using Identity (2.5), we get that ˇ
ˇ ˇ ˇ ˇ ˇ
1
d c
d
Z
c
Gw.x; t /f 0.t /dt 1 .d c/2
d
Z
c
Gw.x; t /dt
d
Z
c
f 0.t /dt ˇ ˇ ˇ ˇ ˇ ˇ 1
4.U .c; d; x/ L .c; d; x// .B2 B1/ : But
U .c; d; x/ L .c; d; x/DV .c; d; x/ ; which implies that
ˇ ˇ ˇ ˇ ˇ ˇ 2 4 1 2
0
@
d
Z
c
w .t / dt 1
A.f .x/Cf .aCb x// I .c; d; x/H .c; d / 3 5
b
Z
a
w .t / f .t / dt ˇ ˇ ˇ ˇ ˇ ˇ 1
4V .c; d; x/ .B2 B1/ .d c/ :
Corollary 1. Let 1< c < d <1andfWŒc; d !R be a differentiable function on.c; d /. Suppose thatwis a weight function overŒc; d andwis symmetric about
cCd
2 : Iff 02L1Œc; d andB1f 0.t /B2, for allt 2Œc; d ;where B1; B2 are constants, then the inequality
ˇ ˇ ˇ ˇ ˇ ˇ 1 2
0
@
d
Z
c
w .t / dt 1
A.f .x/Cf .cCd x//
d
Z
c
w .t / f .t / dt ˇ ˇ ˇ ˇ ˇ ˇ
1 2max
8 ˆˆ
<
ˆˆ :
x
Z
c
w .t / dt;
cCd 2
Z
x
w .t / dt 9
>>
=
>>
;
.B2 B1/ .d c/
1 2
0 B B
@
cCd
Z2
c
w .t / dt 1 C C A
.B2 B1/ .d c/ : (2.6)
holds for allx2h
c;cC2di :
Proof. Note that fort2Œ0; 1andx2h
c;cC2di
, we have
min 8 ˆˆ
<
ˆˆ :
c
Z
x
w .t / dt;
x
Z
cCd 2
w .t / dt 9
>>
=
>>
;
Gw.x; t /max 8 ˆˆ
<
ˆˆ :
x
Z
c
w .t / dt;
cCd 2
Z
x
w .t / dt 9
>>
=
>>
; :
(2.7) Applying Lemma2, and following same argument used in the proof of Theorem
3, the result follows.
Remark2. If we setw .t /D1;Inequality2.6becomes same as Inequality1.6. Theorem 4. LetfWIR!R be a differentiable mapping onI0;the interior of the intervalI;and letc; d2I withc < d:Letwbe a weight function overŒc; d :If f 02L1Œc; d withB1f 0.t /B2for allt2Œc; d , whereB1; B2are constants, then for eachx2h
c;cC2di
, we have
ˇ ˇ ˇ ˇ ˇ ˇ 2 4 1 2
0
@
d
Z
c
w .t / dt 1
A.f .x/Cf .aCb x//
.B1CB2/
2 I .c; d; x/
Zd
c
w .t / f .t / dt ˇ ˇ ˇ ˇ ˇ ˇ
B2 B1
2
d
Z
c
jGw.x; t /jdt: (2.8)
whereI .c; d; x/D
d
R
c
Gw.x; t /dt:
Proof. Using Lemma1, we have
d
Z
c
Gw.x; t /
f 0.t / B1CB2
2
dt
D1 2
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
.B1CB2/ 2
d
Z
c
Gw.x; t /dt
d
Z
c
f .t / w .t / dt:
Note that, for eacht2Œc; d ; B1 B2
2 f 0.t / B1CB2
2 B2 B1
2 which implies that
max
t2Œc;d
ˇ ˇ ˇ ˇ
f 0.t / B1CB2
2 ˇ ˇ ˇ
ˇB2 B1
2 :
Thus,
ˇ ˇ ˇ ˇ ˇ ˇ
d
Z
c
Gw.x; t /
f 0.t / B1CB2
2
dt ˇ ˇ ˇ ˇ ˇ ˇ
max
t2Œc;d
ˇ ˇ ˇ ˇ
f 0.t / .B1CB2/ 2
ˇ ˇ ˇ ˇ
d
Z
c
jGw.x; t /jdt
.B2 B1/ 2
d
Z
c
jGw.x; t /jdt;
which implies that ˇ
ˇ ˇ ˇ ˇ ˇ 2 4 1 2
0
@
d
Z
c
w .t / dt 1
A.f .x/Cf .cCd x// .B1CB2/
2 I .c; d; x/
3 5
d
Z
c
w .t / f .t / dt ˇ ˇ ˇ ˇ ˇ ˇ
.B2 B1/ 2
d
Z
c
jGw.x; t /jdt:
Corollary 2. LetfWI R!R be a differentiable mapping onI0;the interior of the intervalI;and letc; d2I withc < d:Suppose thatwis a weight function over Œc; d andwis symmetric about cC2d:Iff 02L1Œc; d withB1f 0.t /B2for all t2Œc; d , whereB1; B2are constants, then for eachx2Œc; d , we have
ˇ ˇ ˇ ˇ ˇ ˇ 1 2
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
d
Z
c
f .t / w .t / dt ˇ ˇ ˇ ˇ ˇ ˇ
B2 B1
2
d
Z
c
jGw.x; t /jdt: (2.9)
Proof. Using Lemma2and applying Theorem4, the result follows.
Theorem 5. LetfWIR!R be a differentiable mapping on.c; d /I:Suppose that w is a weight function over Œc; d and w is symmetric about cC2d. If f 0 2 L1Œc; d withB1f 0.t /B2for allt2Œc; d , whereB1; B2are constants, then for eachx2h
c;cC2di
, we have
ˇ ˇ ˇ ˇ ˇ ˇ 1 2
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
d
Z
c
f .t / w .t / dt ˇ ˇ ˇ ˇ ˇ ˇ .d c/
f .d / f .c/
d c B1
sup
t2Œc;d jGw.x; t /j; (2.10) and
ˇ ˇ ˇ ˇ ˇ ˇ 2 4 1 2
d
Z
c
w .t / dt 3
5Œf .x/Cf .aCb x/
d
Z
c
f .t / w .t / dt ˇ ˇ ˇ ˇ ˇ ˇ .d c/
B2
f .c/ f .d /
d c
sup
t2Œc;d jGw.x; t /j: (2.11) Proof. Using Lemma2
b
Z
a
Gw.x; t /dtD0;
which implies that
d
Z
c
Gw.x; t /f 0.t / dtD
d
Z
c
Gw.x; t /dt f 0.t / B1 dt;
and
d
Z
c
Gw.x; t /f 0.t / dtD
d
Z
c
Gw.x; t /dt f 0.t / B2 dt:
Using Lemma1and the triangle inequality we get ˇ
ˇ ˇ ˇ ˇ ˇ 1 2
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
d
Z
c
f .t / w .t / dt ˇ ˇ ˇ ˇ ˇ ˇ
d
Z
c
ˇˇGw.x; t / f 0.t / B1ˇ ˇdt;
and
ˇ ˇ ˇ ˇ ˇ ˇ 1 2
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
d
Z
c
f .t / w .t / dt ˇ ˇ ˇ ˇ ˇ ˇ
d
Z
c
ˇˇGw.x; t / f 0.t / B2ˇ ˇdt:
Note that
d
Z
c
ˇˇGw.x; t /dt f 0.t / B1ˇ ˇdt
sup
t2Œc;d
jGw.x; t /j
d
Z
c
ˇ
ˇf 0.t / B1
ˇ ˇdt and
d
Z
c
ˇ
ˇf 0.t / B1
ˇ ˇdtD
d
Z
c
f 0.t / B1
dt Df .d / f .c/ B1.d c/
D.d c/
f .d / f .c/
d c B1
: Similarly,
d
Z
c
ˇˇGw.x; t / f 0.t / B2ˇ ˇdt
sup
t2Œc;d
jGw.x; t /j
d
Z
c
ˇˇf 0.t / B2ˇ ˇdt:
and
d
Z
c
ˇˇf 0.t / D2
ˇ ˇdtD
d
Z
c
B2 f 0.t / dt
DB2.d c/ .f .d / f .c//
D.d c/
B2
f .d / f .c/
d c
: Therefore,
ˇ ˇ ˇ ˇ ˇ ˇ 1 2
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
d
Z
c
f .t / w .t / dt ˇ ˇ ˇ ˇ ˇ ˇ .d c/
f .d / f .c/
d c B1
sup
t2Œc;d jGw.x; t /j; and
ˇ ˇ ˇ ˇ ˇ ˇ 1 2
2 4
d
Z
c
w .t / dt 3
5Œf .x/Cf .cCd x/
d
Z
c
f .t / w .t / dt ˇ ˇ ˇ ˇ ˇ ˇ .d c/
B2
f .d / f .c/
d c
sup
t2Œc;d
jGw.x; t /j:
3. SOME APPLICATIONS
Before introducing the first two applications, recall that a tagged partitionP of a finite intervalŒc; d is a finite sequence of numberscDx0< x1< < xnDd, with corresponding tagsti 2Œxi 1; xi, foriD1; : : : ; n.
Theorem 6. Let 1< c < d <1andfWŒc; d !Rbe a differentiable function on.c; d / ;andPWcDx0< x1< < xnDdbe a tagged partition with correspond- ing valuesti 2Œxi 1;xi 12Cxi, foriD1; : : : ; n. Suppose thatwis a weight function overŒc; d . Iff 02L1Œc; d andB1f 0.t /B2, for allt 2Œc; d ;whereB1; B2
are constants, then we have the quadrature formula
d
Z
c
w .t / f .t / dtDA .f; P /CR .f; P / ;
where
A .f; P /D
n 1
X
iD0
1
2W .xi; xiC1/ .f .tiC1/Cf .xiCxiC1 tiC1//
I .xi; xiC1; tiC1/H .xi; xiC1/ ;
W .xi; xiC1/D
xiC1
Z
xi
w.t /dt; 0in 1;
I .xi; xiC1; tiC1/D
xiC1
Z
xi
Gw.tiC1; t /dt; 0i n 1;
H .xi; xiC1/Df .xiC1/ f .xi/ xiC1 xi
; 0in 1;
and the remainder satisfies the inequality
jR .f; P /j .B2 B1/ 4
n 1
X
iD0
V .xi; xiC1; tiC1/ .xiC1 xi/ ; where
V .xi; xiC1; tiC1/Dmax 8 ˆ<
ˆ:
xiCxiC1 tiC1
Z
tiC1
w .t / dt; W .xi; xiC1/
xiCxiC1 tiC1
Z
tiC1
w .t / dt 9
>=
>; Proof. For each0i n 1;applying Theorem3onŒxi; xiC1withxDtiC1; we get that
ˇ ˇ ˇ ˇ ˇ ˇ
xiC1
Z
xi
w .t / f .t / dt 1
2W .xi; xiC1/ .f .tiC1/Cf .xiCxiC1 tiC1//
I .xi; xiC1; tiC1/H .xi; xiC1/
ˇ ˇ ˇ ˇ ˇ ˇ .B2 B1/
4 V .xi; xiC1; tiC1/ .xiC1 xi/ : Using the triangle inequality, we find that
ˇ ˇ ˇ ˇ ˇ ˇ
n 1
X
iD0
0
@
xiC1
Z
xi
w .t / f .t / dt 1
2W .xi; xiC1/ .f .tiC1/Cf .xiCxiC1 tiC1//
I .xi; xiC1; tiC1/H .xi; xiC1/
1 A ˇ ˇ ˇ ˇ ˇ ˇ
.B2 B1/ 4
n 1
X
iD0
V .xi; xiC1; tiC1/ .xiC1 xi/ :
But
n 1
X
iD0 xiC1
Z
xi
w .t / f .t / dt D
d
Z
c
w .t / f .t / dt;
which implies that
d
Z
c
w .t / f .t / dt DA .f; P /CR .f; P / and
jR .f; P /j .B2 B1/ 4
n 1
X
iD0
V .xi; xiC1; tiC1/ .xiC1 xi/ :
Theorem 7. LetfWIR!R be a differentiable mapping onI0;the interior of the intervalI; c; d 2I withc < d, andP WcDx0< x1< < xnDd be a tagged partition with corresponding valuesti 2Œxi 1;xi 12Cxi, fori D1; : : : ; n. Suppose thatw is a weight function overŒc; d :Iff 02L1Œc; d withB1f 0.t /B2 for allt2Œc; d , whereB1; B2are constants, then we have the quadrature formula
d
Z
c
w .t / f .t / dtDA .f; P /CR .f; P / ; where
A .f; P /D
n 1
X
iD0
1
2W .xi; xiC1/ .f .tiC1/Cf .xiCxiC1 tiC1// BI .xi; xiC1; tiC1/
;
W .xi; xiC1/D
xiC1
Z
xi
w.t /dt; ; 0in 1;
i .xi; xiC1; tiC1/D
xiC1
Z
xi
Gw.tiC1; t /dt; ; 0in 1;
BDB1CB2
2 ;
and the remainder satisfies the inequality
jR .f; P /j .B2 B1/ 2
n 1
X
iD0
Z .xi; xiC1; tiC1/ ;
where
Z .xi; xiC1; tiC1/D
xiC1
Z
xi
jGw.tiC1; t /jdt:
Proof. For each0in 1;applying Theorem4onŒxi 1; xiwithxDtiC1; we get that
ˇ ˇ ˇ ˇ ˇ ˇ
xiC1
Z
xi
w .t / f .t / dt 1
2W .xi; xiC1/ .f .tiC1/Cf .xiCxiC1 tiC1//
BI .xi; xiC1; tiC1/
ˇ ˇ ˇ ˇ ˇ ˇ
.B2 B1/ 2
xiC1
Z
xi
jGw.tiC1; t /jdt
D.B2 B1/
2 Z .xi; xiC1; tiC1/ : Using the triangle inequality, we find that
ˇ ˇ ˇ ˇ ˇ ˇ
n 1
X
iD0 xiC1
Z
xi
w .t / f .t / dt 1
2W .xi; xiC1/ .f .tiC1/Cf .xiCxiC1 tiC1//
BI .xi; xiC1; tiC1/
ˇ ˇ ˇ ˇ ˇ ˇ
.B2 B1/ 2
n 1
X
iD0
Z .xi; xiC1; tiC1/ : But
n 1
X
iD0 xiC1
Z
xi
w .t / f .t / dt D
d
Z
c
w .t / f .t / dt and
n 1
X
iD0 xiC1
Z
xi
jGw.x; t /jdt D
d
Z
c
jGw.x; t /jdt;
which implies that
d
Z
c
w .t / f .t / dt DA .f; P /CR .f; P / and
jR .f; P /j .B2 B1/ 2
n 1
X
iD0
Z .xi; xiC1; tiC1/ :
Before we introduce the next application, recall that ifX is a random variable with values in a finite intervalŒc; d ,c < d;andf WŒc; d !Œ0; 1is a probability density function, then the cumulative distribution function with respect tof is denoted byF and is defined as:
F .x/D Z x
c
f .t / dt for eachx2Œc; d : Sincef satisfies the conditionRd
c f .x/ dxD1; F .c/D1:
Theorem 8. LetXbe a random variable with values in a finite intervalŒc; d ,c <
d;andf WŒc; d !Œ0; 1be a probability density function. Letwbe a differentiable weight function overŒc; d such thatw .d /D1andwbe symmetric about cC2d:Let F be the cumulative distribution function with respect tof . Iff 2L1Œc; d and B1f .t /B2, for allt2Œc; d ;whereB1; B2are constants, then the inequality
ˇ ˇ ˇ ˇ ˇ ˇ 1 2
0
@
d
Z
c
w .t / dt 1
A.F .x/CF .cCd x// .d EG/ ˇ ˇ ˇ ˇ ˇ ˇ
1 4
2 6 6 4
cCd 2
Z
c
w .t / dt 3 7 7 5
.B2 B1/ .d c/ : (3.1)
holds for allx2h
c;cC2di
;where EGDd
Z d c
w .t / F .t / dt:
Proof. Define the functionG overŒc; d as follows:
G .x/D Z x
c
d
dt.wF / dt; x2Œc; d : Note that
G .c/D0 and
G .d /D Z d
c
d
dt .wF / dt
Dw .d / F .d / w .c/ F .c/
D1:
Let
EGD Z d
c
t d
dt.wF / dt:
Using integration by parts, we get that
EGDdw .d / F .d / cw .c/ F .c/
Z d c
w .t / F .t / dt Dd
Z d c
w .t / F .t / dt;
which implies that
Z d c
w .t / F .t / dtDd EG: Applying Corollary1onF, we get that
ˇ ˇ ˇ ˇ ˇ ˇ 1 2
0
@
b
Z
a
w .t / dt 1
A.F .x/CF .cCd x//
d
Z
c
w .t / F .t / dt ˇ ˇ ˇ ˇ ˇ ˇ
1 4
2 6 6 4
cCd 2
Z
c
w .t / dt 3 7 7 5
.B2 B1/ .d c/ ;
which implies that ˇ ˇ ˇ ˇ ˇ ˇ 1 2
0
@
d
Z
c
w .t / dt 1
A.F .x/CF .cCd x// .d EG/ ˇ ˇ ˇ ˇ ˇ ˇ
1 4
2 6 6 4
cCd 2
Z
c
w .t / dt 3 7 7 5
.B2 B1/ .d c/ : (3.2)
REFERENCES
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Authors’ addresses
S. Obeidat
Department of Basic Sciences, University of Hai’l, Saudi Arabia E-mail address:obeidatsofian@gmail.com
M. A. Latif
Department of Basic Sciences, University of Hai’l, Saudi Arabia E-mail address:muhammad.mlatif@gmail.com
A. Qayyum
Department of Mathematics, Institute of Southern Punjab, Pakistan E-mail address:atherqayyum@gmail.com
