Second Derivatives Belong to Lp (A,B) and Applications
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Cerone, Pietro, Dragomir, Sever S and Roumeliotis, John (1998) An Ostrowski Type Inequality for Mappings whose Second Derivatives Belong to Lp (A,B) and Applications. RGMIA research report collection, 1 (1).
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RGMIA Research Report Collection, Vol. 1, No. 1, 1998
http://sci.vut.edu.au/ rgmia/reports.html
AN OSTROWSKI TYPE INEQUALITY FOR
MAPPINGS WHOSE SECOND DERIVATIVES
BELONG TO L
P
(AB) AND APPLICATIONS
P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS
Abstract. An inequality of the Ostrowski type for twice dierentiable map- pings whose derivatives belong toLp(ab) (p>1) and applications in Numerical Integration are investigated.
1 Introduction
The following inequality is well known in the literature as Ostrowski's integral inequality (see for example 1, p. 468])
Theorem 1.1.
Letf :IR
!R
be a dierentiable mapping onI(Iis the interior of I) and let ab 2 I with a< b:If f0 : (ab)!R
is bounded, i.e.,kf 0
k
1:= sup
t2(ab)jf
0(t)j<1 then we have the inequality:
f(x); 1
b;a b
Z
a
f(t)dt
"
14 +
;
x; a+2b
2
(b;a)2
#
(b;a)kf0k1 (1.1)
for allx2(ab):
The constant 14 is the best possible.
For a simple proof and some applications of Ostrowski's inequality to some special means and some numerical quadrature rules, we refer the reader to the recent paper 2] by S.S. Dragomir and A. Wang.
In 3], the same authors considered another inequality of Ostrowski type for
kk
p
;norm (p>1) as follows:
Theorem 1.2.
Let f : IR
!R
be a dierentiable mapping on I andab 2I
with a < b. If f0 2Lp(ab) p>11p +1q = 1 then we have the inequality:
f(x); 1
b;a b
Z
a
f(t)dt 1
b;a
"
(x;a)q+1+ (b;x)q+1
q+ 1
#1
q
kf 0
k
(1.2) p
Date.November, 1998
1991 Mathematics Subject Classication. Primary 26D15 Secondary 41A55.
Key words and phrases.Ostrowski's Inequality, Numerical Integration.
43
for allx2ab], where
kf 0
k
p :=
0
@ b
Z
a
jf(t)jpdt
1
A 1
p
is the Lp(ab);norm.
They also pointed out some applications of (1:2) in Numerical Integration as well as for special means.
In 1976, G.V. Milovanovic and J.E. Pecaric proved a generalization of Os- trowski inequality for n;times dierentiable mappings (see for example 1, p.
468]). The case of twice dierentiable mappings 1, p. 470] is as follows:
Theorem 1.3.
Let f : ab]!R
be a twice dierentiable mapping such thatf
00 : (ab)!
R
is bounded on (ab) i.e., kf00k1 := supt2(ab)jf
00(t)j <1: Then we have the inequality:
12
f(x) + (x;a)f(a) + (b;x)f(b)
b;a
;
1
b;a b
Z
a
f(t)dt (1.3)
kf
00
k
4 (1 b;a)2
2
4 1 12 +
x; a+2b2
(b;a)2
3
5
for allx2ab]:
In this paper, we point out an inequality of Ostrowski type for twice dier- entiable mappings which is in terms of thekkp-norm of the second derivative
f
00 and apply it in Numerical Integration.
2 Some Integral Inequalities
The following inequality of Ostrowski type for mappings which are twice dier- entiable, holds:
Theorem 2.1.
Letf : ab]!R
be a twice dierentiable mapping on(ab) andf 00
2L
p(ab) (p>1):Then we have the inequality:
f(x); 1
b;a b
Z
a
f(t)dt;
x; a+b
2
f 0(x) (2.1)
2(b;a)(21 q+ 1)1q
h(x;a)2q+1+ (b;x)2q+1i1qkf00kp
(b;a)1+1qkf00kp 2(2q+ 1)1q for allx2ab]where 1p +1q = 1:
Ostrowski for Bounded Second Derivative Mappings in Lp(ab) 45
Proof. Let us dene the mappingK() : ab]2!
R
given byK(xt) :=
8
>
>
>
<
>
>
>
:
(t;a)2
2 ift2ax] (t;b)2
2 ift2(xb]
:
Integrating by parts, we have successively,
b
Z
a
K(xt)f00(t)dt=
x
Z
a
(t;a)2
2 f00(t)dt+
b
Z
x
(t;b)2
2 f00(t)dt
= (t;a)2 2 f0(t)
x
a
; x
Z
a
(t;a)f0(t)dt+ (t;b)2 2 f0(t)
b
x
; b
Z
x
(t;b)f0(t)dt
= (x;a)2
2 f0(x);
2
4(t;a)f(t)jxa;
x
Z
a
f(t)dt
3
5
;
(b;x)2
2 f0(x);
2
4(t;b)f(t)jbx;
b
Z
x
f(t)dt
3
5
= 12h(x;a)2;(b;x)2if0(x)
;(x;a)f(x) +
x
Z
a
f(t)dt+ (x;b)f(x) +
b
Z
x
f(t)dt
= (b;a)
x; a+b
2
f
0(x);(b;a)f(x) +
b
Z
a
f(t)dt from which we get the integral identity
b
Z
a
f(t)dt= (b;a)f(x);(b;a)
x; a+b
2
f 0(x) +
b
Z
a
K(xt)f00(t)dt (2.2)
for allx2ab]:
RGMIA Research Report Collection, Vol. 1, No. 1, 1998
Using (2:2), we have, by Holder's integral inequality, that
f(x); 1
b;a b
Z
a
f(t)dt;
x; a+b
2
f 0(x) (2.3)
= 1
b;a b
Z
a
K(xt)f00(t)dt 1
b;a 0
@ b
Z
a K
q(xt)dt
1
A 1
q
kf 00
k
p
= 1
b;a 2
4 x
Z
a
(t;a)2q 2q dt+
b
Z
x
(t;b)2q 2q dt
3
5 1
q
kf 00
k
p
= 1
b;a
"
(x;a)2q+1
2q(2q+ 1) +(b;x)2q+1 2q(2q+ 1)
#1
q
kf 00
k
p
= 1
2(b;a) 1 (2q+ 1)q1
h(x;a)2q+1+ (b;x)2q+1i1qkf00kp
and the rst inequality in (2:1) is proved. The second inequality is obvious taking into account that
(x;a)2q+1+ (b;x)2q+1(b;a)2q+1 for allx2ab]:
The following particular case for euclidean norms is interesting
Corollary 2.2.
Letf : ab]!R
be as above andf002L2(ab):Then we have the inequality:f(x); 1
b;a b
Z
a
f(t)dt;
x; a+b
2
f 0(x) (2.4)
(b;a)32 2
"
80 +1 1 2
;
x; a+2b
2
(b;a)2 +
;
x; a+2b
4
(b;a)4
# 1
2
kf 00
k2:
Proof. Apply inequality (2:1) for p=q= 2to get
f(x); 1
b;a b
Z
a
f(t)dt;
x; a+b
2
f 0(x) (2.5)
Ostrowski for Bounded Second Derivative Mappings in Lp(ab) 47
2(b;1a)p5
h(x;a)5+ (b;x)5i12kf00k2: Denote t:=x;a+2b:Then
x;a=t+b;2ab;x= b;2a;t:
Let us compute
I := (x;a)5+ (b;x)5=
t+b;2a
5
+
b;a
2 ;t
5
:
We know that, for numbersAB2
R
,we haveA
5+B5= (A+B);A4;A3B+A2B2;AB3+B4
= (A+B)A4+B4;AB;A2+B2+A2B2
= (A+B)h;A2+B22;A2B2;AB;A2+B2i: Now, if we put A:=t+b;a2 B:= b;a2 ;tthen we get
A2+B2= 2t2+ (b;a)2
2 AB= (b;a)2 4 ;t2 and then
J :=;A2+B22;A2B2;AB;A2+B2
=
"
2t2+ (b;a)2 2
#2
;
"
t2;
(b;a)2 4
#2
;
"
(b;a)2 4 ;t2
#"
2t2+ (b;a)2 2
#
= 5t4+ 52 (b;a)2t2+ (b;a)4 16 = 5
"
t4+ 2
b;a
2
2
t2+ 15
b;a
2
4#
:
Consequently,
I = (b;a)
"
5
x; a+b
2
4
+ 52 (b;a)2
x; a+b
2
2
+ (b;a)4 16
#
= 5(b;a)5
"
80 +1 1 2
;
x; a+2b
2
(b;a)2 +
;
x; a+2b
4
(b;a)4
#
:
Finally, using the inequality (2:5), we get the desired result (2:4)
Remark 2.1.
Let f : ab]!R
be as above . Then we have the midpoint inequality:f
a+b 2
;
1
b;a b
Z
a
f(t)dt 1
8(2q+ 1)1q (b;a)1+1qkf00kp: (2.6)
RGMIA Research Report Collection, Vol. 1, No. 1, 1998
Taking into account the fact that the mapping
h: ab]!
R
h(x) = (x;a)2q+1+ (b;x)2q+1 has the property thatinf
x2ab]h(x) =h
a+b 2
= (b;a)2q+1 22q and
sup
x2ab]h(x) =h(a) =h(b) = (b;a)2q+1
then, the best estimation we can get from (2:1) is that one for whichx= a+2b obtaining the inequality (2:6):
Remark 2.2.
If in(2:1) we choose x=awe getf(a); 1
b;a b
Z
a
f(t)dt+b;2af0(a) (b;a)1+1q 2(2q+ 1)1q kf
00
k
p
and puttingx=bwe also get
f(b); 1
b;a b
Z
a
f(t)dt;b;2af0(b) (b;a)1+1q 2(2q+ 1)1q kf
00
k
p :
Summing the above two inequalities, using the triangle inequality and dividing by 2, we get the perturbed trapezoid formula
f(a) +f(b)
2 ;
b;a
4 (f0(b);f0(a)); 1
b;a b
Z
a
f(t)dt (b;a)1+1q 2(2q+ 1)1q kf
00
k
p :
(2.7)
Remark 2.3.
Ifp=q= 2, then we get for the euclidean norm, from (2:6)f
a+b 2
;
1
b;a b
Z
a
f(t)dt
p5(b;a)32 40 kf00k2 (2.8)
and, from(2:7)
f(a) +f(b)
2 ;b;4 (a f0(b);f0(a)); 1
b;a b
Z
a
f(t)dt (2.9)
p5(b;a)32
10 kf00k2:
Ostrowski for Bounded Second Derivative Mappings in Lp(ab) 49
3 Applications in Numerical Integration
Let In : a = x0 < x1 < ::: < xn;1 < xn = b be a division of the interval,
i
2xixi+1] (i= 0:::n;1):We have the following quadrature formula:
Theorem 3.1.
Let f : ab]!R
be a twice dierentiable mapping on (ab) whose second derivativef00: (ab)!R
belongs toLp(ab) (p>1) i.e.,kf 00
k
p:=
0
@ b
Z
a jf
00(t)pjdt
1
A 1
p
<1:
Then the following perturbed Riemann type quadrature formula holds:
b
Z
a
f(x)dx=A(ff0 In) +R(ff0 In) (3.1)
whereA(ff0 In) is given by
A(ff0 In) :=n;X1
i=0
h
i
f(i);n;X1
i=0
f 0(i)
i
; x
i+xi+1
2
h
i
and the remainder satises the estimation:
jR(ff0 In)j (3.2)
2(2q1+ 1)1q
n;X1
i=0(i;xi)2q+1+ (xi+1;i)2q+1
! 1
q
kf 00
k
p
2(2q+ 1)1 1=q
n;X1
i=0
h
2q+1
i
! 1
q
kf 00
k
p
for alli 2xixi+1] (i= 0:::n;1):
Proof. Apply inequality (2:1) on the interval xixi+1] (i= 0:::n;1) to get
f(i)hi;
x
i+1
Z
xi
f(t)dt;
i
; x
i+xi+1
2
f 0(i)hi
2(2q+ 1)1 1=q
h(i;xi)2q+1+ (xi+1;i)2q+1i1q
0
@ x
i+1
Z
xi jf
00(t)jpdt
1
A 1
p
for alli2f0:::n;1g:
RGMIA Research Report Collection, Vol. 1, No. 1, 1998
Summing over i from 0 to n;1 using the generalized triangle inequality and Holder's discrete inequality, we get:
jR(ff0 In)j
n;X1
i=0
f(i)hi;
i
; x
i+xi+1
2
f
0(i)hi;
xi+1
Z
x
i
f(t)dt
2(2q+ 1)1 1=q
n;X1
i=0
h(i;xi)2q+1+ (xi+1;i)2q+1i1q
0
@ xi+1
Z
xi jf
00(t)jpdt
1
A 1
p
2(2q+ 1)1 1=q
n;1
X
i=0
h(i;xi)2q+1+ (xi+1;i)2q+1i1q
q
! 1
q
0
B
@ n;1
X
i=0
0
B
@ 0
@ x
i+1
Z
xi jf
00(t)jpdt
1
A 1
p 1
C
A p
1
C
A 1
p
= 1
2(2q+ 1)1=q
n;X1
i=0
h(i;xi)2q+1+ (xi+1;i)2q+1i
! 1
q
kf 00
k
p
and the rst inequality in (3:2) is proved.
The last part is obvious from the fact that
(i;xi)2q+1+ (xi+1;i)2q+1h2iq+1 for alli2f0:::n;1g:
Now, if we consider the midpoint formula
M(fIn) :=n;X1
i=0
f
x
i+xi+1
2
h
i
then we have
b
Z
a
f(t)dt=M(fIn) +R(fIn) (3.3)
and the remainderR(fIn) can be estimated in terms of thep;norm of f00as follows:
jR(fIn)j 1 8(2q+ 1)1=q
n;1
X
i=0
h
2q+1
i
! 1
q
kf 00
k
(3.4) p
Ostrowski for Bounded Second Derivative Mappings in Lp(ab) 51
which is, in a certain sense, the best estimation we can obtain from (3:2): Also, we can construct the following perturbed trapezoid formula
T
p(ff0In) := 12n;Xi=01
f(xi) +f(xi+1)
2 hi+ 14n;Xi=01
h2
i(f0(xi);f0(xi+1)): Then we have
b
Z
a
f(t)dt=Tp(ff0In) +Rp(ff0In) (3.5)
and the remainder can be estimated (see the inequality (2:7) ) as follows:
jR
p(ff0In)j 1 2(2q+ 1)1=q
n;1
X
i=0
h
2q+1
i
! 1
q
kf 00
k
p
(3.6) :
Remark 3.1.
To derive the corresponding results for the euclidean normkf00k2, we put in the above p=q= 2:We omit the details.
Remark 3.2.
The reader can obtain the corresponding quadrature formulae for equidistant partitioning by choosingxi=a+ib;an (i= 0:::n;1):Remark 3.3.
If we consider equidistant partitioning ofab] then the perturbed trapezoid formula we considered above will involve the calculation forf0 only at the endpoints aandb , which is a good advantage for practical applications.RGMIA Research Report Collection, Vol. 1, No. 1, 1998
References
1] D.S. MITRINOVIC, J.E. PECARIC and A.M. FINK, Inequalities for Func- tions and Their Integrals and Derivatives, Kluwer Academic, Dordrecht, 1994.
2] S.S. DRAGOMIR and S. WANG, Applications of Ostrowski's inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett.,
11
(1998), 105-109.3] S.S. DRAGOMIR and S. WANG A new inequality of Ostrowski's type in
L
p
;norm, submitted.
School of Communications and Informatics, Victoria University of Technology, PO Box 14428, Melbourne City MC, Victoria 8001, Australia.
E-mail address: fpc, sever, johnrg@matilda.vut.edu.au
