Vietnam J Math (2014) 42-141-151 DOI 10.1007/s 10013-013-0038-y
Some Weighted Ostrowsid Type Inequalities
Zheng Liu
Received. 12 May 2013 / Accepted 17 August 2013 /Pubhshed online 4 October 2013
© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract Some weighted Ostrowski type inequalities are estabhshed via the weighted Montgomery identity. A generalized weighted Montgomery identity is also considered, and thus a different proof of an Ostrowski type mequality is presented.
Keywords Montgomery identity • Ostrowski inequality • Weighted Montgomery identity • Weighted Ostrowski type inequality
Mathematics Subject Classification (2010) 26D15
1 Introduction
The following result is known as the Ostrowski inequality [9].
Theorem 1 Let f : [a.b] -s- K i^e differentiable on (a.b). whose derivative f : (a.b) —* R is bounded on (a.b), i.e., ||/'||oo •=sup,£(u,(,) |/'(r)l <-Hoc. Then tve have the inequality
for all X e [a, b] The constant | is Ihe best possible in the sense tliat it cannot be replaced by a smaller one.
A simple proof of diis dieorem can be given by using the Montgomery identity ([8, Chap, XVIIL p. 565]),
-j f{t)dt + j^ P{x.t)f'{t)dt,
Z Liu ( ^ )
Institute of Apphed Mathematics. School of Science. University of Science and Technology Liaoning, Anshan 114051, Liaoning. China
e-mail. Iewzheng@163 net
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142 Z. Lm where P{;i:, f) is the Peano kernel, defined by
I
T=T, a<t<x,(3) i ^ , x<t<b
Moreover, by applying the Montgomery identity (2) for / ' ( - ) , one can prove the following Ostrowski type inequality for mappings whose second derivatives are bounded as in [5, Theorem 17] or [6, Theorem 2.1].
Theorem 2 Let / : [a, ft] ^ - H be twice differentiable on {a, b), whose second derivative f" : (a, b) -^ R IS bounded on (a, b), i.e., \\f"\\co '•= su-p,^f^i,j \f"(T)\ <:-|-oo Thenwehave the inequality
. ^ l i r L
for all X e [a.b]
In [3], we see that the first mam result presented by Anastassiou is the following Os- trowski type inequahty.
Theorem 3 Let f [a,b]^R be three limes differentiable on (a.b), whose third derivative f" (a, b)^R IS bounded on (a, b), i.e., ||/'"|loc '- sup,^(„^j,, \f"'{t)\ < -hoc. Then -we have the inequality
2(b-a) - (h-a) for every X ^\a,h\ where
b~aj, b-a \ 2 /'(i.)-/'(a)r/ a + bV (b-^V
I-AM
AU):= abx' - ''-aVx + ifl'ftx'- - ab'x' - ^-aVx + -ab^x' + aVx'
.\aV-hV-
3 3
-g Sprii
Some Weighted Ostrowski Type Inequalities The inequality (5) is attained by
f(x) = ix-a)'' + ib-x)\
m that case both sides ofthe inequality equal zero.
In [31 inequality (5) was proved by using a generalization of Montgomery identity as
/w-^r/(.,).,i-^<^^^/%(.,-,)..
a -a Jg tl -a J a
- £ < ' | ^ ^ / T p ( . . , i ) P ( r i , « . M . b-a J^L
P(X,tOPiti,r2}PU2.t2)f"'(t3)dlydt2dti and the error bound (6) is given by using Mathematica-4 and Maple-6,
Now, we suppose w : [a, b] —»• [0, oo) is some probability density function, i e.. an inte- grable function satisfying/j'i(;(Oi^? = L a n d W(l) = f^w(x)dx forte la,b], W(0 =^ 0 for t < a, and W(l) = 1 for r > ft. In [2] we see the following weighted generalization of the Montgomery identity
f(x)= f w(t)f{t)dt+ [ PM,l)f'(t)dt. (7) where the weighted Peano kernel is
I
W(t). a<t<x.- - (8) W(t)-l. x<t<b.
The purpose of this paper i,s to give some weighted generalization of inequalities (1) and (4) via the weighted Montgomery identity (7), and a further generalization of the weighted Montgomery identity is also considered which leads to provide a different proof of the inequality (5) with a neat expression of the error bound.
2 Weighted Generalization of Inequalities (1) and (4)
We first provide the following weighted Ostrowski type inequality for functions, whose derivauves are bounded
Theorem 4 Lel f :[a,b]^R be differentiable on (a. b). whose derivative f ; (a.b) -» M is hounded on (a.b), i.e., | | / ' I U = sup,^,,,;,, | / ' ( r ) | < +oo. Ifw : [a. b] -^ [0, co) is some probability density function, tlien we have the inequality
\f(x) - r w{t)fit)dt\ < U \x - t\w(t)dt^ | | / ' | | ^ f By (7) and (8). we get
|/(.v)- f woyfiOdtl^K"pux.t)f(ndt\<\[ iPAx.Dldt^Wf'W^
I J„ \ \Jii I '-•'" -I
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I \PM,')\dt= r W{t)dt+ j [\-W{t)]dt
^ I \ I w{u)du\dt + \ w(u)du\dt
— / (x-u)w{u)du+ I {u—x)w(u)du
- f \x-t\w(i)dt
by interchanging the order of integration on the third line. Consequently, the inequality (9)
follows, • Remark 1 It is clear that for ui(;) = ^ , t e[a, fo], the inequality (9) reduces to the inequal-
ity (1), and so Theorem 3 can be regarded as a generalization of Theorem 1.
Now, we consider the weighted Ostrowski type inequality for mappings whose second denvatives are bounded.
Theorem 5 Let f . [a.b] —> R be twice differentiable on (a,b). whose second deriva- tive f" : ia.b) -* R is bounded on (a.b), i.e., ||/"||,o : = sup,^,^ ^) | / " ( 0 | < H-oo. //
w:[a.b]—^ [0, oo) is some probability density function, then we have the inequality / ( A : ) - /" w(t)f(t)dt- f (x-t)w(t)dt I w{t)f'(t)dt\
\t -s\w(s)w{u)dsdtdu
+ fff \!-s\w(s)w(u)dsdtdu'\\\f"\\^. (10)
Proof Applying the weighted generalization of the Montgomery identity (7) for /'(•) we can state
f'{t)=j w(s)f'(s)ds+ j P,,(t.s)f"(s)ds.
Substituung / ' { / ) in the right hand side of (7) we get
/ ( . v ) - y wU)f(t)dt+j P„.(x,t)\ j w(s)f'(s)ds+ f P,,(t.s)f"(s)ds]dl
= j w(t)f(t]dt +j PAx.Ddt f w(s)f'(s)ds + j j PAx.t)P^(t.s)f"{s)dsdt
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Some Weighted Ostrowski Type Inequaliti,
= / l«l.t)f(l)dl +j (x-l-)w(t)dt j w{t)f\t)dl + [] PM.l)PUl,s)f"Wdsdl and
j j \F^(.x,t)\\P,(t.s)\dsdt
= j \P,<.x,l)\\f \t-s\w(s)ds\d,
= J " ' ( ' ) / ' \t-s\w(s)ds]dl+ I [l~W(t)]\f |/-s|i»(s)<Jjl<;r
= / / u)(u)du\ \t-s\u>(s)ds\dl+ j j ui{u)du\ j [r-s|i«(s)iisl<<r
= \t-s\w(s)iii(u)dsdrdu+ j I j tl-s\ii:(s)w(u}dsdldu.
by interchanging the order of integration on the fourth line. Consequently, die inequahty (10)
follows. D Remark 2 If we take U)(r) — ^ , r e [c.fc], then
j I j ll-s\w(s)u>(ii)dsdldu+ k j I |(-i|i»(i)uj(u)rfi<i/rfi<
2 IL ( 6 - 0 ) ' 4 j ^ 1 2 )
We see that the inequality (4) is recaptured in a different way and hence Theorera 4 may be legaitled as a generalization of Theorem 2.
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146
3 Further Generalization
We first state and prove the following generalized weighted Montgomery identity.
Theorem 6 Let / : [a, A] ->- R i e n times differentiable on [a, b] (n € N), whose nth deriva- tive / ' " ' : [a, b]^}-R is mtegrable on [a, b]. Ifw : [a.b] -»• [0, oo) is some probability den- sity function, then for all x e [a, b] we have the identity:
f{x)=j w{li)f(t,)dlt-¥j^t w(t,)f^'\t,)dt.
Ja ,^, Ja
^ I •• I Pu,ix,tOPUtuT2)---PM--u',)dtx---dti
+ 1 ---j Pw(x.t,)P,,(t,.t2)---P^(l„-l.l„)f'"Ht„)dtj---dt„, ( I I ) where P^,(-. -) isdefinedin (8)
Proof Let us prove by mathematical induction.
For « — 1. (11) is just the same as the weighted Montgomery identity (7).
Assume that (11) holds for a natural numbers and let us prove it for/i-|- 1, i.e., we have to prove the identity:
fU)^j Mtt)f(ti)di,-i-J2( w(t,)f''\t,)dt,
X / ••• / Pw(x,tj)P^(li.t2)---P^(t,_i.li)dt,-- dt,
+ j j P^(j:,fi)---Pu,(;„_,,/„)P^(r„,(„+,)/'"+"(Wi)dri---rf/„+i (12) Using the weighted Montgomery identity (7) we can state that
f"\tn)^ j lt.(r„+|)/""(;„ + , ) ^ W l + j P.,:(tn,tn+\)f"^^\t„+y)dt„ + y
- y ui(t„)f\i„)dt„+j P ^ ( r „ , W i ) / " ' + " ( r „ + i ) d W ] - By the mathematical induction hypothesis, we get
Mti)f(ii)dtf+J2 w(t,)f"Ut,)dt.
^ 1 ' " i ^"'^•^•'i"''"('i-'2>---^"'(''-i-'')'^'i---'^'.-
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Some Weighted Ostrowski Type Inequalities
+ f ---[ /'^(Jt.f,) ••P,(r„_,,r„)
x [ / " w(t„)f'">(t„)di„+ f P^(t„,t„+i)f'''+^>(t„+i)dt„+Adti-- dt„
= I w(tt)f(ti)du+Y,l w(t,)f'\t.)dl,- Ja ,^, Ja
X / -- j Pu>(x,U)P^(tul2)---PAt,-t,t,)dtt---dt,
+ I •- f PAX,li)---PA!n-l,tn)Pu,(tn,ln+l)f''^'H'n+l)dt,---dl„+u I.e., identity (12) holds, and identity (11) is thus proved.
Denote
R„(w,f,x):= f I PAx,ti) • - - P,^(l„.u'n)f*"Ht„)dti - - -di„
We are interested in poinUng out some upper bounds for the absolute value of R„(w. f. x),
xe[a,b]. D The following general result holds.
Theorem 1 Let f : [a,b] -^Rbe n times differentiable on [a.b], whose nth derivative / ' " ' : {a,b) -^ E is bounded on (a.b), ie., ||/<"'IU •= sup,^,^^, |/'"'(f)l <-Hoc, Ifw : [a.b]-^ [0, oo) is some probability density function, then for all x &[a,b] one has the estimate:
| « . ( i " , / . « ) | < ' „ ( i » , ' ) | | / " " I L . (13) where
l,{w,x)= I I | P „ ( ; t , i , ) | | / > „ ( r , , i 2 ) | . - | P , ( i . - i . / „ ) | < / ' i ' - r f l . (14) and
' . - i ( i " . ' » ) = / |/'..(ri.'n-i)l'—i-i(«'-'i+i)<"n-i
= / T r ; , - , - i ( i « , I , + i ) * i + i \vif)du
+ [ \ l '.-1-1(1".'»+i)'"l+i «'(")«'" < " ' fork = 0 /( - 1 witli IQ = X and
; i ( i » . ( . - , ) = f | P , ( / , - i . t . ) | r f ' . = f l l . - i - ' . l u ' C . ) ' ' ' .
= / | ( , - i - « | i « ( « ) r f " . (16)
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Proof It is immediate by mathematical induction. Q Remarks A similar resuh ofthe generalized weighted Montgomery identity (11) has been
considered previously in [3, Theorem 2] and [1, Theorem 9]. The special case for w(t) =
^ , t s [a,b] has also been considered in [4, Theorem 3], However, (13)-(16) provide a convement and simple way to estimate the remainder. We will show it in proving the following theorem which IS just Theorem 3 with a neat expression of the error bound Theorems Let f :[a,b]~^ R be three times differentiable on (a, b), whose third derivative f" : (a, &) -» E is bounded on (a. b), i.e., ||/"'|[co :^ sup,g,^ ^11/"'(0[ < +co Then we have the inequality
U,--^rmd.-^^q^^(x-'-±^)
I b-a J^ b-a \ 2 ) r(b)-f{a)\( a + bS^ tb-
- - » ) ' 1 |12 JI
<l^,^,x, + ls'(x, + l3\xi + ^]ib-anrL
for every X E [a, b], where
b-a ' Proof For w(t) ^ ^ , I e [a, b] and « ^ 3 in Theorem 6, we get
/(.).^f/(,.,*..^«^^/V(.ri,..
f'(b)-f'(a) f'-f'
+ ^ yj__ P(x.h)P(r,.,2)dl,dt2
-* j j j P('.'i)P('t,t2)P(h,h}f"(h)dt,dt2dt,.
where P( , •) is defined in (3) Denote
RlU.x)-- j j j P(x,t,)P{t,.l2)P(t2.t,}f"'(l,)dl,dl2dt,
''^'^'^jJJ \''^''''>\\''f'<-'2'>\\P('2-'l)idl,dl2dl,.
(18)
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Some Weighted Ostrowski Type iDequalities Then we have
\R3(f.x)\<hix)\\f"'\\^.
From Remark 2 we see that
1 [/ a + bV (b-a)^i a-i-bV 7 , ."I
and so
hix)= j\P(x.n)\l2(h'>dt,
Note that
and put
[\p(x.n)\i2
i, i, ''"•""'("-">''"* = 5 | . ( . ' - ^ "~i2-J
2 ( 6 - o ) ' U , ""J. LV 2 ; '" 2 V 2 /
- 2 ) 2
'e dien get the first inequality in (17).
The second inequality in (17) is obvious by the fact diat
SorMxEla.b]. ^ Remark 4 It should be noticed that Theorem 8 has been proved in [7) in a different way
TheoT^em 9 Ut f .[a.b]-*R be four limes differentiable on {a.b). whose fourth deriva- tive f" : (.a.b) -* B is bounded on (a. b). i.e.. | | / » ' I U := sup,,,.,,, | / " ' ( / ) l < +oo. Then
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we have the inequality
fm-fia)/ a + b\ f'(.b)-f\a)
r , a' + b'+Aabl— 6 i b ^ ^ ' - ' - ' ' \ ' - ^ ^ ' - ' ' ^
for every X e [a.b], where &(x) is defined m (18).
Proof For )/;(r) ^ ^ , r € [o, t ] and n ^ 4 in -Theorem 6, we get
f{x) = -^fft.„)dt.^f^^fp(x.„dt,
b-a J^ b-a J^
f(b)-f'(a) [''['•
^ b-a j j P('.'l)P('<:l2)dt,dl2 rm - f"(a)
j j j P(x.t,)P{l,,l2)P(t2.t,)dl,dt2dt, P(x. i,)Pl.t,.l2)P(t2.t3}Pl.t,, tt)f">{t,)dl, dt2dl,dt,.
where P(.. •) is defined in (3).
Denote
* * ' ' ^ ' ^ ' " / / / / ' ' ' ' • ' ' ' ' • < ' ' - ' 2 > ' ' ( ' 2 - ' 3 > ' ' ( ' ! . ' « ) / ' * ' ( ' 4 ' ' " l < " 2 ' " j ' " 4
and
' * ' ' ' • " / / / / \Pl'.h)\\P(l,.t2)\\P(t2.t,)\\Pi.l,,t,)\dt,dt2dl,dlt.
Then we have
iR4(/..<)|s;4(:<)||/»'||„
and
'4U) = j \P(x.tt)\l,(t,)dl,
" ^ 1 / [/ ''^'>'"]''" + [[l"h(t)dtjdu^
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Some Weighted Ostrowski Type Inequalili
b^!r-/"[('-T)4-«K'-Tr 4,.-«,'r,-^u^.-«.']..f..f[(-^)'
23„ ,2/ a + b\' 7 , / (H-6\' 41 ,] 1
^[(.-^)V(.-4-^y
iir"'-
f l ) P 0 l . « f « 2 . < 3 ) r f t l r f t j * = 7 [ ( ' * - ^ ) - "* . ° ' " ] 1 / o + 6 \a n d p u t
w e t h e n g e t t h e first m e q u a l i t y in ( 1 9 ) , T h e s e c o n d i n e q u a l i t y in ( 1 9 ) is o b v i o u s .
R e f e r e n c e s
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3 Anaslassiou, G.A,. Univariate Osu-owskiinequaliues, revisited, Monaishelte Math 135. 175-189(2002) 4 Bamelt, N,S., Dragomir. S.S., An identity for ii-ume differentiate functions and applicauons for Os-
trowski type inequalities RGMlARes Rep. Collect 6.7(2003)
5. Ccrone, P., Dragomir. S S . : Mid-point type rules from an inequalities point of view In Anastassiou. 0 A.
(ed,) Applied Mathematics Handbook of Analytic-Computational Methods, pp. 135-200. CRC Press.
New York (2000)
O, Dragomir, S S . Bamen. N.S,, An Ostrowski type inequality for mappings whose second denvaUves are bounded and applications J Indian Math Soc 66,237-245(1999)
7 Liu, Z • A note on an Ostrowski type inequality Tamsui Oxf. J. Math Sci. 24. 7-10 (2008) 8 MitrinoviS, D S , PeCand, J E . Fink. A.M, Inequaliues Involving for Functions and Their Integrals and
Derivatives, Kluwer Academic, Dordrecht (1991)
9 Ostrowski, A,, Uber die Absolutabweichung einer differentierbaren Funktion von ihrem iniegralmiiielw- crt. Comment, Math, Helv 10. 226-227 (1938)
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