known in the literature as the Ostrowski inequality. Definition 1. Let P ( a ) x# < x$ < ... < xn ) b be any partition of *a, b+ and let f!xi" ) f!xi"$" f!xi" Then f!x" is said to ...
ON GENERALIZATION OF DRAGOMIR’S INEQUALITIES HUSEYIN BUDAK AND MEHMET ZEKI SARIKAYA Abstract. In this paper, we establish some generalization of weighted Ostrowski type integral inequalities for mappings of bounded variation.
1. Introduction Let f : [a; b] ! R be a di¤erentiable mapping on (a; b) whoose derivative f 0 : (a; b) ! R is bounded on (a; b) ; i.e. kf 0 k1 := sup jf 0 (t)j < 1: Then we have the t2(a;b)
inequality (1.1)
f (x)
1 b
a
Zb
"
f (t)dt
a
x 1 + 4 (b
# a+b 2 2 (b 2 a)
a) kf 0 k1 ;
for all x 2 [a; b][13]. The constant 41 is the best possible. This inequality is well known in the literature as the Ostrowski inequality. De…nition 1. Let P : a = x0 < x1 < ::: < xn = b be any partition of [a; b] and let f (xi ) = f (xi+1 ) f (xi ) Then f (x) is said to be of bounded variation if the sum n X i=1
is bounded for all such partitions.
j f (xi )j
Let f be of bounded variation on [a; b], and
P
(P ) denotes the sum
i=1
corresponding to the partition P of [a; b]. The number b _
(f ) := sup
a
nX
n X
j f (xi )j
o (P ) : P 2 P ([a; b]) ;
is called the total variation of f on [a; b] : Here P ([a; b]) denotes the family of partitions of [a; b] : In [7], Dragomir proved following Ostrowski type inequalities related functions of bounded variation: Theorem 1. Let f : [a; b] ! R be a mapping of bounded variation on [a; b] : Then Zb
f (t)dt
(b
a) f (x)
1 (b 2
a
a) + x
a+b 2
b _
(f )
a
2000 Mathematics Subject Classi…cation. 26D15, 26A45, 26D10, 41A55. Key words and phrases. Bounded Variation, Ostrowski type inequalities, Riemann-Stieltjes. 1
2
HUSEYIN BUDAK AND M EHM ET ZEKI SARIKAYA 1 2
holds for all x 2 [a; b] : The constant
is the best possible.
In [9], Dragomir gave a simple proof of following Lemma: Lemma 1. Let f; u : [a; b] ! R: If f is continious on [a; b] and u is bounded variation on [a; b] ; then ! Zb Zb b _ jf (t)j d (u) f (t)du(t) a
a
a
"
b _ a
# q1 8Zb !9 p1 b < = _ 1 1 p (u) jf (t)j d (u) if p > 1; + = 1; : ; p q a a
max jf (t)j
t2[a;b]
b _
(u):
a
In [5], Dragomir obtained following Ostrowski type inequality for functions of bounded variation: Theorem 2. Let Ik : a = x0 < x1 < ::: < xk = b be a division of the interval [a; b] and i (i = 0; 1; :::; k + 1) be k + 2 points so that 0 = a; i 2 [xi 1 ; xi ] (i = 1; :::; k) ; k+1 = b: If f : [a; b] ! R is of bounded variation on [a; b] ; then we have the inequality: (1.2)
Zb
f (x)dx
k X
a
1 (h) + max 2 (h)
(
i+1
i ) f (xi )
i=0
b _
i+1
xi + xi+1 ; i = 0; 1; :::; k 2
1
b _
(f )
a
(f )
a
where (h) := max f hi j i = 0; :::; n 1g ; hi := xi+1 b _ (f ) is the total variation of f on the interval [a; b] :
xi (i = 0; 1; :::; k
1) and
a
For some recent results connected with functions of bounded variation see [1][4],[6],[8],[10]-[12],[14]-[18]. The aim of this paper is to obtain some generalization of weighted Ostrowski type integral inequalities for functions of bounded variation. 2. Main Results Firstly, we will give the following notations which are used in main Theorem: Let In : a = x0 < x1 < ::: < xn = b be a partition of the interval [a; b] , i (i = 0; 1; :::; n + 1) be n + 2 points so that 0 = a; i 2 [xi 1 ; xi ] (i = 1; :::; n) ; n+1 = b: Let w : [a; b] ! (0; 1) be continious and positive mapping on (a; b), and (h) := max f hi j i = 0; :::; n
1g ; hi := xi+1
xi (i = 0; 1; :::; n
1) ;
ON GENERALIZATION OF DRAGOM IR’S INEQUALITIES
(L) := max f Li j i = 0; :::; n
1g ; Li =
3
x Zi+1
w(u)du (i = 0; 1; :::; n
1) :
xi
Theorem 3. If f : [a; b] ! R is of bounded variation on [a; b] ; then we have the inequalities 1 0 i+1 Zb Z n X A @ (2.1) f (t)w(t)dt w(u)du f (xi ) i=0
a
i
1 (h) + max 2
kwk1;[a;b]
kwk1;[a;b] (h)
b _
xi + xi+1 ; i = 0; 1; :::; n 2
i+1
1
b _ a
(f )
a
and (2.2) n X i=0
2
0
1 Zi+1 @ w(u)duA f (xi ) i
4 1 v(L) + max 2 i2f0;1;:::;n v(L)
b _
1 1g 2
Zb
f (t)w(t)dt
a
Zi+1 w(u)du
xi
x Zi+1 i+1
w(u)du 5
(f )
a
where
b _
(f ) is the total variation of f on the interval [a; b] :
a
Proof. Let us consider the mappings K de…ned by 8 Zt > > > > w(u)du; t 2 [a; x1 ) > > > > > 1 > > > Zt > > > > > w(u)du; t 2 [x1 ; x2 ) > > > > > < 2 .. K(t) = . > > > Zt > > > > w(u)du; t 2 [xn 2 ; xn 1 ) > > > > > > n 1 > > > Zt > > > > w(u)du; t 2 [xn 1 ; b] : > > : n
3
b _ a
(f )
(f )
4
HUSEYIN BUDAK AND M EHM ET ZEKI SARIKAYA
Integrating by parts , we obtain Zb
(2.3)
K(t)df (t)
a
=
n X1 i=0
3 2 xi+1 Z 4 K(t)df (t)5 xi
1 3 2 xi+10 t Z n X1 Z 4 @ = w(u)duA df (t)5 i=0
xi
i+1
1 1 0 i+1 20 xi+1 Z Z n X1 4@ w(u)duA f (xi ) w(u)duA f (xi+1 ) + @ = i=0
=
n X i=1
x Zi+1
xi
xi
i+1
1
0
1
0
i+1 Zxi n X1 Z @ @ w(u)duA f (xi ) + w(u)duA f (xi )
i=0
i
Zb
xi
3
f (t)w(t)dt5
f (t)w(t)dt:
a
In last equality in (2.3), we have
(2.4)
n X i=1
0x 1 0 b 1 0x 1 i Zi Z n X1 Z @ w(u)duA f (xi ) = @ w(u)duA f (b) + @ w(u)duA f (xi ); i
i=1
n
i
and similarly
(2.5)
n X1 i=0
0
1 0 1 0 i+1 1 Zi+1 Z1 n X1 Z @ @ w(u)duA f (xi ) = @ w(u)duA f (a) + w(u)duA f (xi ): xi
i=1
a
xi
Adding (2.4) and (2.5) in (2.3), we get the equality Zb
(2.6)
K(t)df (t)
a
0
= @
Zb n
1
w(u)duA f (b) +
0 1 Z1 + @ w(u)duA f (a) a
=
n X i=0
0
n X1 i=1
1 Zi+1 @ w(u)duA f (xi )
Zb
i
f (t)w(t)dt
a
1 Zi+1 @ w(u)duA f (xi ) i
0
Zb a
f (t)w(t)dt:
ON GENERALIZATION OF DRAGOM IR’S INEQUALITIES
5
On the other hand, taking modulus in (2.6) and using triangle inequality we have 0 i+1 1 Zb Z n X @ A (2.7) f (t)w(t)dt w(u)du f (xi ) i=0
Zb
=
a
i
K(t)df (t)
a
n X1
=
i=0
1 3 2 xi+10 t Z Z 4 @ w(u)duA df (t)5 xi
i+1
0 t 1 xi+1 Z n X1 Z @ w(u)duA df (t) i=0
xi
i+1
kwk1;[a;b]
x Zi+1
n X1
(t
i=0
i+1 ) df (t)
xi
Using Lemma 1 in last inequality in (2.7), we have x Zi+1
(2.8)
(t
i+1 ) df (t)
xi
xi+1
sup t2[xi ;xi+1 ]
jt
i+1 j
_
(f )
xi
xi+1
=
=
max f
i+1
1 (xi+1 2
xi ; xi+1
xi ) +
Putting (2.8) in (2.7), we obtain 0 i+1 1 Z n X @ (2.9) w(u)duA f (xi ) i=0
kwk1;[a;b] kwk1;[a;b]
n X1 i=0
1 (xi+1 2
max
i2[0;:::;n 1]
i+1
Zb
_
(f )
xi
xi + xi+1 2
xi+1
_
(f ):
xi
f (t)w(t)dt:
a
i
kwk1;[a;b]
i+1 g
xi ) +
1 (xi+1 2
1 (h) + max 2 i2[0;:::;n
1]
i+1
xi ) +
i+1
xi + xi+1 2
i+1
xi+1
_
(f )
xi
xi + xi+1 2
xi + xi+1 2
b _ a
(f ):
n i+1 X1 x_ i=0
xi
(f )
6
HUSEYIN BUDAK AND M EHM ET ZEKI SARIKAYA
This completes the proof of …rst inequality in (2.1). On the other hand, in last inequality in (2.9), we have
(2.10)
i+1
xi + xi+1 2
1 hi and 2
max
i+1
i2[0;:::;n 1]
xi + xi+1 2
1 (h): 2
Adding (2.10) in last inequality in (2.9), we obtain inequality (2.1). Finally, for proof of inequality (2.2), taking modulus in (2.6), we have
n X
(2.11)
i=0
Zb
=
1 Zi+1 @ w(u)duA f (xi ) 0
i
Zb
f (t)w(t)dt:
a
K(t)df (t)
a
n X1
=
i=0
2 xi+10 t 1 3 Z Z 4 @ w(u)duA df (t)5 xi
i+1
0 t 1 xi+1 Z n X1 Z @ w(u)duA df (t) : i=0
xi
i+1
Using Lemma 1 for the last integral of (2.11), we have
0
x Zi+1
(2.12)
xi
Zt
@
i+1
sup t2[xi ;xi+1 ]
1
w(u)duA df (t) Zt
xi+1
w(u)du
(f )
xi
i+1
=
_
8 i+1 9 x Zi+1 i+1
