The extension of Montgomery identity via Fink identity with applications
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new extension of the weighted Montgomery identity is given by using Fink identity and is used to obtain some Ostrowski-type inequalities and estimations of the difference of two integral means. 1. Introduction The following Ostrowski inequality is well known [10]: 2 b x − (a + b)/2 1 1 f (x) − ≤ f (t)dt + (b − a)L, b−a a 4 (b − a)2
x ∈ [a,b],
(1.1)
where f : [a,b] → R is a differentiable function such that | f (x)| ≤ L, for every x ∈ [a,b]. The Ostrowski inequality has been generalized over the last years in a number of ways. Milovanovi´c and Peˇcari´c [8] and Fink [6] have considered generalizations of (1.1) in the form b n −1 1
1 f (x) + Fk (x) − f (t)dt ≤ K(n, p,x) f (n) p n b−a a
(1.2)
k=1
which is obtained from the identity
n−1
1 1 f (x) + Fk (x) − n b − a k=1
b a
1 f (t)dt = n!(b − a)
b a
(x − t)n−1 k(t,x) f (n) (t)dt, (1.3)
where Fk (x) =
n − k f (k−1) (a)(x − a)k − f (k−1) (b)(x − b)k , k! b−a t − a, a ≤ t ≤ x ≤ b, k(t,x) = t − b, a ≤ x < t ≤ b.
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:1 (2005) 67–80 DOI: 10.1155/JIA.2005.67
(1.4)
68
The extension of Montgomery identity
In fact, Milovanovi´c and Peˇcari´c have proved that K(n, ∞,x) =
(x − a)n+1 + (b − x)n+1 , n(n + 1)!(b − a)
(1.5)
while Fink gave the following generalizations of this result. Theorem 1.1. Let f (n−1) be absolutely continuous on [a,b] and let f (n) ∈ L p [a,b]. Then inequality (1.2) holds with
K(n, p,x) =
(x − a)nq+1 + (b − x)nq+1 n!(b − a)
1/q
1/q
B (n − 1)q + 1, q + 1
,
(1.6)
where 1 < p ≤ ∞, 1/ p + 1/q = 1, B is the Beta function, and K(n,1,x) =
(n − 1)n−1 max (x − a)n ,(b − x)n . n n n!(b − a)
(1.7)
Let f : [a,b] → R be differentiable on [a,b] and f : [a,b] → R integrable on [a,b]. Then the Montgomery identity holds [9]: f (x) =
1 b−a
b a
b
f (t)dt +
a
P(x,t) f (t)dt,
(1.8)
where P(x,t) is the Peano kernel defined by t−a , b P(x,t) = − a t−b , b−a
a ≤ t ≤ x, (1.9) x < t ≤ b.
Now, we suppose w : [a,b] → [0, ∞ is some probability density function, that is, an int b tegrable function satisfying a w(t)dt = 1, and W(t) = a w(x)dx for t ∈ [a,b], W(t) = 0 for t < a, and W(t) = 1 for t > b. The following identity (given by Peˇcari´c in [12]) is the weighted generalization of the Montgomery identity: f (x) =
b a
b
w(t) f (t)dt +
a
Pw (x,t) f (t)dt,
(1.10)
where the weighted Peano kernel is W(t),
a ≤ t ≤ x, Pw (x,t) = W(t) − 1, x < t ≤ b.
(1.11)
A. Agli´c Aljinovi´c et al.
69
The aim of this paper is to give the extension of the weighted Montgomery identity (1.10) using identity (1.2), and further, obtain some new Ostrowski-type inequalities, as well as the generalizations of the estimations of the difference of two weighted integral means (generalizations of the results from [1, 3, 7, 11]). 2. The extension of Montgomery identity via Fink identity Theorem 2.1. Let f : [a,b] → R be such that f (n−1) is an absolutely continuous function
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