THE EXTENSION OF MONTGOMERY IDENTITY VIA FINK IDENTITY WITH APPLICATIONS

The 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.


Introduction
The following Ostrowski inequality is well known [10]: 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. Milovanović and Pečarić [8] and Fink [6] have considered generalizations of (1.1) in the form (1.2) which is obtained from the identity where 1 < p ≤ ∞, 1/ p + 1/q = 1, B is the Beta function, and . Then the Montgomery identity holds [9]: (1.8) where P(x,t) is the Peano kernel defined by The following identity (given by Pečarić in [12]) is the weighted generalization of the Montgomery identity: (1.10) where the weighted Peano kernel is 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]).

The extension of Montgomery identity via Fink identity
is an absolutely continuous function on [a,b] for some n ≥ 1. If w : [a,b] → [0,∞ is some probability density function, then the following identity holds: Now, by putting this formula in the weighted Montgomery identity (1.10), we obtain Now, if we replace n with n − 1, we will get (2.1). This identity is valid for n − 1 ≥ 1, that is, n > 1.

Remark 2.2.
We could also obtain identity (2.1) by applying identity (1.3) such that we multiply this identity by w(x) and than integrate it to obtain If we subtract this identity from (1.3) we will obtain (2.1). (2.6) A. Aglić Aljinović et al. 71 We denote (2.7) Then we have where Further, and then (2.13) So, we get the generalized trapezoid identity (2.14) Similarly, applying identity (2.1) with x = (a + b)/2, we get We can regard this as the second Euler-Maclaurin formula (the generalized midpoint identity).

Ostrowski-type inequalities
We denote, for n ≥ 2, (3.1) Assume (p, q) is a pair of conjugate exponents, that is, , the following inequality holds: a P w (x,t)(t − y) n−2 k(y,t)dt| q dy) 1/q is sharp for 1 < p ≤ ∞ and is the best possible for p = 1.

Proof. From Theorem 2.1 we have
We denote C 1 (y) = (1/(n − 2)!(b − a)) b a P w (x,t)(t − y) n−2 k(y,t)dt. We use identity (2.1) and apply the Hölder inequality to obtain For the proof of the sharpness of the constant ( b a |C 1 (y)| q dy) 1/q , we will find a function f for which the equality in (3.2) is obtained.