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Generalizations of Ostrowski inequality via biparametric Euler harmonic identities for measures. (English) Zbl 1194.26025

A sequence of functions P n :[a,b] is called a μ-harmonic sequence of functions if P 1 (t)=c+μ 1 * (t), t[a,b] for some c and P n+1 (t)=P n+1 (a)+ a t P n (s)ds, where μ 1 * (t)=μ([a,t]). If μ is a real Borel measure on [a,b] and if P n is a μ-harmonic sequence, then for a function f:[a,b] such that f (n-1) is a continuous function of bounded variation the following identity holds:

[a,b] f x,y (t)dμ(t)-μ({a})f(a+y-x)+S n (x,y)=(-1) n [a,b] K n (x,y,t)df (n-1) (t),

where f x,y (t)=f(y-x+t) for t[a,b+x-y] and f x,y (t)=f(a-b+y-x+t) for t(b+x-y,b] and

S n (x,y)= k=1 n (-1) k P k (b+x-y)[f (k-1) (b)-f (k-1) (a)]+ k=1 n (-1) k f (k-1) (a+y-x)[P k (b)-P k (a)]·

In the rest of the paper, the authors use the above-mentioned identity to prove Ostrowski-type inequalities which hold for a class of functions f whose derivatives f (n-1) are either L-Lipschitzian or continuous and of bounded variation. Analogous results are obtained for a class of functions f with derivatives f (n) L p [a,b].

MSC:
26D15Inequalities for sums, series and integrals of real functions
28A25Integration with respect to measures and other set functions
26D20Analytical inequalities involving real functions