@article{1194.26025,
author="Civljak, A. and Dedic, Lj.",
title="{Generalizations of Ostrowski inequality via biparametric Euler harmonic
    identities for measures.}",
language="English",
journal="Banach J. Math. Anal. ",
volume="4",
number="1",
pages="170-184",
year="2010",
abstract="{A sequence of functions $P_n:[a,b] \rightarrow {\Bbb R}$ is called a
    $\mu$-harmonic sequence of functions if $P_1(t)=c+ \mu^{\ast}_1(t)$, $t\in
    [a,b]$ for some $c\in {\Bbb R}$ and
    $P_{n+1}(t)=P_{n+1}(a)+\int_a^tP_n(s)\,ds$, where
    $\mu^{\ast}_1(t)=\mu([a,t])$. If $\mu$ is a real Borel measure on $[a,b]$
    and if $P_n$ is a $\mu$-harmonic sequence, then for a function
    $f:[a,b]\rightarrow {\Bbb R}$ such that $f^{(n-1)}$ is a continuous function
    of bounded variation the following identity holds: 
 $$\int_{[a,b]}
    f_{x,y}(t)d\mu(t)-\mu(\{a\})f(a+y-x)+S_n(x,y)= (-1)^n
    \int_{[a,b]}K_n(x,y,t)\,df^{(n-1)}(t),$$ 
 where $f_{x,y}(t)=f(y-x+t)$ for
    $t\in [a,b+x-y]$ and $f_{x,y}(t)=f(a-b+y-x+t)$ for $t\in (b+x-y,b]$ and 
    $$\multline
    S_n(x,y)=\sum_{k=1}^n(-1)^kP_k(b+x-y)[f^{(k-1)}(b)-f^{(k-1)}(a)]\\
    +\sum_{k=1}^n (-1)^k f^{(k-1)}(a+y-x)[P_k(b)-P_k(a)].\endmultline$$ 
 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 $f^{(n)}\in L_p[a,b]$.}",
reviewer="{Sanja Varo\v sanec (Zagreb)}",
keywords="{Ostrowski inequality; harmonic sequences; biparametric Euler
    identities}",
classmath="{*26D15 (Inequalities for sums, series and integrals of real
    functions)
28A25 (Integration with respect to measures and other set functions)
26D20 (Analytical inequalities involving real functions)
}",
}