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Generalizations of Ostrowski inequality via biparametric Euler harmonic identities for measures. (English) Zbl 1194.26025
A sequence of functions $P_n:[a,b] \rightarrow {\Bbb R}$ 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 $f^{(n)}\in L_p[a,b]$.
26D15Inequalities for sums, series and integrals of real functions
28A25Integration with respect to measures and other set functions
26D20Analytical inequalities involving real functions
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