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In this paper we establish some new bounds for the companion of Ostrowski'sinequality for the case when $f'\in L^1[a,b]$, $f"\in L^2[a,b]$ and $f'\inL^2[a,b]$, respectively. We point out that the results in the first and thirdcases are sharp and that some of these new estimations can be better than theknown results. Some applications to composite quadrature rules, and toprobability density functions are also given.

New bounds for the companion of Ostrowski's inequality and applications