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

In this paper we establish some new bounds for the companion of Ostrowski’s inequality for the case when $f^{\mathrm{\prime }}\in L^1[a,b]$, $f\mathrm{"}\in L^2[a,b]$ and $f’\in L^2[a,b]$, respectively. We point out that the results in the first and third cases are sharp and that some of these new estimations can be better than the known results. Some applications to composite quadrature rules, and to probability density functions are also given.