  In this paper we establish some new bounds for the companion of Ostrowski's
inequality for the case when $f'\in L^1[a,b]$, $f"\in L^2[a,b]$ 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.
