Bessack [2] in 1979 used Holder's inequality to obtain an integral inequality which has as special cases Opial's and Hardy's. Here, using mainly Jensen's inequality for convex functions, with a non-negative, non-decreasing function in the operator, we obtain an integral inequality which is similar to Bessaack's but now containing a refinement term. When $l'(x)$ in Bessack [2] and $f$ in Imoru and Adeagbo-Sheikh [4] are restricted to being non-decreasing, these two inequalities become special cases of our results.