An identity for the difference between two integral means is obtained in terms of a Riemann-Stieltjes integral. This enables bounds to be procured when the integrand is of bounded variation, Lipschitzian and monotonic. If f is absolutely continuous, bounds are also obtained for f  Lp[a, b], 1 ≤ p < ∞, the usual Lebesgue norms. This supplements earlier results involving f′  L∞ [a, b].