An inequality for the Riemann-Stieltjes integral

Let g and h be real valued and continuous on the interval (a, b), and suppose that the variation, V(h), of h on (a, b) is finite. By completely elementary methods, it is shown that V(h)-supaia<^it(s(fi)—g(a)) is an upper bound for/0 (h—inf h)dg. Several writers have recently obtained upper bounds for integrals of the form f"a hdg, where h is of bounded variation on the interval (a, b) and g is continuous there ((l), (2), (3, p. 573), (4)). It is our purpose to establish the following extension by completely elemen- tary methods. Theorem. If h is real and of bounded variation on the interval (a, b) and g is real and continuous there, then