An inequality for the Riemann-Stieltjes integral

Abstract

Let g g and h h be real valued and continuous on the interval [ a , b ] [a,b] , and suppose that the variation, V [ h ] V[h] , of h h on [ a , b ] [a,b] is finite. By completely elementary methods, it is shown that V [ h ] ⋅ sup a ≦ α > β ≦ b ( g ( β ) − g ( α ) ) V[h] \cdot {\sup _{_{a \leqq \alpha > \beta \leqq b}}}(g(\beta ) - g(\alpha )) is an upper bound for ∫ a b ( h − inf h ) d g \int _a^b {(h - \inf h)dg} .