Hadamard's inequality and trapezoid rules for the Riemann-Stieltjes integral (Q929572)

For a convex function \(f\) on \([a,b]\), the approximations of \(\int_a^bf(x)dx\) given by the Midpoint Rule and the Trapezoid Rule satisfy Hadamard's inequality \[ f\Big(\frac{a+b}{2}\Big)(b-a)\leq\int_a^bf(x)dx\leq\frac{f(a)+f(b)}{2}(b-a). \] The author obtains Midpoint and Trapezoid Rules for the Riemann-Stieltjes integral which engender a natural generalization of Hadamard's inequality. Error terms are then obtained for these rules and other related quadrature formulas.