## Hadamard’s inequality and trapezoid rules for the Riemann-Stieltjes integral.(English)Zbl 1147.26013

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.

### MSC:

 26D15 Inequalities for sums, series and integrals 65D32 Numerical quadrature and cubature formulas