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.


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