Sharp Integral Inequalities of the Hermite–Hadamard Type Original Research Article

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanović and Pecarić, and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning Lp estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds.