Note on an Iyengar type inequality

Using Hayashi's inequality, an Iyengar type inequality for functions with bounded second derivative is obtained. This result improves a similar result from [N. Elezovi´c, J. Pečari´c, Steffensen's inequality and estimates of error in trapezoidal rule, Appl. In 1938 Iyengar proved the following inequality in [1]: Theorem 1. Let function f be differentiable on [a, b] and | f (x)| ≤ M. Then 1 b − a b a f (x) dx − f (a) + f (b) 2 ≤ M(b − a) 4 − (f (b) − f (a)) 2 4M(b − a). (1) Through the years, Iyengar's inequality (1) has been generalized in various ways. Set I = b a f (x) dx − 1 2 (b − a)(f (a) + f (b)) + 1 8 (b − a) 2 (f (b) − f (a)).