A note on Simpson's inequality for function of bounded variation (Q2764883)

The authors establish some generalizations of certain inequalities of Simpson type for functions of bounded variation due to S. S. Daragomir, as well as J. E. Pečarić and S. Varošanec. We quote only the following result: Let \(f:[a,b]\to \mathbb{R}\) be of bounded variation on \([a,b]\) and let \(k\in[{1\over 2},1]\). Then NEWLINE\[NEWLINE\Biggl|\int^b_a f(x) dx+ f\Biggl({a+b\over 2}\Biggr)(2k- 1)(a- b)- (1- k)(b- a)[f(a)+ f(b)]\Biggr|\leq g(k)(b- a)\bigvee^b_a (f),NEWLINE\]NEWLINE where \(g(k)= 1-k\) for \(k\in [{1\over 2}, {3\over k}]\), and \(g(k)= k-{1\over 2}\) for \(k\in [{3\over 4}, 1]\).