Mean value integral inequalities (Q2392526)

The main result in this interesting paper is a mean value theorem in the spirit of the paper by \textit{J. J. Kohilal} [Am. Math. Mon. 116, No. 4, 356--361 (2009; Zbl 1229.26009)]: if \(F[a,b]\mapsto\mathbb R\) is absolutely continuous then both sets \(\bigl\{c: F'(c)(b-a) \leq F(b) - F(a)\bigr\}\) and \(\bigl\{c: F'(c)(b-a) \geq F(b) - F(a)\bigr\}\) have positive measure. An easy deduction from this result is that if \(F'(x)=0\) almost everywhere then \(F\) is a constant. This result can easily be written as an integral mean value theorem for the Lebesgue integral and the author gives analogue for the Riemann integral that has the corollary: if \(F\) is a Riemann integral with \(F'(x) = 0\) on a dense set then \(F\) is a constant.