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Mean value integral inequalities. (English) Zbl 1277.26016

The main result in this interesting paper is a mean value theorem in the spirit of the paper by 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.

MSC:

26A42 Integrals of Riemann, Stieltjes and Lebesgue type
26D15 Inequalities for sums, series and integrals

Citations:

Zbl 1229.26009
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References:

[1] Y. Katznelson and K. R. Stromberg, Everywhere differentiable, nowhere monotone, functions , Amer. Math. Monthly 81 (1974), 349-354. · Zbl 0279.26007 · doi:10.2307/2318996
[2] J. Koliha, Mean, Meaner, and the Meanest Mean Value Theorem , Amer. Math. Monthly 116 (2009), 356-356. · Zbl 1229.26009 · doi:10.4169/193009709X470227
[3] K. R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth Inc., Belmont, California, 1981. · Zbl 0454.26001
[4] B. S. Thomson, Characterization of an indefinite Riemann integral , Real Anal. Exchange 35 (2010), 487-492. · Zbl 1222.26009
[5] W. F. Trench, Introduction to real analysis , free edition downloaded from http://ramanujan.math.trinity.edu/wtrench/misc/index.shtml (previously published by Pearson Education, 2003). · Zbl 1204.00023
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