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Sharp Grüss-type inequalities for functions whose derivatives are of bounded variation. (English) Zbl 1142.26012

The Grüss-type error functional
\[ D(f,u)=\frac{1}{b-a}\int_a^b f(t)\,du(t)-\frac{u(b)-u(a)}{b-a}\int_a^bf(t)\,dt \]
is considered. Bounds for \(D(f,u)\) are given assuming that \(u\) is absolutely continuous with \(u'\) of bounded variation or Lipschitzian, \(f\) is of bounded variation or nondecreasing or Lipschitzian. These results are applied to estimate the well-known Chebyshev functional
\[ C(f,g)=\frac{1}{b-a}\int_a^b f(t)g(t)\,dt -\frac{1}{b-a}\int_a^b f(t)\,dt\cdot\frac{1}{b-a}\int_a^b g(t)\,dt. \]
The nicest-looking result of this part seems to be the following: if \(f\), \(g\) are of bounded variation on \([a,b]\) then
\[ \left| C(f,g)\right| \leq\frac{1}{4}\bigvee_a^b(f)\cdot\bigvee_a^b(g) \] with \(1/4\) the best possible constant.
The inequalities proved in the paper are sharp. However, the sharpness of one inequality is declared to be an open problem.
The reader must be careful because there are some misprints in the paper which may be confusing. The most important one is a bad definition of \(D(f,u)\) (the correct is given in this review) at the very beginning of the paper.

MSC:

26D15 Inequalities for sums, series and integrals
26D10 Inequalities involving derivatives and differential and integral operators
41A55 Approximate quadratures
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