Generalized fraction rules for monotonicity with higher antiderivatives and derivatives (Q6579985)

The Gromov's Theorem says the following. Let \(I\) be an interval in \([-\infty,\infty]\), let \(c\) be a point in \(I\) and let \(f\) and \(g\) be two locally Lebesgue integrable functions \(I\to\mathbf{R}\) with \(g\) preserving Lebesgue integrability almost everywhere a non-zero sign. If \N\[\N\frac{f}{g}\colon I\to[-\infty,\infty] \N\]\Nis Lebesgue integrable almost everywhere (strictly) monotonic, then \N\[\N\frac{\int_cf(t)dt}{\int_cg(t)dt}\colon I\setminus\{c\}\to\mathbf{R}\N\]\Nis (strictly) monotonic of the same (strict) monotonicity. A very similar L'Hôpital's rule for monotonicity is also well known. The main purpose of the authors is to state and prove the generalizations of those two results to higher antiderivatives and derivatives, respectively.