TITLE: non-trivial result about integrals due Gromov
QUESTION [16 upvotes]: Gromov proved that if $$
f,g:\left[ {a,b} \right] \to R
$$
are integrable functions, such that the function $$
t \to \frac{{f\left( t \right)}}
{{g\left( t \right)}}
$$
is also integrable, and decreasing.  Then the function $$
r \to \frac{{\int\limits_a^r {f\left( t \right)dt} }}
{{\int\limits_a^r {g\left( t \right)dt} }}
$$
is decreasing.
I could not proved, and I could not find a proof )=

REPLY [5 votes]: The geometric interpretation of the result is fairly clear if you draw the picture of a particle with velocity vector $(f(t), g(t))$ that at time $t=a$ is at $(0,0) \quad$ (assume $g(t) > 0$ so that the particle moves to the right at all times).  Decreasing $f(t)/g(t)$ means the path of the particle is convex, curving downward.  This implies the second property if the particle goes through $0$; the slope of the velocity vector when $t>a$ is always less than the slope of the line from the particle to $0$ so that continued motion forces the latter to decrease.