In many of the types of problems Ive looked at the following quantity keeps arising and I was wondering if anyone knew any references I could look at to learn some its properties.
Take any function $f$ and two points $a>b$, Define
$V(f,a,b)=\frac{f(a)-f(b)}{a-b}-f'(a)$
Now its obvious that if $f$ is concave (convex) then this is positive (negative). Its also true that if $a$ is on the concavification of $f$ then this is positive for all $b$.
Im more interested in how this would behave when $f$ is a density function. More specifically lets say $y\;distributed\;f(y,e)$ where $f$ has the monotone likelihood ratio in $y,e$. How would the above behave when we fix $y$ and consider two $e_1>e_0$. And take,
$V=\frac{f(y,e_1)-f(y,e_0)}{e_1-e_0}-f_e(y,e_1)$
How does this change with $y$ or with $e$ can we make regularity assumptions on the distribution such that this is negative and then positive as we increase $y$. Any thoughts or references would be very helpful.