I am reading this result which I am not able to prove:
$$ \frac{f'(x)}{f'(f^{-1}(f(x)+a))}-1 $$ is negative for all x and, and for all $a\geq1$ if and only if $f$ is convex. $f'$ is the derivative of $f$ wrt $x$ and $f^{-1}$ is the inverse function. I am actually blanking and do not have an idea about how to prove this result. Can anyone help?
Try drawing a picture, say of $f(x) = x^2$. To make it invertible, you'll need to restrict your attention to either the right half ($x \ge 0$) or the left half $(x \le 0)$. The top and bottom of that fraction can be seen as slopes of two different tangent lines. Which is steeper?