Suppose $\varphi $ is a convex function on the real line. I wonder if the following is true? For $h>0 $
$\frac{\varphi(c)-\varphi(c-h)}{h} \leq \frac{\varphi(c+h)-\varphi(c)}{h}$
This seems like a trivial fact that should follow from convexity if one considers the intuitive meaning of a convex function as having increasing slopes for consecutive points on its graph.
But I got stuck on which numbers to pick and then use in the convex property. Help would be appreciated!