# Show convex function is increasing in both variables of difference quotient using alternative definition of convexity

Let $$\phi$$ be a function that satisfies $$\frac{\phi (t) – \phi (s)}{t – s} \leq \frac{\phi (u) – \phi (t)}{u – t}$$

where $$s < t < u$$.

Is it possible to directly use this definition of convexity to prove that $$\phi$$‘s difference quotients are increasing in each variable, i.e., $$\frac{\phi (u) – \phi (s)}{u – s} \leq \frac{\phi (u) – \phi (t)}{u – t}$$ and $$\frac{\phi (t) – \phi (s)}{t – s} \leq \frac{\phi (u) – \phi (s)}{u – s}$$

where again $$s < t < u$$.

Background: $$\phi$$ is convex and using the usual definition of convexity the proof is fairly direct. So I’m curious if there’s a direct proof from this definition of convexity.