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.