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proof of $\frac{f(t)-f(s)}{t-s}\leq\frac{f(u)-f(s)}{u-s}\leq\frac{f(u)-f(t)}{u-t}$ for convex $f$

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proof of $\frac{f(t)-f(s)}{t-s}\leq\frac{f(u)-f(s)}{u-s}\leq\frac{f(u)-f(t)}{u-t}$ for convex $f$

Since you're using three variables s<x<u, I suggest to add
$x = \lambda u + (1 - \lambda)s$
in order to your exposition be clearer.

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