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Inequality for convex function

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Let a < b be real numbers and f: [a,b] -> R a convex function (equivalently a concave upward function). Define

F(s,t) = f (s) + f(t) - 2f((s+t)/2)

Prove that F(s,t) &#8804; F(a,b) for every s,t &#8712;[a,b].

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Solution Summary

In this solution we are given a convex function in two variables and we are proving that it satisfies an inequality based on that fact alone.

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Please see attachment.

Proof. First, we observe that for each s, t (please see the attached file) [a,b] we have that F(s, t) = F(t, s).
Therefore, without loss of generality we can always assume that t <= s. Also since convex means that f"(x) <= 0 we conclude that f is differentiable and ...

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