Let be $f:[0,1]\rightarrow\mathbb{R}$ a continuous function such that far all $a,b\in [0,1]$ with $a<b$
$$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_a^b f(x)\,dx.$$
How to prove that $f$ is a convex function? Thank you for your help.
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put on hold as off-topic by Douglas Zare, Robert Israel, Jeremy Rouse, Stefan Kohl, S. Carnahan♦ 6 hours agoThis question appears to be off-topic. The users who voted to close gave this specific reason:
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