Question Number 30557 by abdo imad last updated on 23/Feb/18
![if ϕ convexe and f continue on [a,b] prove that ϕ( (1/(b−a)) ∫_a ^b f(t)dt)≤ (1/(b−a)) ∫_a ^b ϕof(t)dt.](https://www.tinkutara.com/question/Q30557.png)
$${if}\:\varphi\:{convexe}\:{and}\:{f}\:{continue}\:{on}\:\left[{a},{b}\right]\:{prove}\:{that} \\ $$$$\varphi\left(\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:{f}\left({t}\right){dt}\right)\leqslant\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:\varphi{of}\left({t}\right){dt}. \\ $$