# Jensen type inequality for a non-convex function.

I suppose that a function $$f$$ is $$\geq 0$$ on $$[-1,1]$$, decreasing and $$f(t)(1+t)$$ is concave. Moreover for every $$a,b \in [-1,1]$$, $$a, we have a (simple positive) measure $$\mu_{a,b}$$ such that $$f(\int_a^b s \ \mathrm{d}\mu_{a,b}(s)) \geq 1/2 \ f((a+b)/2)$$.

I would like to prove that there exists $$c>0$$ such that for all $$a,b \in [-1,1]$$, $$a, we have $$\int_a^b f(s) \ \mathrm{d}\mu_{a,b}(s) \geq c \ f((a+b)/2) .$$ whith $$\int_a^b \mathrm{d}\mu_{a,b}(s)=1$$.

If $$f$$ is convex, this is the Jensen inequality. Indeed, we have $$\int_a^b f(s) \ \mathrm{d}\mu_{a,b}(s) \geq f(\int_a^b s \ \mathrm{d}\mu_{a,b}(s)) \geq c \ f((a+b)/2)$$ with $$c=1/2$$.