I suppose that a function $f$ is $\geq 0$ on $[-1,1]$, decreasing and $f(t)(1+t)$ is concave. Moreover for every $a,b \in [-1,1]$, $a<b$, 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 $a,b \in [-1,1]$, $a<b$, 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 $c=1/2$.


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