Let $f$ be Riemann integrable such that $\int_a^b f(t) \ dt = 1$ and $f \geq 0$ on $[a,b]$. If $\sigma \in C^2$ is convex, show
$$\sigma\left(\int_a^b tf(t) \ dt\right)\leq\int_a^b f(t)\sigma(t) \ dt $$
and, in addition, discuss when equality holds given the tighter condition $f>0$.
I'm assuming this is related to probability theory, since $f$ meets the conditions to be a probability density function.
I was able to prove this using Jensen's inequality; however, my professor insists that isn't used. I tried a few other approaches, like fixing one variable and differentiating with respect to the other, but this doesn't seem to lead anywhere. Please help!