Let $f\colon[0, 1] \to \mathbb R$ be an $m$-strongly convex function and $\mu$ be a probability measure on $[0,1].$ For any $t<1$, the goal is to find a lower bound on $\int\_{0}^t f^2(x) d\mu(x)$ in terms of $t$, $m$, and $\mu$ (and nothing else). We currently have the following bound
$$\int\_{0}^t f^2(x) d\mu(x) \ge \frac{ m^2 t^4}{36} \mu[0,t].$$
We do not know if our bound is tight. Moreover, our proof is really long and messy. A clean/simple proof of such an elementary result would be helpful.



Lower bound on $L^2$ norm of a strongly convex function
The left-hand side, $\int\_{0}^t f^2(x)\, dx$, of your inequality does not contain $\mu$. Since you wanted "to find lowerbound on $\int\_{0}^t f^2(x) d\mu(x)$", it appears that your desired inequality is
$$\int\_0^t f^2(x)\,\mu(dx)\ge cm^2t^4 \mu[0,t] \tag{1}$$
for some real $c>0$.


However, (1) is obviously false in general. E.g., let $t=1/2$, $m=1$, $f(x)\equiv x^2/2$, and let $\mu$ be the Dirac measure supported on $\{0\}$. Then $f$ is $m$-strongly convex and $\int\_0^t f^2(x)\,\mu(dx)=0$, so that (1) fails to hold for any $c>0$.


