I've got a problem in Statistical Inference which I've reduced to showing
$\displaystyle\sum_{i=1}^k\pi_i\int_{R_i} f_i(x)dx\ge\sum_{i=1}^k\pi_i^2$
Here $\pi_i>0\:\forall\:1\le i\le k,$ and $\sum_{i=1}^k\pi_i=1$. Also, $f_i$ are distinct non-negative functions having integral $1$ over $\mathbb R$. $$R_i:=\{x\in\mathbb R\mid\pi_if_i(x)\ge\pi_jf_j(x)\:\forall\:j\ge1\}$$
Also, $\{R_i\mid1\le i\le k\}$ is a partition of $\mathbb R$.
I tried using $\pi_if_i\ge\frac1k\sum_{j=1}^k\pi_jf_j$ in LHS (and then interchanging sums, of course!), but then I get $\frac1k$ as a bound instead of the desired RHS. I couldn't proceed then.
For each $j\in \{1,\ldots, k\}$,
$$\sum_{i = 1}^k\pi_i \int_{R_i} f_i(x)\, dx = \sum_{i = 1}^k \int_{R_i} \pi_i f_i(x)\, dx \ge \sum_{i = 1}^k \int_{R_i} \pi_j f_j(x)\, dx = \int_{\mathbb{R}} \pi_j f_j(x)\, dx = \pi_j$$ Multiplying by $\pi_j$ and summing from $j = 1$ to $k$, we obtain $$\sum_{i = 1}^k\pi_i \int_{R_i} f_i(x)\, dx \ge \sum_{j = 1}^k \pi_j^2$$ since the $\pi_j > 0$ and $\sum_{j = 1}^k \pi_j = 1$.