Let $f,g$ positive measurable functions where $\mu$ is a positive measure.
Is it true that $$\left(\int f d \mu\right)^2 + \left(\int g d \mu\right)^2 \leq \left(\int \sqrt{f^2+g^2}d \mu\right)^2$$?
By monotone convergence theorem, it suffices to prove this for simple functions. Write
$$f= \sum_i a_i \chi_{E_i}, \quad g = \sum_j b_j \chi_{F_j}$$ with the sets on the indicators pairwise disjoint
The LHS is $$\left(\sum_i a_i \mu(E_i)\right)^2 + \left(\sum_j \mu(F_j)\right)^2$$
The RHS is
$$\left(\int \sqrt{\sum_i a_i^2 \chi_{E_i} + \sum_j b_j^2 \chi_{F_j}}d \mu\right)^2$$
How can I compare these two?
Hint:
See the reverse inequality (for $p<1$) of Minkowski inequality. Let $p=1/2$ and apply the inequality to $f^2$ and $g^2$.