Is it true that $(\int f d \mu)^2 + (\int g d \mu)^2 \leq (\int \sqrt{f^2+g^2} d \mu)^2$

Question

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?

Answer 1

Hint:

See the reverse inequality (for $p<1$) of Minkowski inequality. Let $p=1/2$ and apply the inequality to $f^2$ and $g^2$.