Is true that $|\int_X fd\mu|^p \le \int_X|f|^pd\mu?$

Question

Let $X$ a measure space and $f \in L^p(X), p \in [1,\infty).$ Then:

I am trying to prove one property for a certain space and if I have the following it will solve the problem.

Is true that $|\int_X fd\mu|^p \le \int_X|f|^pd\mu?$

If yes, how can I prove it?

Answer 1

The inequality is not true in general. For instance consider the harmonic sequence then the LHS of the inequality diverges while the RHS is finite (When $p>1$).

Possibly the best inequality you can achieve in this direction is $$\left| \int fd\mu\right|^p \leq ||f||_1^{p-1} \int |f|^pd\mu $$ which can be proven with Jensen's inequality. (Assuming $f$ is also in $L^1$)