If $0 < p < 1$, then $\int_X \log|f| \,d\mu \le \log \|f\|_p$ using Jensen's inequality

Question

I think this should be a really simple example of Jensen's inequality, but I'm really struggling to prove that if $f$ is a positive function such that both $f$ and $\log f$ are integrable, then $\int_X \log f \,d\mu \le \log \|f\|_p$ for $0 < p < 1$ when $X$ is a measure space such that $\mu (X) = 1$. I know I should use the concavity of the $\log$ function but I'm really struggling to see how.

Answer 1

With no further assumptions, this is false.

Let $\mu$ be a probability measure on $\Omega$ and $X \subseteq \Omega$ such that $\mu(X) < 1$ (hence $\mu(X)^\frac{1}{p}< 1$) as well as $f \equiv 1$ then

$$\int_X \log f \,d\mu = \int_X 0 \,d\mu = 0 > \log(\mu(X)^\frac{1}{p}) = \log \|f\|_p$$