If $M_n$ is a Martingale, How to Show that $\log M_n$ is a Supermartingale? I think that the problem should use Jensen's inequality.
This is because my textbook states that with Jensen's inequality, if $\phi$ is a convex function and $M_n$ a martingale, then $\phi(M_n)$ is a submartingale.
The problem is that $\log$ is not a convex function and I am trying to show the opposite direction.

You have that $-\log x$ is a convex function. Then
$\mathbb{E}[-\log M_{n+1} | \mathcal{F_n}] \geq -\log(\mathbb{E}(M_{n+1} | \mathcal{F}_n) = -\log(M_n)$
(I used Jensen in the first inequality)
Then, $-\log(M_n)$ is a submartingale and then $\log(M_n)$ is a supermartingale!
