My problem is, to apply Jensen's Inequality (below), we need $P(X_n \in (0,\infty))=1$ i.e, $P(|Y|=0)=0$, which is not necessarily the case. Am I missing something?
EDIT: I have thought of a way: We work on new measure space, where $\hat{\Omega}= \{ Y >0\}$, $\hat{\mathcal{F}} := \{ \hat{\Omega}\cap F \, : \, F \in \mathcal{F} \}$ and $\hat{P}(B):= \frac{P(B)}{P(\hat{\Omega})}.$ Applying above argument to this result, then translating back, $$ \Big( \int |Y|^p dP \Big)^{\frac{1}{p}} \le P(\hat{\Omega} )^{\frac{1}{p} - \frac{1}{r} } \Big( \int |Y|^r \, dP \Big)^{\frac{1}{r} }. $$ which is what we wanted since $\frac{1}{p}-\frac{1}{r} > 0 $. This seems really messy, (or maybe this is wrong), I hope there is a neat explanation.
EDIT 2: A nice solution would be to show $c(x):= |x|^a$ is convex for all $a \ge 1$, $x \in \mathbb{R}$! Is there a neat way to show this is convex?