I'm trying to prove the Minkowski inequality for the $l_p$ norm: $$ \| f + g\|_p \le \|f\|_p + \|g\|_p $$ where $f,g : \mathbb{R}^n \rightarrow \mathbb{R}$ are Lebesgue measurable functions and $p \in [1,\infty)$.
The case $p=1$ and $p=\infty$ follow from the triangle inequality, so the last case to prove is when $p\in [1,\infty)$. The hint for this question is to use the Holder inequality, $$ \|fg\|_1 \le \|f\|_p\|g\|_q, $$ and apply it to $$ |f+g|^p \le |f+g|^{p-1}(|f| + |g|). $$
Can anyone provide a (further) hint as to how to prove the Minkwoski inequality using the hint given?
Firstly, note that $p-1=p(1-\frac{1}{p})=p\frac{1}{q}=\frac{p}{q}$
So you have $$ |f+g|^p \le |f+g|^{p/q}(|f|+|g|)=|f||f+g|^{p/q}+|g||f+g|^{p/q} $$
Now apply Holder's inequality to both of the terms on the RHS separately. Then you will see that you can take the $|f+g|$ term out common. Then divide LHS and RHS by that common term. The RHS will now look like the RHS of Minikowski. Finally, you will again use the fact that $p/q=p-1$ to get the LHS term looking like it does in Minikowski.