Explanation for the proof of Minkowski's inequality, Theorem 3.5, Rudin's RCA

Question

I understand how can we get the second factors on the right of $(10)$ from the second factors on the right of $(9)$. But I don't know how can we get the first factor on the right of $(10)$, i.e., $$\big\{\int(f+g)^p\big\}^{1/q}.$$ I think he omitted some step here. It probably has another inequality between $(9)$ and $(10)$. *

• I omitted the proof for $(1)$.

*

Answer 1

He explains it:$$\left\{\int(f+g)^{(p-1)q}\right\}^{1/q}=\left\{\int(f+g)^p\right\}^{1/q},$$since $(p-1)q=p$.

Answer 2

\begin{align*} \int(f+g)^{p}&=\int f\cdot(f+g)^{p-1}+g\cdot(f+g)^{p-1}\\ &\leq\left\{\int f^{p}\right\}^{1/p}\left\{\int(f+g)^{(p-1)q}\right\}^{1/q}+\left\{\int g^{p}\right\}^{1/p}\left\{\int(f+g)^{(p-1)q}\right\}^{1/q}\\ &=\left\{\int(f+g)^{p}\right\}^{1/q}\left[\left\{\int f^{p}\right\}^{1/p}+\left\{\int g^{p}\right\}^{1/p}\right]. \end{align*}