Please forgive my ignorance. I have a quick silly question about a statement given without proof in Convex Optimization by Boyd and Vandenberghe (page 87).
Suppose $\mathbb{R}_+^n$ is the set of vectors $v$ such that every coordinate of $v$ is real-valued and non-negative. Suppose $z \in \mathbb{R}_+^n$. Suppose that $z_i$ refers to the $i^{th}$ coordinate of $z$. Suppose $x$ is a scalar such that $x \geq 0$ and suppose that $p$ is a scalar such that $0 < p \leq 1$.
Consider the functions,
\begin{align} i(z) & = \sum_{i=1}^{n} {z_i^p} \\\\ j(x) & = x^{\frac{1}{p}} \end{align}
Boyd and Vandenberghe claim without proof that the function,
$$h(z) = j(i(z))=\left( \sum_{i=1}^{n} {z_i^p} \right)^{\frac{1}{p}}$$
is a concave function of $z$. Why is this true?
I understand that $z_i^p$ is a non-decreasing concave function of $z_i$. Therefore $i(z)$ is a concave function of $z$. I also understand that $j(x)$ is a non-decreasing convex function of $x$.
However, this is where my understanding breaks down. Since $i(z)$ is concave and non-decreasing in each $z_i$, but $j(x)$ is convex, I don't know of any composition rule I can apply to arrive at the conclusion that $h(z)$ is concave. What am I missing?