Let $x=(x_1,...,x_n) \in \mathbb R^n$ and
$$g(p)=\sqrt[p]{\frac{1}{n}\sum_{k=1}^{n} |x_k|^p)}$$
- Using Hölder's inequality, show that $g(p)$ is increasing on $(0,\infty)$.
For a sequence with positive weights $w_k$ with sum $\sum w_k=1$, we define the weighted generalized mean as $g(p)=(\sum_{k=1}^{n}w_k|x_k|^p)^\frac{1}{p}$ To prove (1), we need to show that for any $p<q$, the following inequality holds:
$$\sqrt[p]{(\sum_{k=1}^{n}w_k|x_k|^p)} \leq \sqrt[q]{(\sum_{k=1}^{n}w_k|x_k|^q)}$$
I'm stuck after this point. Most of the helpful notes I've found online use Jensen's inequality for this proof, so any help regarding Hölder's inequality would be appreciated. Since we are asked to use Hölder's, I would assume that only the case for positive $p$ and $q$ needs to be proved.
- Find $\lim_{p \rightarrow \infty}g(p)$.
I know that it's supposed to be $\max(x_1,...,x_n)$, but I have no idea how to show this.
Hölder's inequality can be written as:
Let $1<p,q<\infty$ be conjugate exponents ($\frac{1}{p}+\frac{1}{q}=1$), and $x=(x_1,...x_n) \in \mathbb R^n$, $y=(y_1,...,y_n) \in \mathbb R^n$. Then,
$$\sum_{k=1}^{n}|x_k||y_k| \leq \sqrt[p]{\sum_{k=1}^{n}|x_k|^p}\cdot\sqrt[q]{\sum_{k=1}^{n}|y_k|^q}$$
If $0 < p < q < \infty$, use Holder's inequality with conjugate exponents $\frac{q}{p}$ and $\frac{q}{q-p}$ to get $$\sum_{k = 1}^n |x_k|^p \le n^{1 - \frac{q}{p}} \left(\sum_{k = 1}^n |x_k|^q\right)^{\frac{q}{p}}$$ Rearrange the inequality to obtain $g(p) \le g(q)$.
To prove that $\lim_{p \to \infty} g(p) = \max\{|x_1|\,\ldots, |x_n|\}$, show that for every $\varepsilon > 0$, $$n^{-1/p}(\max\{|x_1|\,\ldots, |x_n|\} - \varepsilon) \le g(p) \le \max\{|x_1|\,\ldots, |x_n|\} \quad (p \ge 1)$$