What is the tightest bound that relates the L-1 norm with the L-p norm, for $p \geq 1$ and $n\rightarrow \infty$?

Question

I have a situation where is cheaper to compute $y = \sum_{i=1}^n x_i$ instead of $y_p = \sum_{i=1}^n x_i^p$, with $x_i \in \mathbb{R}_{>0}$ and $p\geq 1$. From this answer, using Hölder's inequality, I have

\begin{equation} \sum\limits_{i=1}^n x_i^p \geq n^{1-p}\left(\sum\limits_{i=1}^n x_i\right)^p \end{equation}

However, for large $n$ (say >5 million), that lower bound becomes trivial (0). Is there another result that I can use to approximate $y_p$ in terms of $y$?

Answer 1

The lower bound you have is the best. The bound is attained when $x_i=1$ for all $i$. The case $p=1$ is trivial, so I will assume that $p >1$. Suppose \begin{equation} \sum\limits_{i=1}^n x_i^p \geq c_n\left(\sum\limits_{i=1}^n x_i\right)^p \end{equation}

Then, putting $x_i=1$ for all $i$ we see that $n \geq c_nn^{p}$ or $c_n \leq n^{1-p}$ so $c_n $ must tend to $0$ as $n \to \infty$.