inequality in $l^p$ space

Question

I have to check the inequality $\sum |x_k+y_k|^p\leq C\cdot(\sum |x_k|^p+\sum |y_k|^p)$. It is true for $p=2$ with $C=2$. I assume $C=2^p-2$ in general but cannot prove. The hypothesis relies on the $2^p$ as the sum of all binomial coefficients. However, I have no idea how to extend it for all real $p\geq1$.

Answer 1

\begin{align*} \left(\dfrac{|x_{k}|+|y_{k}|}{2}\right)^{p}\leq\dfrac{1}{2}|x_{k}|^{p}+\dfrac{1}{2}|y_{k}|^{p} \end{align*} by Jensen's inequality. So \begin{align*} |x_{k}+y_{k}|^{p}\leq(|x_{k}|+|y_{k}|)^{p}\leq 2^{p-1}(|x_{k}|^{p}+|y_{k}|^{p}), \end{align*} so \begin{align*} \sum_{k}|x_{k}+y_{k}|^{p}\leq 2^{p-1}\left(\sum_{k}|x_{k}|^{p}+\sum_{k}|y_{k}|^{p}\right). \end{align*}

Answer 2

The simplest argument is as follows: $|x+y|^{p} \leq (|x|+|y|)^{p} \leq (2 max\{|x|,|y|\}^{p} =2^{p} max\{|x|,|y|\}^{p} \leq 2^{p} \{|x|^{p}+|y|^{p}\}$. This is a crude estimate. You can get $C=2^{p-1}$ by using Jensen's inequality as user284331 hs done; note that I use no theorem whatsoever.