$|a_1|+|a_2| \leq |c_1|+|c_2| $ implies $|a_1|^p+|a_2|^p \leq |c_1|^p+|c_2|^p $?

Question

If $|a_1|+|a_2| \leq |c_1|+|c_2| $ then does this implies that $$|a_1|^p+|a_2|^p \leq |c_1|^p+|c_2|^p $$ for all $p\geq 1$

Answer 1

No: try $a_1=2,a_2=0$ and $c_1=c_2=1$. Then $a_1+a_2=2=c_1+c_2$, but $a_1^p+a_2^p=2^p$ will be larger than $c_1^p+c_2^p=2$ for $p>1$.

Answer 2

It's wrong:

$2+3\geq4+1$, but $2^2+3^2\geq4^2+1^2$ is wrong.