I am stuck with this question: How to prove$|a-b|^p\leq \max(1,2^{p-1})(|a|^p+|b|^p)$
I forgot to say a ,b are both complex number
I am stuck with this question: How to prove$|a-b|^p\leq \max(1,2^{p-1})(|a|^p+|b|^p)$
I forgot to say a ,b are both complex number
$p \geq 1$ is addressed here. For $p < 1$, and assuming that $a \geq b$, set $t = a/b>1$. Then we want to prove that $$(t+1)^p \leq t^p + 1$$ $$f(t) = t^p + 1 - (t+1)^p \implies f'(t) = p\left(t^{p-1} - (t+1)^{p-1} \right) > 0$$ Hence, we get that $$t^p + 1 - (t+1)^p \geq f(0) = 0$$ This proves it for $p < 1$.
Hint: Without loss of generality, suppose $a > b$, and let $c = a - b$ and use the binomial theorem.