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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

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    a,b are complex number and 02012-10-28

2 Answers 2

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$p \geq 1$ 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 $p < 1$.

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Hint: Without loss of generality, suppose $a > b$, and let $c = a - b$ and use the binomial theorem.