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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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Could you clarify what $a$, $b$ and $p$ are? I'm guessing $a$ and $b$ are complex numbers (from the tags), and $p$ is an integer? – Douglas S. Stones Oct 28 '12 at 5:18
a,b are complex number and $0<p<\infty$ – user46262 Oct 28 '12 at 5:21
up vote 0 down vote accepted

$p \geq 1$ 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.

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