- #36
Fredrik
Staff Emeritus
Science Advisor
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Yes, I agree. [tex]|z^4-4z^2+3|=|z^2-3|\,|z^2-1|\geq \text{something}\cdot\text{something}[/tex] is definitely the way to go.I like Serena said:IMHO ehild's method of factoring, followed by the triangle inequalities, is the nicest one.
Pranav-Arora, the method I suggested doesn't work. We get [itex]|z^4-4z^2+3|\geq -3[/itex] if we do it exactly the way I suggested. We need +3 on the right, not -3, so this result is useless. Even if we change the first step into [itex]|z^4-4z^2+3|\geq 3-|z^4-4z^2|[/itex], we're getting something useless. I apologize for misleading you.
The only solution I have found that is similar to my original idea is to start with [tex]|z^4-4z^2+3|\geq|z^4-4z^2+4|-1=|z^2-2|^2-1,[/tex] but as you can see, the only point of doing it this way would be to make the factorization a bit easier. I don't see a way to avoid doing a factorization.
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