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How do I use the reverse triangle inequality for this function? |z|=R

$$|z^4 +5z^2+4|$$

From constructing a triangle, I obtain that $$|z^4|< |z^4 +5z^2+4| + |5z^2| +|4|$$

And thus $$|z^4 +5z^2+4| > |z^4| -|5z^2| - |4| = R^4-5R^2-4$$

However my teacher approached the problem like this:

$$|z^4 +5z^2+4| = |z^2-4| |z^2-1| > (R^2-4)(R^2-1)$$

These two results differ by the constant term ±4 at the end of the polynomial, what method is correct, and how did my teacher obtain his result step by step?

I assume $$|z^2-4| > R^2-4$$

$$|z^2-1| > R^2-4$$

And then multiplied them, not sure if that is allowed though...?

jensen paull
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1 Answers1

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Sure. $$z^4 + 5z^2 + 4 = (z^2 - 4)(z^2 - 1)$$ Which should be obvious. Then, we put absolute value around it. $$|z^4 + 5z^2 + 4| = |(z^2 - 4)(z^2 - 1)| = |z^2 -4)| |(z^2 - 1)|$$ If you are confused why we can do this, think of what it is we are doing. $|a| > 0$ if $a$ is a real number, so let $a = b \cdot c$, Then if $b$ and $c$ to be positive, $|a| = b \cdot c$, and we can guarantee that $b$ and $c$ are both positive by $|a| = |b| |c|$. Above, $a = z^4 + 5z^2 + 4$, $b = z^2 - 4$ and $c = z^2 -1$. Now for the last step:

$$|z^2 -4| \geq R^2 - 4 \\ |z^2 - 1| \geq R^2 -1 \\ |z^2 - 4||z^2 - 1| \geq (R^2 -4)(R^2-1)$$ If your teacher used $>$ and not $\geq$, your teacher made an error. However, you are both correct, just your teacher's method produced a better inequality since the bound is tighter.

Robert Murray
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