So, this far I have stumbled upon only one tool to solve complex number absolute value inequalities, but now I have a problem it isn't effective against.

$$\frac{|z+i|}{|z^2+1|} \geq 1 $$

My method was to write $|z|$ as $\sqrt{a^2+b^2}$ and then try to calculate $b(a)$. Is there any other effective tool to solve such inequality?


The denominator factors as $(z - i)(z + i)$. Hence if $z \ne \pm i$ this inequality is equivalent to

$$\frac{1}{|z - i|} \ge 1.$$

Alternatively, this is $|z - i| \le 1$, which has a good geometric interpretation.


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