Let $z$ be a complex number that satisfies $|z+(4/z)|=2$ then find $\max|z|$ and $\min|z|$ .
I just need to know that how I should approach this problem.
How to find the $\max|z|$ and $\min|z|$ for $|z+(4/z)|=2$?
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complex-numbers
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0http://www.cut-the-knot.org/arithmetic/algebra/ComplexNumberInequalities.shtml – 2017-02-04
1 Answers
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The Maximum Modulus
We have by the Triangle Inequality that $$2=\left|z+\frac{4}{z}\right|\geq |z|-\frac{4}{|z|}\,.$$ Thus, $$|z|^2-2|z|-4\leq 0\,.$$ This means $$-\sqrt{5}+1\leq |z|\leq \sqrt{5}+1\,.$$ Hence, $$|z|\leq \sqrt{5}+1\,.$$ The equality holds iff $z=\pm(\sqrt5+1)\,\text{i}$.
The Minimum Modulus
We have by the Triangle Inequality that $$2=\left|z+\frac{4}{z}\right|\geq \frac{4}{|z|}-|z|\,.$$ Thus, $$|z|^2+2|z|-4\geq 0\,.$$ This means $$|z|\geq \sqrt{5}-1\text{ or }|z|\leq -\sqrt{5}-1\,.$$ Hence, $$|z|\geq \sqrt{5}-1\,.$$ The equality holds iff $z=\pm (\sqrt{5}-1)\,\text{i}$.