Determining whether one function has a greater complex modulus than another in a particular region of the complex plane?

Question

I want to determine whether a particular function has a greater modulus than another function in a certain region of the complex plane. For example, how can I tell whether $|g(z)| < |f(z)|$ on the closed disk $|z| < 2$, where $$ f(z) = z^7, $$ and $$ g(z) = -2z^3 + 7. $$ How can I determine whether $|g(z)| < |f(z)|$ in this case?

Answer 1

It's actually easier to compare $|f(z)|^2$ to $|g(z)|^2$, since $|g(z)|<|f(z)|\iff|g(z)|^2<|f(z)|^2$.

It's simple to calculate $|f(z)|^2$ since it is equal to $$f(z)\cdot\overline{f(z)} = z^7 \cdot \overline{z^7} = (z\cdot\overline z)^7 = |z|^{14}$$

It's not much more difficult to calculate $|g(z)|^2$ as well.

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However, in your case, it might also just be useful to look at the values of $f$ and $g$ near points where you know that $|f|$ is small... say... the roots of $f$.