Rouché's theorem example verification

Question

I have to use Rouché's Theorem to check how many zeros in $D(0,2)$(disk with center 0 and radius 2) do the following functions have

• $z^3+6z-1$
• $z^3+6z+1$

Now, the first one: $f(z) = 6z, g(z) = z^3+6z-1$ so for $|z| = 2$ $$ |f(z)-g(z)| = |-z^3+1| \leq |-z^3| + |1| \leq 9 < 12 = |6z| = |f(z)| $$ Thus Rouché's Theorem is satisfied and $z^3+6z-1$ has one zero at $D(0,2)$, because $f(z)=6z$ has just one zero. Is that correct?

Is the second example any different?

Answer 1

Your proof for $f_1(z) = z^3 + 6z -1$ is correct.

$f_2(z) = z^3 + 6z + 1$ can be treated in the same way, or simply by noting that $$ f_2(z) = - f_1(-z) \, . $$