I have to find how many zeros $3e^z - z$ has in $abs(z) < 1$.
I know a function has a zero of order m if $f(z) = (z-z_0)^mg(z)$, where $g(z)$ does not equal 0.
I was thinking of maybe applying Rouche's theorem, by finding another function that is analytic in the same region of $3e^z - z$. I am not sure what to choose though. I may not be on the right path either.
You basically need to solve the equations $$3e^x \cos y=x,\ 3e^x\sin y=y,\ x^2+y^2<1$$ Hence you need to have $$e^x<\frac{1}{3}$$ But $x^2+y^2<1\Rightarrow 1>x> -1\Rightarrow e^x>e^{-1}>\frac{1}{3}$ since $2<e<3$. So no solution is possible.