How many zeros does $g(z)= z^4+iz^3 +1$ have in the first quadrant?

Question

Let $g(z)= z^4+iz^3 +1$. How many zeros does $g$ have in $\{z\in \Bbb{C}: \text{Re }(z), \text{Im }(z)>0\}$?

I tried comparing the number of zeros of $g$ to that of $z\mapsto z^4$ and $z\mapsto iz^3$ using Rouché's theorem applied to the path that first walks the real axis from $0$ to some $R\in\Bbb{R}$, then a quarter circle to $iR$ and then down the imaginary axis. However, Rouché's theorem didn't apply and I don't know what else to try.

Thanks for any help.

Answer 1

Put $f(z)=z^4+1$ and $g(z)=iz^3$.

Step 1: On $|z|=2$, $|g(z)|=8<15=|z^4|-1\le|f(z)|$. What does this imply?

Step 2: Consider $\gamma_1=\{z=x+iy:0\le x\le2,y=0\}$. Is it true that $|g(z)|<|f(z)|$ on $\gamma_1$?
(Hint: $|g(z)|=x^3$ and $|f(z)|=x^4+1$, what is the minimum of $x^4+1-x^3$ on $[0,2]$?)

Step 3: Do the same for $\gamma_2=\{z:|z|=2,0\le \arg z\le\pi/2\}$ and $\gamma_3=\{z=x+iy:x=0,0\le y\le2\}$.

Answer 2

Hint: surely zeroes are inside some big quartercircle with radius R, than use argument principle.

Answer 3

Although the question is tagged (complex-analysis), here's another way to look at the problem:

Setting $f(z)=z^4+iz^3+1$, we might as well look for the roots of the real polynomial $$g(z):=f(iz)=z^4+z^3+1,$$ in the quadrant $Q:=\{z\in\Bbb{C}:\operatorname{Re}(z)>0>\operatorname{Im}(z)\}$. As $g$ is everywhere positive it has no real roots, hence it has two pairs of complex conjugate roots, say $\alpha,\overline{\alpha},\beta,\overline{\beta}\in\Bbb{C}$. This already shows that $g$ has at most two roots in $Q$. Looking at the coefficient of $g(z)$ at $z$ we see that $$0=\alpha\overline{\alpha}\beta+\alpha\overline{\alpha}\overline{\beta}+\alpha\beta\overline{\beta}+\overline{\alpha}\beta\overline{\beta}=2|\alpha|^2\operatorname{Re}(\beta)+2|\beta|^2\operatorname{Re}(\alpha),$$ which shows that $\operatorname{Re}(\alpha)$ and $\operatorname{Re}(\beta)$ have opposite signs. This shows that $g$ has precisely $1$ root in $Q$, and hence $f$ has precisely $1$ root in the first quadrant.