I used the Argument Principle and applied Rouche's Theorem to show that a polynomial with real coefficients had 4 zeroes inside the unit disk. I then argued that, since these roots must come in conjugate pairs - symmetric points with respect to the real axis - then there must be precisely 2 zeroes of the polynomial in the upper half of the unit disk.
But I feel there could be a flaw in my argument. What if there were roots in this unit disk, but they are on the real axis? How would they come in conjugate pairs? Or, I must rule out the case where there are roots on the real line, and then I am safe to use my symmetry argument to conclude that there are exactly 2 roots in the upper half of the unit disk?
(By inspection, it was obvious that my polynomial has no real roots in the unit disk, but I'm not sure how to handle the case when I do get a polynomial (with real coefficients) with real roots in the disk.)
Thanks,