My question relates to a proof of the following lemma, presented in 'Dynamical Systems and Numerical Analysis, Volume 8' by Stuart and Humphries. 
An excerpt from the proof of this lemma is given below. 
I am having trouble understanding why, if $C_1 = 0$, the winding number of the map is greater than one and why it is then possible to choose $\xi$ arbitrarily with the desired properties listed at the bottom of this proof. If anyone can provide a fuller explanation for these facts, I would be very grateful.
Reduction of the power series to its leading term
As the series expansion of a rational function, the right side has a positive radius of convergence. If you make the radius small enough, the first term will dominate the tail of the series. That is, after removing the leading terms with zero coefficients you end up with a right side $$ \sum_{k=m}^\infty C_k(\xi-\xi_0)^k,~~ C_m\ne 0 $$ There exists some $\bar r$ so that for any $r\in[0,\bar r]$ the smallness of the tail can be quantified as $$ \frac{|C_m|}2\ge r\sum_{k=0}^\infty |C_{m+1+r}|r^k. $$
Winding number quantified leads to the Rouché theorem
Now recapitulate the geometric situation. $z=iy$ is on the imaginary axis and $f(\xi)=\frac{\rho(\xi)}{\sigma(\xi)}-z$ has a root at $\xi_0$ with $|\xi_0|=1$.
Consider any $w$ with $Re(w)<0$ and $|w|<\frac12|C_m|r^m$.
Following the Rouché theorem for the functions $f(\xi)-w$ and $g(\xi)-w$, where $g(\xi)=C_m(\xi-\xi_0)^m$, on the circle $|\xi-\xi_0|= r$, one gets $$ |(f(\xi)-w)-(g(\xi)-w)|\le \frac{|C_m|}2r^m<|g(\xi)-w| $$ so that $f(\xi)=w$ and $g(\xi)=C_m(\xi-\xi_0)^m=w$ have the same number of solutions inside the disk $B_r(\xi_0)$.
This kind of argument and implications is summarized in the cited statement "the winding number ... is greater than $1$".
How close are the roots
Now what is further implied but should not fall under a blanket "winding number" argument is that the roots of $f(\xi)=x$ are also increasingly close to the roots $\xi_k$, $k=1,...,m$ of $g(\xi)=0$ as $r\to 0$. But these latter roots lie on a regular $m$-gon around $\xi_0$, only half of them can lie inside the unit disk $D=B_1(0)$. At least one root, say $\xi_1$, is outside the unit disk, and if $r$ is small enough, also the close-by root of $f(\xi)=w$ will be outside the unit disk.
To make this a little more quantitative, regard this as a perturbation problem. Set $w=r^m\alpha$. Then $f(\xi)=w$ gives the equation for $\xi=\xi_0+r\zeta$ $$ α=C_mζ^m+rζ^{m+1}\sum_{k=0}^\infty C_{m+1+k}(rζ)^k $$ from where it follows that $ζ$ has a regular expansion into a perturbation series in $r$, so that the roots of $f(\xi)=w$ and $\xi_k$ are at distance $O(r)$ from $\xi_0$ and their difference from each other is $O(r^2)$.