$\textbf{Problem setup}$:
Let $F : R_{\tau} \rightarrow R_{\tau}$ be a biholomorphism onto its image (i.e. $F : R_{\tau} \rightarrow V$ is a holomorphic function with holomorphic inverse, here $V = F(R_{\tau}) \subset R_{\tau}$ is open), $R_{\tau} = \{ w \in \mathbb{C} : \Re(w) > \tau \}$ where $\tau > 0$ is some fixed real number. Moreover assume that $F$ extends holomorphically to a slightly larger right half plane $R_{\tau’}$ with $\tau’ < \tau$ and is again a biholomorphism onto its image.
Suppose $F(w) = w + 1 + \eta(w^{-\frac{1}{n}})$ (holds in $R_{\tau’}$), where $\eta(\zeta) = b_1 \zeta + b_2 \zeta^2 + ...$, is some analytic function in a neighbourhood of $0$, defined on $|\zeta| < r$ for some fixed $r > 0$. Also assume $|\eta(\zeta)| \leq \frac{1}{2}$ for all $|\zeta| < r$.
Now suppose $\tau > \frac{1}{r^n} + 1$ is fixed, and let $x \geq \tau$ be a fixed real number.
Define $E_{0,x} = \{ w \in \mathbb{C} : \Re(w) = x \}$ and $E_{1,x} = \{w \in \mathbb{C} : \Re(w) = x+1 \}$.
Then setting $H_x(w) = w$ if $w \in E_{0,x}$ and $H_{x}(w) = \Phi(w) = w + \eta((w-1)^{-\frac{1}{n}}) $ if $ w \in E_{1,x}$
we see that since $\Phi(w) = F(w-1)$, $H_{x}(w)$ is injective on both $E_{0,x}$ and $E_{1,x}$.
Moreover since $\Re(H_{x}(w)) \geq \Re(w) - \frac{1}{2} = x+1 -\frac{1}{2} = x + \frac{1}{2}$ for $w \in E_{1,x}$, the images of $E_{0,x}$ and $E_{1,x}$ under $H_{x}(w)$ do not intersect, thus $H_{x}(w)$ is injective.
$\textbf{The problem}$:
Suppose we introduce a complex parameter $c \in \Delta$ where $\Delta = \mathbb{D}$ into $\eta(\zeta)$ as follows:
Define $\eta(c,\zeta) = \eta(c r \zeta (x-1)^{\frac{1}{n}})$ for $|c| < 1$ and $|\zeta| \leq (x-1)^{-\frac{1}{n}}$. Since $|c r \zeta (x-1)^{\frac{1}{n}}| < r$, it follows that $\eta(c,\zeta)$ is a convergent power series and that $|\eta(c,\zeta)| \leq \frac{1}{2}$ for $|c| < 1$ and $|\zeta| \leq (x-1)^{-\frac{1}{n}}$.
Analogously to before, set $H_{x}(c,w) = w $ if $ (c,w) \in \Delta \times E_{0,x}$
and $H_{x}(c,w) = \Phi(c,w) = w + \eta(c,(w-1)^{-\frac{1}{n}})$ if $ (c,w) \in \Delta \times E_{1,x}$
Apparently now we can conclude by Rouche's theorem that for a $\textbf{fixed}$ $ |c| < 1$, $H_{x}(c,.)$ is injective on $E_{0,x}$ and $E_{1,x}$.
I am lost as to how one is supposed to apply Rouche's theorem to see this.