What I am attempting to show (constructively)
- That if the point which maps to the maximum modulus on the contour $\partial{K}$ is known then the region $K$ can be expanded in the neighborhood of that point so that a region $K'$ obtains $\ni$ Rouche's theorem still holds on $\partial{K'}$ (provided it held for the same pair of functions on $\partial{K}$).
Assumptions:
- That multiple classes of functions similar to the class $\Phi$ can be found where $\Phi$ is defined as follows: $\begin{equation}\forall\phi(z):\phi\in\Phi\mspace{5mu}\mathrm{and}\mspace{5mu} z \equiv re^{i\theta} \begin{cases} & \text{The chief characteristic of $\phi$ is}\\ & \text{that the following functions}\\ & \text{$(2)$ and $(3)$ can easily be}\\ & \text{ derived from the definition }\\ & \text{of $\phi$ and that a single functional }\\ & \text{or transform maps each member}\\ & \text{function to $(2)$ and $(3)$.}\end{cases}\tag{1}\end{equation}$ where
$\begin{equation}m_\phi(r_0)\equiv |\phi(z_0)|\mid z\in|z|=r_0\tag{2}\end{equation}$
and
$\begin{equation}\theta_\phi(r_0)\equiv\mathrm{Arg}(z_0)\tag{3}\end{equation}$
Again, $z_0 \equiv r_0e^{i\theta_0}$ where $z_0 = \max_{\zeta:|\zeta|=r}\phi(\zeta)$ reaches a maximum at $z_0$, i.e., the maximum modulus of $\phi$ on the circle of radius $r_0$ with focus at the origin. In other words this would be the maximum of $|\phi(z)|$ as a function of $r$, the radius at which $\theta$ is allowed to vary as $z$ traverses the circle.
Motivation:
I would like to be able to form a closed inner product space so that [for partial sums where each term dominates the next term and where Rouche's theorem holds on the initial boundary] I can safely conclude that the final boundary obtained from a series representation is in fact zero-free (except any roots of the the first term in the series within the initial boundary).
Actual problem description:
Assume complex-valued functions f and g are holomorphic inside and on the closed contour $\partial K$ forming the boundary of $K$, a simply-connected region with $|g(z)| < |f(z)|$ on $\partial K$. Assume also that $\partial K$ is continuous.
For what manner of functions $f$ and $g$ can $K$ be (relatively easily) expanded in the neighborhood of the point $\zeta$ on $\partial K$ (where $|f|$ attains a maximum) to a larger region $K'$ on which Rouche's theorem still holds?
