Suppose $A,B:\mathbb{C}\rightarrow\mathbb{C}^{n\times n}$ are non-constant and invertible as matrices everywhere, and satisfy that $B$ is an entire holomorphic mapping and $A(z)=B(z)+O(1/z)$ as $|z|\rightarrow\infty$. I.e., there is a constant matrix $C$ such that, for large enough $z$, we have in, e.g., matrix norm $\|A(z)-B(z)\|\leq\|C\|/|z|$.
I want to construct similar asymptotics for $A(z)^{-1}$, i.e., find decaying $f(z)$ such that $A(z)^{-1}=B(z)^{-1}+O(f(z))$. (Intuitively, such an $f$ ought to exist!)
Dropping the $z$'s from the matrices for simplicity, I quite easily found the identity \begin{align}\|A^{-1}-B^{-1}\|\leq\|A^{-1}\|\|B^{-1}\|\|A-B\|\leq\frac{\|A^{-1}\|\|B^{-1}\|\|C\|}{|z|}.\end{align}
I have no idea how to take this further, since, e.g., the inequality $1\leq\|T^{-1}\|\|T\|$ is of no use in improving the RHS, and I can think of no way of ensuring that $A^{-1}$ and $B^{-1}$ are bounded in $z$ and remain non-constant (for $B^{-1}$, thanks to Liouville's theorem).
Any ideas are most welcome! :-)