On Inner Radii Estimates for Mutually Non-overlapping Domains via Green's Functions

According to the Riemann mapping theorem, for an arbitrary simply connected domain $B\subset \mathbb{C}$ and a point $a\in B$, there exists a unique regular function $f(z)$ that univalently maps the given domain onto an open disk centered at the origin such that $f(a)=0$ and $f'(a)>0$. The radius of the resulting disk is defined as the conformal radius of the domain $B$ at the point $a$. In the context of multiply connected domains, the concept of an inner radius serves as a crucial analogue, typically expressed through the Green function of the domain. This research focuses on the extremal problems associated with the products of inner radii for systems of non-overlapping domains, which remains a significant topic in geometric function theory. The study yielded new upper estimates for the products of inner radii of mutually non-overlapping domains containing fixed points on the unit circle. Taking into account these estimates we derived analytical formulas for the potential generated by a system of point sources located in specified configurations. These formulas allow for a formal description of the relationship between the positions of the points $a_k$ and the maximum possible values of the inner radii products. The obtained estimates extend existing theorems in the field of univalent functions and potential theory. These findings have potential applications in mathematical physics, particularly in problems related to electrostatics and fluid dynamics where Green’s functions are fundamental.