On Conformal Radii of Non-Overlapping Simply Connected Domains

The paper deals with the following open problem stated by V.N. Dubinin. Let $a_{0}=0$, $|a_{1}|=\ldots=|a_{n}|=1$, $a_{k}\in B_{k}\subset \overline{\mathbb{C}}$, where $B_{0},\ldots, B_{n}$ are disjoint domains. For all values of the parameter $\gamma\in (0, n]$ find the exact upper bound for $r^\gamma(B_0,0)\prod\limits_{k=1}^n r(B_k,a_k)$, where $r(B_k,a_k)$ is the conformal radius of $B_k$ with respect to $a_k$. For $\gamma=1$ and $n\geqslant2$ the problem was solved by V.N. Dubinin. In the paper the problem is solved for $\gamma\in (0, \sqrt{n}\,]$ and $n\geqslant2$ for simply connected domains.The paper deals with the following open problem stated by V.N. Dubinin. Let a0 = 0, ιa1ι =...= ιanι = 1, ak ∈ Bk ⊂ , where B0, ..., Bn are disjoint domains. For all values of the parameter γ∈ (0; n] find the exact upper bound nfor rγ(B0; 0) ∏ r(Bk; ak), where r(Bk; ak) is the conformal radius of Bk with respect to ak. For γ = 1 k=1 and n ≥ 2 the problem was solved by V.N. Dubinin. In the paper the problem is solved for γ ∈ (0; √n ] and n ≥ 2 for simply connected domains.

Improving the Accuracy of the Charge Simulation Method for Numerical Conformal Mapping

In this paper, we present a method to improve the accuracy of the charge simulation method for numerical conformal mapping. The method constructs the constraint equation by using the charge simulation method. The charges and the conformal radius are computed by using the Runge-Kutta method based on the dynamic system of the constraint equation. By using this method, we can obtain new approximated conformal mapping function and improve the accuracy of numerical conformal mapping compared to the charge simulation method for numerical conformal mapping proposed by Amano. Furthermore, the corresponding numerical results are shown to illustrate the performance of the proposed method.