Maximally convergent rational approximants of meromorphic functions (Q2791814)

In the paper [Publ. Inst. Math. (Beograd) (N.S.) 96 (110), 31--39 (2014; \url{doi:10.2298/PIM1410031B}] the author proved the following theorem:NEWLINENEWLINE NEWLINELet \(E\) be compact in \(\mathbb{C}\) with regular connected complement, \({\{m_n\}}_{n\in\mathbb{N}}\) a sequence in \(\mathbb{N}\) with \(m_{n}=o(n)\) as \(n\rightarrow \infty\), \(\lim_{n\rightarrow\infty}m_{n}=\infty\), and let \(f\in M(E)\), the class of functions that are meromorphic in some open neighborhood of \(E\), and \(\{r_{n,m_{n}}\}_{n\in\mathbb{N}}, r_{n,m_{n}}\in R_{n,m_{n}}\), be a sequence of rational approximants of \(f\) such that \(\lim_{n\rightarrow\infty}\sup||f-r_{n,m_{n}}||^{\frac{1}{n}}_{\partial E}\leq\frac{1}{\rho(f)}\) holds. Then \(\{r_{n,m_{n}}\}_{n\in\mathbb{N}}\) converges in capacity to \(f\) inside \(E_{\rho(f)}\). Moreover, there exists a subsequence \(\Lambda\subset\mathbb{N}\) such that the subsequence \(\{r_{n,m_{n}}\}_{n\in\Lambda}\) converges uniformly in capacity to \(f\) inside \(E_{\rho(f)}\), where \(E_{\rho(f)}\) is the maximal Green domain of meromorphy and \(1<\rho(f)<\infty\).NEWLINENEWLINE NEWLINEIn the present paper the author showed that in above the theorem \(m_{1}\)-uniform convergence to the function \(\tilde{f}\) (extension of \(f\)) is \(m_{1}\)-equivalent to \(f\) and the result in the above theorem on uniform convergence in capacity can be sharpend to obtain maximal geometric convergence in capacity, but only for a subsequence \(\Lambda\subset\mathbb{N}\). Finally, upper bounds for the growth of \(\{r_{n,m_{n}}\}_{n\in\mathbb{N}}\) outside \(E_{\rho(f)}\) of Walsh's type have been proved.NEWLINENEWLINEFor the entire collection see [Zbl 1337.41001].