Convergence region inclusion theorems for continued fractions \(K(a_ n/1)\) (Q1900955)

Let \(- {\pi \over 2} < \alpha < {\pi \over 2}\) and \(M > 0\), and let \[ P (\alpha, M) : = \biggl\{ a \in \mathbb{C} : |a|- \text{Re} \bigl(ae^{-2i \alpha} \bigr) \leq{1 \over 2} \cos^2 \alpha, \quad \text{and} \quad |a|\leq M \biggr\}. \] The uniform parabola theorem due to \textit{W. J. Thron} and \textit{L. Jacobson} [Math. Z. 69, 173-182 (1958; Zbl 0081.05704)] says that continued fractions \(K(a_n/1)\) with all \(a_n \in P(\alpha,M)\) converge to finite values. The convergence is uniform with respect to \(\{a_n\}\). Similarly, \textit{L. J. Lange} [Ill. J. Math. 10, 97-108 (1966; Zbl 0141.06502)] proved that also the limaçon \[ \Omega (\alpha, t) : = \left\{ a = z^2 : \left|z \pm i \left(te^{i \alpha} - {1 \over 2} \right) \right|\leq t \right\} \] is such a uniform convergence set for continued fractions \(K(a_n/1)\). More recently \textit{W. J. Thron} and the reviewer [Lect. Notes Math. 1199, 90-126 (1986; Zbl 0594.40001)] studied the cartesian oval \[ \begin{multlined} E (\Gamma, \rho) : = \biggl\{ a : \bigl|a (1 + \overline \Gamma) - \Gamma|1 + \Gamma|^2 - \rho^2 \bigr|+ \rho |a|\leq \rho \bigl(|1 + \Gamma|^2 - \rho^2 \bigr) \biggr\} \\ \text{where } |\Gamma|< |1 + \Gamma|\text{ and } 0 < \rho < |1 + \Gamma|. \end{multlined} \] In the present paper the author shows that \(E(\Gamma,\rho) \subseteq P(\alpha,M)\) and \(E(\Gamma,\rho) \subseteq \Omega (\alpha, t)\) for properly chosen \(\alpha\) and \(M\) or \(t\), and thus that also \(E(\Gamma,\rho)\) is a uniform convergence set for continued fraction \(K(a_n/1)\).