A uniform twin parabola convergence theorem for continued fractions (Q1343982)

Let \(M>0\), and let \(P_ 1\) and \(P_ 2\) denote the sets \[ P_ m:= \{a\in \mathbb{C}:\;| a|- \text{Re} (ae^{-i2 \gamma})\leq 2g_ m (1- g_{m+1}) \cos^ 2 \gamma\}; \qquad m=1,2, \] where \(\gamma\) and \(g_ m\) are fixed real numbers such that \(- {\pi \over 2}< \gamma< {\pi \over 2}\) and \(g_ 2= d\in (0, {1\over 2}]\), \(g_ 1= g_ 3= 1-d\). Lange proves that \[ E_ 1:= P_ 1\cap \{a\in \mathbb{C}:\;| a|\leq M\}, \qquad E_ 2:= P_ 2 \] are uniform twin convergence regions for continued fractions \(K(a_ n/1)\). That is, there exists a positive null sequence \(\{\varepsilon_ n\}\) only depending on \(E_ 1\), \(E_ 2\), such that every continued fraction \(K(a_ n/1)\) with \(a_{2n-1}\in E_ 1\), \(a_{2n}\in E_ 2\) for all \(n\) converges to some finite value \(f\) with speed \[ \Bigl| f- K_{m=1}^ n (a_ m/1) \Bigr|\leq \varepsilon_ n \qquad \text{for all } n. \] He actually shows that \[ \begin{aligned} \Bigl| f-K_{m=1}^{2n +2} (a_ m/1) \Bigr| &\leq \Bigl| f-K_{m=1}^{2n +1} (a_ m/1) \Bigr|\\ &\leq {M \over {(1-d)\cos \gamma}} \prod_{k=1}^ n {1 \over {1+c/k}}\leq {M \over {(1-d)\cos \gamma}} \biggl( {{1+c} \over {n+c}} \biggr)^ c \end{aligned} \] for all \(n\geq 0\), where \(c= (\cos^ 2 \gamma)/8M\). These results generalize earlier results by W. J. Thron.