On the growth of meromorphic functions defined by quasi-regular \(c\)-fractions (Q1919493)

Let \(f(z)\) be defined as the value of the continued fraction \[ 1+ {a_1z \over b+ {a_2z \over 1+ \dots}}; \;a_n,b\in\mathbb{C} \backslash \{0\}\quad \text{and} \quad |a_n |\geq |a_{n+1} |\to 0. \] Stieltjes has proved that \(f(z)\) is a well defined meromorphic function in the whole complex plane. The author proves that if \[ \varlimsup_{n\to\infty} |a_n|^{1/ \log n} = e^{1/\rho} \quad \text{for a} \quad \rho\geq 0, \] then \(f(z)\) is a functional of exponential type of order \(\rho\) except for at most two values of the parameter \(b\).