A result on nearness of functions and their regular 𝐶-fraction expansions

Abstract

We shall prove a "nearness"-result of the following type. If f ( w ) f(w) is a holomorphic function in Ω = { w ∈ C ; | arg ⁡ ( 1 + 4 w ) | > π } \Omega = \{ w \in {\mathbf {C}};\left | {\arg (1 + 4w)} \right | > \pi \} and "sufficiently near" the function ( 1 + 4 w − 1 ) / 2 (\sqrt {1 + 4w} - 1)/2 , then f ( w ) f(w) has a regular C-fraction expansion K ( a n w / 1 ) K({a_n}w/1) where a n → 1 {a_n} \to 1 .