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We shall prove a "nearness"-result of the following type. If f(w) is a holomorphic function in ß = {w e C ; |arg(l + 4w)\ < n} and "sufficiently near" the function (\/l +4u>-l)/2 , then f(w) has a regular C-fraction expansion K(anw/l) where a„ —> 1 . 1. The result The present paper deals with regular C-fraction expansions ,, ,. oo (aw\ a.w a-.w aw (u> ä(-H--t>V+..:+-t-+..." an e C, an / 0, of functions f(w). It is well known that (1.1) corresponds to a formal power series (1.2) dxw + d2w2 + ... + dnw" + ..., dn e C. For definition of correspondence we refer to [2, Sec. 5.1, p. 148]. If (1.2) is the Taylor expansion of a function / of a complex variable w, holomorphic in a neighborhood of the origin, we say that ( 1.1 ) is the regular Cfraction expansion of f(w). To a formal power series there always corresponds a C-fraction [2, Cor. 5.3, p. 156], but not always a regular C-fraction (1.1). Necessary and sufficient conditions for such correspondence are known, see e.g. [2, Thm. 7.2, p. 223]. They are expressed in terms of Hankel determinants of the coefficients dn , and are in general impractical in use. The very simplest (infinite) C-fraction is ., „. oo ¡w\ w w w (L3) *(t) = t+t+...+t+...This C-fraction corresponds to the Taylor series expansion of 1 (1.4) -(^1 +4w1), Re^l +4w >0, Received by the editors February 29, 1988 and, in revised form, October 31, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 30B70; Secondary 40A15, 41A20, 41A21.

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