Ox Sixcular Radii of Power

by Let QU denote the class of anal>-tic functions which are regular and unbounded in ;zI < 1. According io D. GAIER and IV. MEYER-K~NIG [I] we call the radius Rq defined bJ+ x = r&c'!, 0 5 r <: 1 singular for f(x), if f(z) is unbounded in any sector /zI < 1, pl-E < arg 2 < < p7 + e Fith E > 0. h radius which is not singular for f(z) is called regular for f(z). In [l] it has been shown that if f(z) belongs to the class QU and the power series of f(z) has HADAMARD-gaps, i. e. then every radius is singular for f(z). Clearly for every j(z) E /,Q, there is at least one singular radius. Tt, is easy to see that if we suppose onlv that the power series (lb) has FABRY-gaps, i. e. if instead of (2a) we suppose only (3b) lim 1 " 1 = 0 , x-+= Xna<x then it, is possible that there is only one singular radius for f(z). 9 simple example is furnished bg (3a) where N,+, 2 N, + E2 (k = 1, 2,. . .), Clearly fI(z) is regular in 121 < 1 and if .x is real, we have lim fI(z) = + 00 w-1-0