On the location of singularities at the boundary of the domain of overconvergence

It is well known that the occurrence of gaps in a series and overconvergence are closely related phenomena; a classical theorem of Ostrowski states that a Taylor series which has gaps of relative length' bounded below away from one has a sequence of partial sums overconvergent in the neighborhood of every regular point on the circle of overconvergence. If the relative length of the gaps tends to infinity, then the sequence of partial sums overconverges to the limit function in its region of regularity. We will improve on the latter result by showing that if the boundary of the region of overconvergence is sufficiently smooth, then the regions of overconvergence and regularity are identical, provided that the relative length of the gaps is greater than some finite number. We will also establish a criterion for the location of some of the singularities of the limit function when these two regions are not identical. Consider a simply connected bounded schlicht domain O containing IzI <1 but not the whole of IzI =1, and let 93 be the family2 of functions having a subsequence of partial sums of their Taylor expansion about the origin, overconvergent in O. Letf(z) be an element of 5, and put f(z) -= Er'Oarzr (necessarily of unit radius of convergence), S.(z) = =,Oarzr and rn(z) =f(z) S,(z). Suppose that the sequence Snk overconverges in 'D, and denote the complement of the closure of D by D*. Let g(z) and G(z) be the Green's functions of OD and D* with respect to the points 0 and oo respectively, then using the method of harmonic majorants we readily obtain the following estimates: