A number of authors ((2), (4), (5)) have recently considered problems relating to the convergence of specified sequences of partial sums of a gap Taylor series on its circle of convergence. For convenience in stating such results we shall standardize the Taylor series as Σanzn with the unit circle as circle of convergence, and denote the partial sums and the sum function by sn(z) and f(z) respectively. Following the notation of ((4)) and (5) we shall define φ(x) for 0 ≤ x < ∞ to be the least concave majorant of log (|an| + 2). One of the recent results in this field ((5), Theorem 1, and see also (2)) may be stated as

THEOREM A. Suppose that f(z) is regular on an arc α < arg z < β. Suppose also that there are sequences of integers nk, Nk → ∞ such that Nk – nk → ∞ and am = 0 if nk < m < Nk.