Integral Functions with Gap Power Series

be an integral function, A,, being a strictly increasing sequence of nonnegative integers. We shall use the notations M (r) =max f (z) , m(r)=min if (z) 1, Izl=r IzI=r μ (r) = max j an rn, n = 0, 1, 2, . . . describing M (r) as the maximum modulus, m (r) as the minimum modulus and μ (r) as the maximum term of f (z) . The present paper is a development of a remark by Pólya (Math . Zeit., 29 (1929), 549-640, last sentence of the paper) that if lim log (An+1 -gym) > i (2) Log A„ 2