Integral Functions with Gap Power Series

P. Erdös, A. J. Macintyee
1954 Proceedings of the Edinburgh Mathematical Society  
1. Let f(z) = ia n z"n (1) 0 be an integral function, A n being a strictly increasing sequence of nonnegative integers. We shall use the notations I 2 I = r \z\~r /z (r) = max | a n \ r", n = 0, 1, 2, ... describing M (r) as the maximum modulus, m (r) as the minimum modulus and fi(r) as the maximum term of f(z). The present paper is a development of a remark by Polya (Math. Zeit., 29 (1929), 549-640, last sentence of the paper) that if then Our first THEOREM 1. If Theorem 1 is clearly a
more » ... s clearly a sharpened form of Polya's result, for from (2) it evidently follows that for sufficiently large n
doi:10.1017/s0013091500021416 fatcat:swmtob74qfhjtd7z7gbjngmdcq