Vol. 18, No. 1, 1966

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Asymptotic values of a holomorphic function with respect to its maximum term

Alfred Gray and S. M. Shah

Vol. 18 (1966), No. 1, 111–120

Let f(z) = n=0anzn be holomorphic with radius of convergence R(0 < R ), and let μ(r) denote the maximum term and ν(r) the central index of f(z). By definition for r > 0, μ(r) = max{|an|rn|n = 0,1,2,} and ν(r) = max{n|μ(r) = |an|rn} so that μ(r) = |aν(r)|rν(r). In previous papers we have investigated the limiting values of the quotient μ(r)∕M(r) as r R. Here, as usual, M(r) denotes the maximum modulus of f(z). Recently Clunie and Hayman have disproved a conjecture of Erdös that if μ(r)∕M(r) tends to a limit, the limit must be zero.

In this paper we consider a more general problem. There are two complex functions μ(z) and m(z) which can be regarded as complex extensions of μ(r) in a natural way. We are led to investigate the limiting values of f(z)∕μ(z) and f(z)∕m(z) along curves tending to |z| = R, and we call these μ and m asymptotic values. We prove that for a class of functions which are either of very slow growth, or have gap power series, there are no μ or m asymptotic values. On the other hand, for the admissible functions of Hayman, is a μ and m asymptotic value along the positive real axis, while 0 is a μ and m asymptotic value along any other path in an angle excluding the positive real axis.

Mathematical Subject Classification
Primary: 30.55
Received: 5 August 1964
Revised: 30 December 1964
Published: 1 July 1966
Alfred Gray
S. M. Shah