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Let f(z) = ∑
n=0∞anzn 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.
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