Toppila 1976 counting function values meromorphic function

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Toppila, Sakari, On the counting function for the $a$-values of a meromorphic function. Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 565--572. The scan prints only "doi:10.5186/aasfm.1976.0235" and no license line; the journal's article page states "Copyright (c) 1976 The Finnish Mathematical Society" and "This work is licensed under a Creative Commons Attribution 4.0 International License" (https://afm.journal.fi/article/view/134314, read 2026-10-02).

Using the Nevanlinna-theory notation n(r, a) for the number of roots of f = a in |z| <= r, n(r) for its maximum over the Riemann sphere and A(r) for its average, Toppila addresses several problems from Hayman's lists. Hayman proved 1 <= lim inf n(r)/A(r) <= e and asked (Research problems in function theory, Problem 1.16) whether e can be replaced by a smaller quantity, in particular by 1. Theorem 1 exhibits an explicit infinite product f, built from h(z) = 4(z-1)/(3z) and g_n(w) = 1 - (1/w)^{2^n}, with lim inf n(r)/A(r) >= 80/79, so e cannot be replaced by 1. Theorem 2 answers Erdos's question (Hayman, New problems, Problem 1.25) affirmatively by constructing a meromorphic function f with lim sup_{r -> infinity} n(r, a)/n(r, b) = infinity for every pair of distinct values a, b, and Theorem 3 does the same with an entire function for every pair of distinct finite values a, b. Theorem 4 shows, for each integer s > 10, that the meromorphic product f(z) = prod_{n >= 1} (1 - z exp(-(2s)^n))^{(-s)^n} satisfies n(r, 0)/A(r^{1 + 1/(5s)}) > s/2 on a set of r-values of lower logarithmic density at least (2s)^{-4}, showing that the characterization of the exceptional set in the theorems of Hayman-Stewart and of Miles is best possible in that the exceptional set may have positive lower logarithmic density. Theorem 5 gives an entire product with lim sup n(r, 0)/A(7r/6) >= 9/5, so that for M < 9/5 the constant K in Rickman's bound lim sup n(r, B)/A(Kr) <= M (B compact and not containing an asymptotic value of f) cannot be replaced by 1, and Theorem 6 gives an entire product with lim sup n(r)/A(Kr) = infinity for every K >= 1, so that the compact set B there cannot be replaced by the whole sphere. Theorems 2 and 3 are the direct resolution of Erdos problem 1116.

Source: https://afm.journal.fi/article/view/134314.

Bears on. #1116

Results to transcribe.

• Theorem 1: An explicit infinite product f has lim inf n(r)/A(r) >= 80/79, so the constant e in Hayman's bound lim inf n(r)/A(r) <= e cannot be replaced by 1.
• Theorem 2: There exists a meromorphic function f with lim sup_{r} n(r, a)/n(r, b) = infinity for every pair of distinct values a, b, answering a question of Erdős.
• Theorem 3: There exists an entire function f with lim sup_{r} n(r, a)/n(r, b) = infinity for every pair of distinct finite values a, b.
• Theorem 4: For each integer s > 10 an explicit meromorphic product satisfies n(r, 0)/A(r^{1+1/(5s)}) > s/2 on a set of lower logarithmic density at least (2s)^{-4}, showing the exceptional sets in the Hayman-Stewart and Miles theorems can have positive lower logarithmic density.
• Theorem 5: An explicit entire product satisfies lim sup n(r, 0)/A(7r/6) >= 9/5, so for M < 9/5 the constant K in Rickman's theorem cannot be replaced by 1.
• Theorem 6: An explicit entire product satisfies lim sup n(r)/A(Kr) = infinity for every constant K >= 1, so the compact set in Rickman's theorem cannot be replaced by the whole sphere.