Goldberg 1979 sets which modulus entire function has

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section_1: Gol'dberg's proof of Hayman's conjecture: if the set where an entire function has modulus greater than some c > 0 has finite planar measure, then the integral of r dr / ln ln M(r,f) to infinity converges.

section_2: For every continuous positive nondecreasing Phi on [0, infinity) with the integral of r dr / Phi(r) from 1 to infinity convergent, some entire f has ln ln M(r,f) = O(Phi(r)) and E(c) of finite measure for every c > 0.

section_3: For every m > 0 there are entire functions for which the set of c > 0 with E(c) of finite planar measure is exactly [m, infinity), and others for which it is exactly (m, infinity), which answers Erdos's question no.

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Golʹdberg, A. A., Sets on which the modulus of an entire function has a lower bound. Sibirsk. Mat. Zh. 20 (1979), no. 3, 512--518, 691.

In Russian. The paper solves Problem 2.40 of Hayman's "New problems" (Canterbury symposium, 1973; published 1974): if f is a non-constant entire function and the planar measure of E(c) = {z : |f(z)| > c} is finite for some c, what is the minimal growth of f? Hayman conjectured that the integral of r dr / ln ln M(r) over [r_0, infinity) converges and that this is best possible. Erdos's variant asked whether finiteness of |E(c)| for one c implies the same for E(c') with c' < c; Gol'dberg inserts the word some into the question (finiteness for some c' < c), since the answer to that form is already negative, and the answer to the for-all form (finiteness for every c' < c) is then a fortiori negative too. Part 1 proves Hayman's convergence claim: using the inequality ln^+ ln^+ M(er,f) >= pi times the integral of dt/l(t) over A(r) minus K (inequality (1)), which Pfluger and Arima proved independently by Carleman's method, where l(r) is the largest arc length of the part of the circle |z| = r lying in E(c), together with the Cauchy-Bunyakovsky inequality on dyadic blocks (steps (3)-(5)), he derives that |E(c)| < infinity implies convergence of the integral of r dr / ln ln M(r,f) (formula (6)). Part 2 shows (6) cannot be sharpened: for any continuous positive nondecreasing Phi with the integral of r dr / Phi(r) convergent (7), there exists an entire f with ln ln M(r,f) = O(Phi(r)) and |E(c)| < infinity for all c > 0, a lemma for which is credited to V. S. Boichuk. Part 3 shows, by Keldysh's approximation theorem, that the set of c with |E(c)| finite can be [m, infinity) or (m, infinity) for any m > 0. A closing note (p. 517), written while the paper was with the editors, records that G. Camera's 1977 London doctoral thesis also proved Hayman's conjecture as established in Parts 1 and 2. The manuscript was received on 6 July 1977.

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Read status: claims checked. The statements of Parts 1--3 and the remark on Erdős's question were read clause by clause on the page images of pp. 512--517; the proofs were read in outline only, the results they import were not checked against their sources, and nothing here is independently reviewed.

Contents

The paper numbers its parts 1°, 2°, 3° and labels no theorem; the result pages take those section numbers. Throughout, $E(c)=\{z:|f(z)|>c\}$, $|E(c)|$ is its planar measure and $M(r,f)=\max_{|z|=r}|f(z)|$.

• Section 1° (pp. 512--513, formula (6) on p. 513): if $f$ is entire and $|E(c)|<\infty$ for some $c>0$, then $\int_{r_0}^\infty r\,dr/\ln\ln M(r,f)<\infty$.
• Section 2° (statement p. 513, lemma p. 514, construction pp. 515--517): for every continuous positive nondecreasing $\Phi$ on $[0,\infty)$ with $\int_1^\infty\{\Phi(r)\}^{-1}r\,dr<\infty$ there is an entire $f$ with $\ln\ln M(r,f)=O(\Phi(r))$ as $r\to\infty$ and $|E(c)|<\infty$ for every $c>0$.
• Section 3° (p. 517): with $T=\{c>0:|E(c)|<\infty\}$, the cases $T=\emptyset$ and $T=(0,\infty)$ occur, and for every $m>0$ there are entire functions with $T=[m,\infty)$ and with $T=(m,\infty)$. The result page also records the remark of p. 512 on Erdős's question.

Bears on. #1118, whose two questions are the two the paper quotes from Hayman's Problem 2.40, the second attributed there to Erdős (p. 512). Section 1° proves the growth bound Hayman conjectured for the first question, and section 2° shows it cannot be sharpened in the sense stated there. For the second question the paper states (p. 512) that the answer is negative even when only some $c'<c$ is asked for; the functions of section 3° with $T=[m,\infty)$ have $|E(m)|<\infty$ and $|E(c')|=\infty$ for every $c'<m$.

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