---

Source. Section 1° (formula (6), p. 513; the section runs pp. 512--513) of A. A. Gol'dberg, Sets on which the modulus of an entire function has a lower bound (Russian), Sibirsk. Mat. Zh. 20 (1979), no. 3, 512--518, 691, the edition named on the source card. The paper numbers its sections 1°, 2°, 3° and gives its results no theorem labels.

Statement

Setting (p. 512). For an entire function $f$ and $c>0$, $E(c)=\{z:|f(z)|>c\}$, $|E(c)|$ is its planar (Lebesgue) measure, and $M(r,f)=\max\{|f(z)|:|z|=r\}$. The problem the paper solves (Hayman's Problem 2.40) concerns entire functions that are not identically constant.

Result of 1° (formula (6), p. 513). Let $f$ be an entire function and let $c>0$ be such that $|E(c)|<\infty$. Then

$$\int_{r_0}^{\infty}\frac{r\,dr}{\ln\ln M(r,f)}<\infty ,$$

where the paper takes the lower limit $r_0>1$ (p. 512). This is the convergence half of Hayman's conjecture; that it cannot be sharpened is section 2°.

Read depth. Claims checked: the statement, the setting and the steps (1)--(6) were read on the page images of pp. 512--513. The inequality (1) that the proof imports from Pfluger and Arima was not checked against their papers, and nothing here is independently reviewed.

Proof pointer

Pages 512--513, outlined here. Write $\Gamma_r$ for the circle $|z|=r$, $A$ for the set of $r>0$ at which $\Gamma_r$ is not contained in $E(c)$, $A(r)=A\cap[1,r]$ for $r>1$, and $l(r)$ for the length of the longest arc of $\Gamma_r\cap E(c)$ when $r\in A$. The proof starts from the Carleman-type estimate (1), proved independently by Pfluger and by Arima: $\ln^+\ln^+M(er,f)\ge\pi\int_{A(r)}dt/l(t)-K$ for $r>1$, with a constant $K$. Finite area of $E(c)$ gives that $[1,\infty)\setminus A$ has finite linear measure $L$ and that $\int_A l(t)\,dt<\infty$ (2). By (1) it suffices to show that $r$ divided by $\int_{A(r)}dt/l(t)$ is integrable at infinity. On each dyadic block $A\cap[2^{j-1},2^j)$, with $2^N>2L$ and $j\ge N+1$, the Cauchy--Bunyakovsky inequality and the lower bound $2^{j-2}$ for the block's measure bound the reciprocal of $\int dt/l(t)$ over the block by $2^{-2j+4}$ times $\int l(t)\,dt$ over the block (steps (3)--(5)); summing over the blocks and using (2) gives (6).

Dependencies

Inequality (1), cited from A. Pfluger, Compt. Rend. Acad. Sci. 229 (1949), 542--543, and K. Arima, J. Math. Soc. Japan 4 (1952), 62--66; the paper notes (p. 512) that Pfluger does not write (1) explicitly and that it follows easily from the stronger inequality he proves.

Bears on

• Problem 1118: the problem's first question asks for the minimal growth of a non-constant entire $f$ for which $E(c)$ has finite measure for some $c$. This page gives the bound $\int^\infty r\,dr/\ln\ln M(r,f)<\infty$, which Hayman conjectured; section 2° shows it is best possible in the sense stated there. The paper states the question as Problem 2.40 of Hayman's list.