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Source. Section 2° (statement p. 513, with condition (7); the section's lemma p. 514; the construction pp. 515--517, ending with formula (22) and the growth estimate on p. 517) 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.

Statement

Setting as in section 1°: $E(c)=\{z:|f(z)|>c\}$, $|E(c)|$ its planar measure, and $M(r,f)=\max\{|f(z)|:|z|=r\}$.

Result of 2° (p. 513). Let $\Phi$ be an arbitrary continuous, positive, nondecreasing function on $[0,\infty)$ such that

$$\int_1^{\infty}\{\Phi(r)\}^{-1}\,r\,dr<\infty . \qquad (7)$$

Then there is an entire function $f$ such that $\ln\ln M(r,f)=O(\Phi(r))$ as $r\to\infty$ and $|E(c)|<\infty$ for every $c>0$.

The paper presents this as showing that relation (6) of 1° cannot be sharpened in this sense.

Read depth. Claims checked: the statement, the lemma and the outline of the construction were read on the page images of pp. 513--517. The estimates (13)--(21) were not rechecked, the cited theorems of Boichuk and Warschawski and the continuation of $f$ to an entire function, which the paper obtains by standard methods with references to Evgrafov and to Gol'dberg and Ostrovskii (p. 516), were not checked against their sources, and nothing here is independently reviewed.

Proof pointer

Pages 514--517, outlined here.

• Lemma (p. 514; the paper says on p. 513 that its proof was communicated to the author by V. S. Boichuk). There is a twice continuously differentiable positive function $L$ on $[0,\infty)$ with $L(r)r^2=O(\Phi(r))$ as $r\to\infty$, $\int_1^\infty dr/(rL(r))<\infty$ (8), and $rL'(r)=o(L(r))$ as $r\to\infty$. The print states the last condition as "$r^2L'(r)=o(L(r))$" [sic]; the proof (p. 515) establishes $r^kL^{(k)}(r)=o(L(r))$ for $k=1,2$, so it is meant as $r^2L''(r)=o(L(r))$. The proof (pp. 514--515) regularizes $\Phi$ through a function of first order and convergence class and uses a theorem of Boichuk on proximate orders.
• Construction (pp. 515--517). With $\theta(x)=1/(x^2L(x))'$, which behaves like $(2xL(x))^{-1}$ by (13), the paper takes the curvilinear half-strips $S(q)=\{x>0,\ |y|<q\theta(x)/2\}$, a conformal map $\zeta$ of $S(1)$ onto the half-strip $\{\xi>0,\ |\eta|<\pi/2\}$ with asymptotics (14) from a theorem of Warschawski, and defines $f$ outside $S(3/4)$ as the Cauchy integral of $F=\exp\exp(2\zeta)$ over the boundary of $S(3/4)$. This extends to an entire function equal to that integral plus $\exp\exp(2\zeta(z))$ inside $S(3/4)$ (17), and $f$ is $O(1/z)$ outside $S(3/4)$ (22). Hence for each $c>0$ the set $E(c)$ lies in a disc together with $S(1)$, whose area $\int_0^\infty\theta(x)\,dx$ is finite by (13) and (8); and $\ln\ln M(r,f)\le(1+o(1))4\pi r^2L(r)=O(\Phi(r))$.

Dependencies

Section 1° supplies the bound this result shows to be sharp. The construction cites V. S. Boichuk, Sibirsk. Mat. Zh. 20 (1979), no. 2, 229--236, and S. E. Warschawski's theorem on conformal maps of infinite strips (Matematika 2 (1958), no. 4, 67--116).

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 $c$. Together with section 1°, this page shows that Hayman's convergence condition is best possible: for every $\Phi$ satisfying (7) some entire $f$ with $\ln\ln M(r,f)=O(\Phi(r))$ has $|E(c)|<\infty$, and indeed for every $c>0$.