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Statement

Notation (p. 22). Throughout Chapter 2, $M(r,f)=\max_{|z|=r}|f(z)|$.

Problem 2.40 (pp. 37--38, quoted). "Let $f(z)$ be a non-constant entire function, and assume that for some constant $c$ the plane measure of the set $E(c)$ where $|f(z)|>c$ is finite. What is the minimum growth rate of $f(z)$? Hayman conjectures that

$$\int_0^\infty\frac{r\,dr}{\log\log M(r,f)}<\infty$$

is true and best possible. If $E(c)$ has finite measure, is the same true for $E(c')$ for $c'<c$?"

The book attributes the problem to P. Erdős, and Table 2 (p. 253) lists it among the problems of the 1974 symposium list.

Update 2.40 (p. 38). The update credits Camera's thesis (the book's [127]: G. Camera, Doctoral thesis, University of London, 1977) with Hayman's conjecture: the integral above is finite, and this is best possible in the sense that for an increasing $\phi(r)$ with $\int_0^\infty r\,dr/\phi(r)=\infty$, as printed, there is an entire $f$ with $\log\log M(r,f)<\phi(r)$ bounded outside a set of finite area. It also credits Camera with the analogue for subharmonic $u$ in $\mathbb{R}^m$, with $B(r)=\sup_{|x|=r}u(x)$ and $\int_0^\infty(r/\log B(r))^{m-1}\,dr<\infty$, best possible in the same sense. It records independent proofs by Hansen (the book's [375]: L. J. Hansen, On the growth of entire functions bounded on large sets, Canad. J. Math. 29 (1977), 1287--1291) and Gol'dberg (the book's [317]: A. A. Gol'dberg, Sets on which the modulus of an entire function has a lower bound, Sibirsk. Mat. Zh. 20 (1979), 512--518), and says Gol'dberg answered the second part with a function for which "$A(c)$ is finite for some $c$, but not for all $c$"; $A(c)$ is not defined there and stands for the measure of $E(c)$.

The printed sharpness condition $\int_0^\infty r\,dr/\phi(r)=\infty$ contradicts the bound it qualifies, as the corpus's Camera claim page explains; that page states the sharpness with $\int^\infty r\,dr/\phi(r)<\infty$, in the form of Gol'dberg's paper.

Source. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (21 September 2018), Chapter 2, pp. 37--38. The edition read is identified on the source card.

Read depth. Claims checked: the problem, its update and the three cited reference entries were read clause by clause on the printed pages. The book proves nothing; it poses and reports.

Proof pointer

None; a problem. Gol'dberg's paper is on the Gol'dberg card.

Dependencies

None.

Bears on

• Problem 1118: the same two questions. The book asks whether $E(c')$ has finite measure for $c'<c$; the site asks whether there must exist some $c'<c$ for which it does. Update 2.40 credits the growth bound and its sharpness to Camera, independent proofs to Hansen and Gol'dberg, and Gol'dberg with an example finite for some $c$ but not for all $c$.