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Statement

Notation (p. 4). For $f$ meromorphic in $|z|<R$ and $0<r<R$, $n(r,a)$ is the number of roots of $f(z)=a$ in $|z|\le r$, counted with multiplicity.

Problem 1.25 (p. 14, quoted). "In the opposite direction to Problem 1.24, does there exist a meromorphic function such that for every pair of distinct values $a,b$, we have

$$\limsup_{r\to\infty}\frac{n(r,a)}{n(r,b)}=\infty\qquad\text{and}\qquad\liminf_{r\to\infty}\frac{n(r,a)}{n(r,b)}=0.$$

"

The book adds that either of the two conditions, for all distinct $a,b$, implies the other; compares the result (1.3) quoted in Problem 1.2, which rules the property out for the $N$-function; and says the question can also be asked for entire functions. It attributes the problem to P. Erdős, and Table 2 (p. 253) lists it among the problems of the 1974 symposium list.

Update 1.25 (p. 14). The update opens "The exceptional sets are necessary." It then credits Gol'dberg (the book's [315]: A. A. Gol'dberg, Counting functions of sequences of $a$-points for entire functions, Sibirsk. Mat. Ž. 19 (1978), 28--36) and Toppila (the book's [754]: S. Toppila, On the counting function for the $a$-values of a meromorphic function, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 565--572) with entire functions for which $\limsup_{r\to\infty}n(r,a)/n(r,b)=\infty$ for every finite unequal pair $(a,b)$, and Toppila [754] with a corresponding meromorphic example.

Source. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (21 September 2018), Chapter 1, p. 14. The edition read is identified on the source card.

Read depth. Claims checked: the notation, the problem, its update and the two 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. The two papers the update credits are on the Gol'dberg card and the Toppila card.

Dependencies

None.

Bears on

• Problem 1116: the problem's wording follows Problem 1.25, which it cites from Hayman's 1974 list; the site asks only for the limsup condition, which by the book's remark is equivalent to the pair. The book counts roots in $|z|\le r$, the site in $|z|<r$. Update 1.25 credits entire examples to Gol'dberg and Toppila and a meromorphic one to Toppila.