Research Problems in Function Theory — Problem 2.28

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A nonconstant meromorphic function $f$ has bounded value distribution if for every $r>0$ there is a finite constant $C_f(r)$, uniform over disc centres and finite target values $w$, bounding the number of solutions of $f(z)=w$ in every disc of radius $r$, counted with multiplicity. One may equivalently impose such a bound at one fixed positive radius. For entire $f$, let $C_f(r)$ denote the least such bound.

Statement recorded in the supplied source.

(a) If an entire function has bounded value distribution, must $C_f(r)=O(r)$ as $r\to\infty$, and hence must $f$ have at most exponential type? (b) For a meromorphic function, is it enough to impose the uniform disc-count bounds for only three distinct finite values $w$ in order to force bounded value distribution for all values? [S2] The paragraph also records Wittich's characterisation for the equation $y^{(n)}+a_1(z)y^{(n-1)}+\cdots+a_n(z)y=0$ with entire coefficients: all its solutions have bounded value distribution exactly when all coefficients are constant. Update 2.28 reports an affirmative answer to (a) and a negative answer to (b). [S2]

Short English statement

Does uniformly bounded local valence force a linear bound as discs grow, and can checking only three target values suffice?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302028 (AMR-022-2028). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.28 and Update 2.28. https://arxiv.org/abs/1809.07200.