Common Borel directions of a function of finite positive order and its q-difference operator $D_qf$

The $q$-difference operator $D_qf(z)=f(qz)-f(z)$, $q\in\mathbb{C}\setminus\{0,1\}$, is the multiplicative-discrete analogue of the derivative, and the normalized quotient $(f(qz)-f(z))/((q-1)z)$ tends to $f'(z)$ as $q\to1$. Classical value distribution theory asks whether a meromorphic function and its derivative share a singular direction: Milloux proved in 1951 that an entire function of finite positive order and its derivative always share Borel directions, while the meromorphic finite positive-order case is still open. Long, Qiao and Yao (2020) posed the $q$-difference analogue and proved only sufficient conditions — a sectoral growth-nontriviality hypothesis for order greater than $1/2$ and growth-regularity hypotheses in the infinite-order regime — and only for real $q$; a 2024 follow-up by the same group added hyper-order and logarithmic-order regimes, still only sufficient conditions and still for real $q$. The open problem stated here is whether every transcendental entire (or meromorphic) function of finite positive order shares a common Borel direction of order $\rho(f)$ with $D_qf$ for every $q\in\mathbb{C}\setminus\{0,1\}$ with $|q|\ne1$, or whether a non-degenerate counterexample exists; rotational parameters with $|q|=1$, where $D_qf$ can even vanish identically, lie outside this existence question. The Julia-direction analogue and the necessity of the known sufficient conditions are companion facets of the same singular-direction problem.

Background

Value distribution theory (Nevanlinna theory) studies how often a meromorphic function takes each value. Growth is measured by the Nevanlinna characteristic $T(r,f)$, a sum of an average magnitude term and a pole-counting term, and the order of $f$ is $\rho(f)=\limsup_{r\to\infty}\log^+T(r,f)/\log r$; 'finite positive order' means $0<\rho(f)<\infty$. For $a$ in the Riemann sphere $\overline{\mathbb{C}}$ and a sector $\Omega(\alpha,\beta)=\{z:\alpha<\arg z<\beta\}$, denote by $n(r,\Omega,a,f)$ the number of $a$-points of $f$ (poles when $a=\infty$) in $\Omega\cap\{|z|<r\}$, counted with multiplicity.

A ray $\arg z=\theta$ is a Borel direction of a meromorphic function $f$ of finite positive order $\rho$ if for every $\varepsilon>0$,

$$\limsup_{r\to\infty}\frac{\log n(r,\Omega(\theta-\varepsilon,\theta+\varepsilon),a,f)}{\log r}=\rho$$

holds for every $a\in\overline{\mathbb{C}}$ with at most two exceptions: it is a direction along which values are assumed with maximal frequency. A ray is a Julia direction if in every angle containing it $f$ takes every complex value, with at most one exception, infinitely often. Borel directions and Julia directions together are called singular directions. Valiron (1928) proved that every meromorphic function of finite positive order has at least one Borel direction.

Because $f$ and its derivative $f'$ always have the same order, Valiron asked whether a meromorphic function of finite positive order and its derivative must share a common Borel direction. Milloux (1951) settled the entire case: if $f$ is entire of finite positive order, then every Borel direction of $f'$ is also a Borel direction of $f$, so common Borel directions exist. For meromorphic $f$ of finite positive order the question remains open, with partial results under extra hypotheses (for example, the existence in the relevant angle of a Borel exceptional value, meaning a value $a$ whose $a$-point counting function grows strictly slower than $T(r,f)$); the infinite-order meromorphic case was settled completely using type functions and proximate orders — a proximate order is a monotone function $\rho(r)$ tending to $\rho$ with $\rho'(r)\,r\log r\to0$, refining the order to capture the second-order growth of $T(r,f)$, and a type function $r^{\rho(r)}$ correspondingly refines the growth constant. Anderson and Clunie (1969) developed the corresponding theory of Julia lines for entire functions of finite order.

The $q$-difference operator is $D_qf(z)=f(qz)-f(z)$ with $q\in\mathbb{C}\setminus\{0,1\}$; it is the multiplicative-discrete analogue of the derivative, since $(f(qz)-f(z))/((q-1)z)\to f'(z)$ as $q\to1$. Nevanlinna theory for $D_q$ was founded by Barnett, Halburd, Morgan and Korhonen (2007), who proved difference analogues of the lemma on the logarithmic derivative and used them for meromorphic solutions of $q$-difference equations; their estimates are radial (whole-plane) statements, and no sectorial or angular version is known. A related but distinct lineage is the work of Ishizaki and Yanagihara (2005) on Borel and Julia directions of meromorphic Schröder functions, that is, solutions of functional equations $f(\lambda z)=R(f(z))$; there the directions of the solution itself are found, not directions common to a function and its $q$-difference.

Long, Qiao and Yao (2020) posed the $q$-difference analogue of the common-direction problem: when do $f$ and $D_qf$ share a common singular direction? For real $q\in\mathbb{R}\setminus\{0,1\}$ they established sufficient conditions — in the finite-order regime, for entire functions of order greater than $1/2$ satisfying a sectoral Phragmén–Lindelöf-type growth-nontriviality condition (a sectorial growth condition in the spirit of the classical Phragmén–Lindelöf principle, which bounds a holomorphic function's growth inside a sector by its growth on the sector's boundary), and in the infinite-order regime under a growth-regularity hypothesis, with proximate order and type preserved under $D_q$; the paper also discusses common singular directions of solutions of $q$-difference equations. Qin, Long and Wang (2024) extended this sufficient-condition programme to extreme growth regimes, using the hyper order $\rho_2(f)=\limsup_{r\to\infty}\log^+\log^+T(r,f)/\log r$ (the order of $\log T(r,f)$, which is positive only for very fast growth) and the logarithmic order $\rho_{\log}(f)=\limsup_{r\to\infty}\log^+T(r,f)/\log\log r$ (which measures very slow, sub-power growth). They proved: for entire $f$ with positive lower hyper order and infinite hyper order ($0<\mu_2(f)\le\rho_2(f)=\infty$, where the lower hyper order $\mu_2$ is defined like $\rho_2$ with $\limsup$ replaced by $\liminf$) satisfying the regularity condition $\lim_{r\to\infty}\log\log T(cr,f)/\log\log T(r,f)=1$ for some $c>1$, $f$ and $D_qf$ share a common Borel direction of hyper proximate order (Theorem 2.1); for entire $f$ of finite hyper order $\rho_2(f)\in(1/2,\infty)$ with non-trivial hyper-order growth, meaning the raywise hyper order $\rho_{2,\theta}(f)$ is not almost everywhere constant, they share a common Borel direction of hyper order (Theorem 3.1); and for entire $f$ of finite logarithmic order $\rho_{\log}(f)\in[1,\infty)$ with non-trivial logarithmic-order growth, they share a common Borel direction of logarithmic order $\rho_{\log}(f)-1$ (Theorem 4.1). All of these results are sufficient conditions for real $q$ and entire $f$ only. Their Lemma 2.1, imported from the 2020 paper, records the order-preservation identity $\rho_2(D_qf)=\rho_2(f)$ for meromorphic $f$ whenever $|q|\ne1$ — the identity that makes a statement about Borel directions of hyper order meaningful, whereas in the finite positive-order regime it is the order-preservation identity $\rho(D_qf)=\rho(f)$ for $|q|\ne1$ (derived in the Problem Statement) that makes a statement about Borel directions of order $\rho(f)$ of $D_qf$ meaningful — and the rotational parameters $|q|=1$ lie outside its scope.

What is missing is any unconditional statement: no general theorem covers all finite-order functions, nothing is proved for complex $q$, no necessary conditions are known, and the Julia-direction common-direction problem is likewise unsettled; the rotational parameters $|q|=1$, where $D_qf$ can even degenerate identically, are likewise untouched.

Problem Statement

Let $f$ be a transcendental entire function of finite positive order $\rho(f)$ — more generally, a transcendental meromorphic function of finite positive order — and let $D_qf(z)=f(qz)-f(z)$ for $q\in\mathbb{C}\setminus\{0,1\}$.

Scope of the parameter $q$. For $|q|\ne1$ the order of $D_qf$ equals that of $f$: the telescoping identity $f(z)=\sum_{n=0}^{m-1}(f(z/q^{n})-f(z/q^{n+1}))+f(z/q^{m})$ (used with $|q|>1$, or rescaled via $1/q$ when $0<|q|<1$) gives $\rho(D_qf)=\rho(f)$, and the hyper-order identity $\rho_2(D_qf)=\rho_2(f)$ for $|q|\ne1$ is Lemma 2.1 of Qin–Long–Wang (2024), taken from the 2020 paper. So for $|q|\ne1$ the phrase "Borel direction of order $\rho(f)$ of $D_qf$" is unambiguous. For $|q|=1$ this can degenerate: if $q$ is a root of unity of order $m$ and $f(z)=g(z^{m})$, then $D_qf\equiv0$ — for instance $f(z)=e^{z^{2}}$ with $q=-1$ — and $D_qf$ then has no Borel directions; for irrational rotations ($|q|=1$, $q$ not a root of unity) no order-preservation statement is known. Such degenerate or rotational parameters motivate the restriction $|q|\ne1$ in the existence question below; a pair with $D_qf\equiv0$ is a triviality, not a resolution of the existence question.

Main question. Decide between the two admissible resolutions of the common-Borel-direction existence problem posed by Long–Qiao–Yao (2020) for the $q$-difference operator:

(i) a general Milloux-type existence theorem: every transcendental function of finite positive order, whether entire or meromorphic, shares a common Borel direction of order $\rho(f)$ with $D_qf$ for every $q\in\mathbb{C}\setminus\{0,1\}$ with $|q|\ne1$ — a parameter class containing every real $q$ except $-1,0,1$ and including the complex $q$ that no result to date covers — without auxiliary hypotheses such as sectoral growth nontriviality, order greater than $1/2$, or growth regularity (both classes are required for this resolution, the meromorphic case being the wider one that contains the entire case); or

(ii) a non-degenerate counterexample: a transcendental entire or meromorphic function $f$ of finite positive order and a parameter $q\in\mathbb{C}\setminus\{0,1\}$ with $|q|\ne1$ — so that $\rho(D_qf)=\rho(f)$ holds automatically — such that the set of Borel directions of order $\rho(f)$ of $f$ and the corresponding set for $D_qf$ are disjoint.

Related open facets. The source's common singular-direction problem is broader than this existence question. Whether the known sufficient conditions — sectoral nontriviality for order greater than $1/2$, the growth-regularity hypotheses in the infinite-order and hyper-order regimes, and non-trivial hyper-order or logarithmic-order growth — are also necessary, the Julia-direction analogue (must $f$ and $D_qf$ share a common Julia direction?), and the behavior of the rotational parameters $|q|=1$ (root-of-unity degeneracies $D_qf\equiv0$ and irrational rotations, for which no order-preservation statement is known) are companion facets of the same singular-direction problem that the literature leaves unaddressed; they are not part of the existence question posed above. Measure-theoretic refinements ("for Lebesgue almost all $q$") are likewise recorded as further open extensions.

References

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