Must a transcendental meromorphic function of finite positive order and its derivative share at least one Borel direction? This is Valiron's 1928 question, one of the oldest open problems of angular value distribution theory (Nevanlinna theory). The surrounding cases are settled: for entire functions of finite positive order, Milloux (1951) proved that every Borel direction of the derivative is a Borel direction of the function; for meromorphic functions of infinite order, Sun (1987) proved that the Borel directions of ff and f′f' coincide; for meromorphic functions of zero order, Chern (2004) constructed an explicit function having no common Borel direction with its derivative, so the statement without growth restrictions is false. The original case — transcendental meromorphic functions of finite positive order in the plane — remains open, with positive results known only under additional exceptional-value or growth hypotheses (Rauch, Chuang, Zhang Guanghou, Yang-Zhang, Ji, Ru, Chuang).

Background

Nevanlinna value distribution theory quantifies how often a meromorphic function ff (holomorphic except for poles) takes its values, with growth measured by the Nevanlinna characteristic T(r,f)T(r,f); the order ρ(f)=lim sup⁡r→∞log⁡T(r,f)/log⁡r\rho(f)=\limsup_{r\to\infty}\log T(r,f)/\log r classifies functions by growth (rational functions have order 00, eze^z has order 11). Angular value distribution refines this to sectors: a ray arg⁡z=θ\arg z=\theta is a Borel direction of order ρ\rho of a function ff of finite positive order ρ=ρ(f)\rho=\rho(f) if in every open sector Ω(θ,ε)={z: ∣arg⁡z−θ∣<ε}\Omega(\theta,\varepsilon)=\{z:\ |\arg z-\theta|<\varepsilon\} around the ray, every value a∈C∪{∞}a\in\mathbb{C}\cup\{\infty\} — with at most two exceptions — is assumed with maximal frequency, i.e. lim sup⁡r→∞log⁡n(r,Ω(θ,ε),a,f)/log⁡r=ρ\limsup_{r\to\infty}\log n(r,\Omega(\theta,\varepsilon),a,f)/\log r=\rho, where nn counts aa-points with multiplicity (poles for a=∞a=\infty) of modulus below rr in the sector. Valiron (1928) proved that every meromorphic function of finite positive order has at least one Borel direction, and posed the question of whether such a function must share a Borel direction with its derivative. Since a transcendental meromorphic function has the same order as its derivative, the question is well posed at the common order ρ\rho. The question has also been stated in a stronger form, asking for a direction common to ff and all its higher derivatives f(k)f^{(k)}, k≥1k\ge1 (Ji 1981); the case of ff and f′f' is the base instance and the one stated in the most recent literature.

The surrounding cases are settled. Milloux (1951) proved that for an entire function of finite positive order, every Borel direction of f′f' is a Borel direction of ff, so ff and f′f' share all of f′f''s Borel directions. For meromorphic functions of infinite order, Sun (1987), using proximate orders ρ(r)\rho(r) (smooth monotone scales tracking T(r,f)T(r,f) when the order is infinite) and type functions (auxiliary scales comparing T(r,f)T(r,f) with rρ(r)r^{\rho(r)}), proved that the Borel directions of ff and f′f' coincide without additional hypotheses. For meromorphic functions of zero order, Chern (2004) constructed an explicit function with no common Borel direction with its derivative (in the logarithmic-order scale appropriate to zero order), so the general statement without growth restrictions is false. The remaining case — meromorphic functions of finite positive order in the plane — is the original question and is open.

For the open case, two structural facts delimit the difficulty. First, the Milloux inclusion provably fails: an example attributed to Steinmetz (communicated in a 1980 letter of W. K. Hayman and documented by Zhang-Dai 1989), f(z)=ez/(1+eiz)f(z)=e^z/(1+e^{iz}), has a Borel direction of f′f' (arg⁡z=0\arg z=0) that is not a Borel direction of ff; hence the entire-function strategy cannot be transplanted, and poles are the essential obstruction. Second, all known positive results for meromorphic ff require additional hypotheses: Rauch (1934) and Chuang (1937) gave conditions in terms of Borel exceptional values in an angle; Zhang Guanghou (1977, parts I-III), Yang-Zhang (1984), Ji (1981), Dai-Ji (1981), Ru (1989), and Chuang (1997) obtained existence of common Borel directions under exceptional-value, deficiency, or slow-growth conditions (a Borel exceptional value is a value assumed with frequency of order strictly below ρ\rho; a Nevanlinna deficiency is a normalized frequency deficit of a value); a standard account is Yang's 1982 monograph. Adjacent but distinct lines include common Borel radii (the unit-disk radial analogue of a Borel direction) of algebroid functions (multi-valued analytic functions satisfying a polynomial equation with holomorphic coefficients) in the unit disk (Wu-Xuan 2009, where the analogue of Valiron's question has positive answers), value distribution with respect to arguments and T-directions, an order-free analogue of Borel directions (Zheng 2011), common Borel directions for the qq-difference operator Dqf(z)=f(qz)−f(z)D_q f(z)=f(qz)-f(z) (Long-Qiao-Yao 2020; Qin-Long-Wang 2024), and existence of singular directions for functions of finite logarithmic order (Hu-Zhang 2025) and for solutions of complex differential equations (Zhang-Yang 2025, entire infinite-order solutions). None of these addresses the finite-positive-order meromorphic case in the plane.

Problem Statement

Let ff be a transcendental meromorphic function in the complex plane of finite positive order ρ=ρ(f)∈(0,∞)\rho=\rho(f)\in(0,\infty), where ρ(f)=lim sup⁡r→∞log⁡T(r,f)/log⁡r\rho(f)=\limsup_{r\to\infty}\log T(r,f)/\log r and T(r,f)T(r,f) is the Nevanlinna characteristic; note that ρ(f′)=ρ(f)\rho(f')=\rho(f). For a ray arg⁡z=θ\arg z=\theta and ε>0\varepsilon>0, write Ω(θ,ε)={z: ∣arg⁡z−θ∣<ε}\Omega(\theta,\varepsilon)=\{z:\ |\arg z-\theta|<\varepsilon\}, and let n(r,Ω(θ,ε),a,f)n(r,\Omega(\theta,\varepsilon),a,f) be the number of aa-points of ff (counted with multiplicity; poles for a=∞a=\infty) in Ω(θ,ε)\Omega(\theta,\varepsilon) with ∣z∣<r|z|<r. The ray arg⁡z=θ\arg z=\theta is a Borel direction of order ρ\rho of ff if for every ε>0\varepsilon>0, lim sup⁡r→∞log⁡n(r,Ω(θ,ε),a,f)/log⁡r=ρ\limsup_{r\to\infty}\log n(r,\Omega(\theta,\varepsilon),a,f)/\log r=\rho holds for every a∈C∪{∞}a\in\mathbb{C}\cup\{\infty\} with at most two exceptions.

Question (Valiron, 1928): must ff and f′f' have at least one common Borel direction of order ρ\rho? Provide either: (a) a proof that for every transcendental meromorphic function ff of finite positive order, ff and f′f' share at least one Borel direction of order ρ(f)\rho(f); or (b) an explicit transcendental meromorphic function ff of finite positive order, together with a rigorous proof that the sets of Borel directions of order ρ(f)\rho(f) of ff and of f′f' are disjoint.

The stronger variant asking for a ray common to ff and all derivatives f(k)f^{(k)}, k≥1k\ge1, is equally open and is not required; an answer settling (a) or (b) for ff and f′f' resolves the problem as posed.

References

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