For a meromorphic function of finite positive order in the unit disk, a radius is a Borel radius if and only if the Nevanlinna characteristic restricted to some sector around it attains the maximal growth exponent; this equivalence makes the T-based and N-based definitions of Borel radius interchangeable. Wu and Xuan (2012), in the paper proving that an algebroid function and its derivative share their Borel radii, noted that the forward implication of this equivalence carries over to ν\nu-valued algebroid functions and posed its converse as an open question: if the angular characteristic of a ρ\rho-order algebroid function in some sector around a radius has growth exponent ρ\rho, must that radius be a Borel radius, i.e. must the distinct-point counting function of every value except at most 2ν2\nu attain the exponent ρ\rho in every sub-sector? The converse is known for meromorphic functions (ν=1\nu = 1) but open in the genuinely multivalued case ν≥2\nu \geq 2.

Background

Nevanlinna value distribution theory measures, for a meromorphic function ff on the plane or the unit disk, how often ff takes each complex value. Its central objects are the proximity function m(r,f)m(r, f) (average closeness of ff to a value on circles), the counting function N(r,f=a)N(r, f = a) (counting aa-points with multiplicity), and their sum, the Nevanlinna characteristic T(r,f)T(r, f), whose growth rate is summarized by the order ρ\rho. A direction, or in the unit disk a radius L(θ)L(\theta), is a Borel direction (radius) of ff if the value distribution of ff is maximal in every sector around it: the counting function of every value aa, with at most two exceptions, has the same order ρ\rho inside the sector as the function has globally. Since Valiron's work (Valiron, 1928) these singular directions have been studied through angular analogues of the Nevanlinna functions, in particular the angular characteristic T(r,Δ(θ,ε),f)T(r, \Delta(\theta, \varepsilon), f) restricted to a sector Δ(θ,ε)\Delta(\theta, \varepsilon).

An algebroid function is a multivalued analytic function ww defined implicitly by an equation Aν(z)wν+Aν−1(z)wν−1+⋯+A0(z)=0A_\nu(z) w^\nu + A_{\nu-1}(z) w^{\nu-1} + \cdots + A_0(z) = 0 whose coefficients are analytic and have no common zeros; generically it takes ν\nu values at each point and lives naturally on a ν\nu-sheeted Riemann surface over the disk. Nevanlinna theory extends to this setting: the characteristic and counting functions are defined by summing contributions over the sheets, and the second fundamental theorem — the theorem bounding how many values a function can take too rarely — holds with at most 2ν2\nu exceptional values instead of 22. This larger exceptional bound is the source of the extra difficulty in almost every question that transfers from meromorphic to algebroid functions.

For meromorphic functions of finite positive order in the unit disk, a radius L(θ)L(\theta) is a Borel radius if and only if there is some sector around it whose angular characteristic has the maximal growth exponent ρ\rho (condition (5.1) below). This equivalence, going back to Zhang Guanghou's work in the plane (Zhang, 1977), was proved for the disk with small-function refinements by Wu and Xuan (2013). It is the tool that makes the T-based and N-based definitions of Borel radius interchangeable, and it underlies the source paper (Wu and Xuan, 2012), whose theorems state that under a growth condition on T(r,w)T(r, w) and, in the infinite-order case, hyperorder 00 (the hyperorder is the growth exponent of log⁡T(r,w)\log T(r, w), i.e. lim sup⁡r→1−log⁡log⁡T(r,w)/log⁡11−r\limsup_{r \to 1^-} \log\log T(r, w)/\log\frac{1}{1-r}), an algebroid function ww and its derivative w′w' have the same Borel radii — a disk version of Valiron's classical program on common Borel directions of a function and its derivative.

The forward implication of the equivalence survives the transfer to algebroid functions verbatim: a Borel radius satisfies condition (5.1). The converse is what Wu and Xuan (2012) left open in Section 5 of their paper. Kong (2012) developed the T-radius theory for algebroid functions in the disk: she proved, under radial growth regularity, that T-radii and Borel radii exist and that a T-radius is a Borel radius when the growth limit exists; but her T-radius condition is a relative-density condition on the integrated counting function, not the sectorial-characteristic growth condition (5.1), so her bridge does not settle the converse. The meromorphic proof of the converse compares the sectorial characteristic with the characteristic of a single-valued function after a conformal map of the sector to a disk; for a ν\nu-valued ww the sheets transform separately and the branch points lying over the sector contribute terms with no single-valued counterpart, so the question is genuinely open in the multivalued case.

Problem Statement

Let ww be a ν\nu-valued algebroid function in the unit disk, given by an irreducible equation Aν(z)wν+Aν−1(z)wν−1+⋯+A0(z)=0A_\nu(z) w^\nu + A_{\nu-1}(z) w^{\nu-1} + \cdots + A_0(z) = 0 with analytic coefficients having no common zeros, and let T(r,w)T(r, w) denote its Nevanlinna characteristic on the associated ν\nu-sheeted Riemann surface. Suppose ww has finite positive order

ρ=lim sup⁡r→1−log⁡T(r,w)log⁡11−r∈(0,∞).\rho = \limsup_{r \to 1^-} \frac{\log T(r, w)}{\log \frac{1}{1-r}} \in (0, \infty).

For θ∈[0,2π)\theta \in [0, 2\pi) and 0<ε<π/20 < \varepsilon < \pi/2 let Δ(θ,ε)\Delta(\theta, \varepsilon) be the sector {z:∣z∣<1, ∣arg⁡z−θ∣<ε}\{ z : |z| < 1,\ | \arg z - \theta| < \varepsilon \}, let T(r,Δ(θ,ε),w)T(r, \Delta(\theta, \varepsilon), w) be the angular characteristic of ww in that sector, and let N‾(r,Δ(θ,ε),w=a)\overline{N}(r, \Delta(\theta, \varepsilon), w = a) be the integrated counting function of the distinct aa-points of ww in the sector on the Riemann surface, not counting multiplicities. Recall that L(θ)L(\theta) is a Borel radius of ww of order ρ\rho if for every ε>0\varepsilon > 0,

lim sup⁡r→1−log⁡N‾(r,Δ(θ,ε),w=a)log⁡11−r=ρ\limsup_{r \to 1^-} \frac{\log \overline{N}(r, \Delta(\theta, \varepsilon), w = a)}{\log \frac{1}{1-r}} = \rho

holds for every a∈C^a \in \hat{\mathbb{C}} except at most 2ν2\nu exceptional values. It is elementary that a Borel radius satisfies the sectorial growth condition

lim sup⁡r→1−log⁡T(r,Δ(θ,ε),w)log⁡11−r=ρfor some ε>0.(5.1)\limsup_{r \to 1^-} \frac{\log T(r, \Delta(\theta, \varepsilon), w)}{\log \frac{1}{1-r}} = \rho \quad \text{for some } \varepsilon > 0. \tag{5.1}

Open question (Wu and Xuan, 2012, Section 5): does condition (5.1) imply that L(θ)L(\theta) is a Borel radius of ww? The case ν=1\nu = 1 of meromorphic functions is known; the essential open case is ν≥2\nu \geq 2. Note on variants: the published version of the source phrases the hypothesis of (5.1) as holding for every sufficiently small ε>0\varepsilon > 0 rather than for some ε>0\varepsilon > 0; the form above (from the arXiv text) has the weaker hypothesis, so an affirmative answer to it implies an affirmative answer to the published form, and a counterexample to either form must satisfy the weaker hypothesis to be conclusive.

References

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