Converse of the angular-characteristic characterization of Borel radii for algebroid functions in the unit disc

For a $\nu$-valued algebroid function $w(z)$ of order $\rho$ in the unit disc, Wu and Xuan (Results in Mathematics 62, 2012) proved that the Borel radii of $w$ and of its derivative $w'$ coincide under natural growth hypotheses, and posed as an open problem the converse of the standard characterization of Borel radii. It is easy to show that a Borel radius $L(\theta)$ forces the angular (sectorial) Nevanlinna characteristic $T(r,\Delta(\theta,\varepsilon),w)$ to have maximal growth exponent $\rho$, and for single-valued meromorphic functions in the disc the converse implication is known; but for genuinely multi-valued algebroid functions it is unknown whether the condition $\limsup_{r\to1^-}\log T(r,\Delta(\theta,\varepsilon),w)/\log(1/(1-r))=\rho$ forces $L(\theta)$ to be a Borel radius. The problem asks for a proof or a counterexample. A later literature on T-points and T-radii (positive-density notions) and on prescribed Borel points of algebroid functions has not settled it.

Background

An algebroid function is a multi-valued analytic function defined by an irreducible polynomial equation $A_\nu(z)w^\nu + A_{\nu-1}(z)w^{\nu-1}+\cdots+A_0(z)=0$ whose coefficients $A_j(z)$ are analytic in the open unit disc $\{z:|z|<1\}$ and have no common zeros. Such a $w$ is $\nu$-valued: its natural domain of definition is a $\nu$-sheeted covering surface $\widetilde{R}_z$ over the disc (a Riemann surface whose points $\widetilde{z}$ project to points $z$ of the disc), and $\nu=1$ recovers ordinary meromorphic functions. Nevanlinna value-distribution theory extends to this setting: for a value $a$ in the Riemann sphere $\widehat{\mathbb{C}}$, the counting function $N(r,a)$ counts $a$-points of $w$ on $\widetilde{R}_z$ (integrated as $\frac1\nu\int_0^r n(t,a)\,dt/t$, with $n$ the number of $a$-points over $|z|\le t$, counted with multiplicity; $\overline{N}$ ignores multiplicity), and the characteristic $T(r,w)$ measures the mean covering of the sphere by the sheets of $\widetilde{R}_z$ (for instance via the mean covering number $S(r,w)=\frac1\pi\iint_{|\widetilde{z}|\le r}\big(\frac{|w'|}{1+|w|^2}\big)^2\,d\omega$ and $T(r,w)=\frac1\nu\int_0^r S(t,w)\,dt/t$, which is equivalent to the proximity-plus-counting definition up to a bounded term). In the unit disc the growth scale is $1/(1-r)$, and the order of $w$ is $\rho(w)=\limsup_{r\to1^-}\frac{\log T(r,w)}{\log\frac1{1-r}}.$

Singular directions describe where value distribution concentrates near the boundary. For a radius $L(\theta)=\{z:\arg z=\theta,\ 0<|z|<1\}$ and an angular domain $\Delta(\theta,\varepsilon)=\{z:|\arg z-\theta|<\varepsilon\}$, Wu and Xuan call $L(\theta)$ a Borel radius of $w$ of order $\rho$ if for every $\varepsilon>0$, $\limsup_{r\to1^-}\frac{\log \overline{N}(r,\Delta(\theta,\varepsilon),w=a)}{\log\frac1{1-r}}=\rho$ holds for every $a\in\widehat{\mathbb{C}}$ with at most $2\nu$ exceptional values (the exceptions possibly depending on $\varepsilon$), where $\overline{N}(r,\Delta(\theta,\varepsilon),w=a)$ is the integrated counting function of distinct $a$-points on the part of $\widetilde{R}_z$ lying over the sector. Borel radii are the unit-disc, boundary-point counterparts of the Borel directions of functions in the plane.

The angular characteristic restricts the characteristic to a sector. On the Riemann surface one takes the sectorial mean covering number $S(r,\Delta(\theta,\varepsilon),w)=\frac1\pi\iint_{\widetilde{\Omega}(r,\theta,\varepsilon)}\Big(\frac{|w'|}{1+|w|^2}\Big)^2\,d\omega,\qquad T(r,\Delta(\theta,\varepsilon),w)=\frac1\nu\int_0^r\frac{S(t,\Delta(\theta,\varepsilon),w)}{t}\,dt,$ where $\widetilde{\Omega}(r,\theta,\varepsilon)$ is the part of $\widetilde{R}_z$ over $\{|z|\le r,\ |\arg z-\theta|<\varepsilon\}$ (for meromorphic functions this is the classical angular characteristic of the unit-disc theory cited by Wu and Xuan; proximity-term variants give the same exponent).

The source paper belongs to a lineage going back to Valiron's 1928 question, recalled by Wu and Xuan, of whether a meromorphic function and its derivative share a common Borel direction. The unit-disc analogue — common Borel radii of $f$ and $f'$ — was settled for meromorphic functions of the disc by Q. Zhang's work cited in the source, and Wu and Xuan extended it to algebroid functions: if a $\nu$-valued algebroid function of order $0\le\rho(w)<\infty$ satisfies $\limsup_{r\to1^-}T(r,w)/\log\frac1{1-r}=\infty$, or if $\rho(w)=\infty$ with hyper order $\rho_2(w)=0$ (hyper order defined with $\log\log T$ in place of $\log T$), then $w$ and $w'$ have exactly the same Borel radii. Their key lemma characterizes Borel radii by conformal pullback: $L(\theta)$ is a Borel radius if and only if, for every sector aperture, the pulled-back function $M(\zeta)=w(z(\zeta))$ on an auxiliary disc has order $\rho$.

For meromorphic functions in the disc it is known (the source recalls this equivalence from the unit-disc meromorphic literature, its cited precursors there being Q. Zhang's papers on Borel radii of meromorphic functions in the disc; the adjacent Chinese-school study of meromorphic T-radii and Borel radii by Kang and Zhang 2009 gives corresponding realization results) that $L(\theta)$ is a Borel radius of a $\rho$-order function if and only if for some $\varepsilon>0$ the angular characteristic has maximal exponent, $\limsup_{r\to1^-}\frac{\log T(r,\Delta(\theta,\varepsilon),f)}{\log\frac1{1-r}}=\rho.$ The forward implication (Borel radius implies maximal angular growth of $T$) is also easy for algebroid functions. The converse is where multi-valuedness bites: a proof would need to transfer the exponent from the sectorial characteristic to sectorial counting functions — an angular second-fundamental-theorem step — and the known second fundamental theorem for algebroid functions holds on the full Riemann surface, while the available angular versions carry branch-point counting terms and large error terms. Whether the converse holds for algebroid functions is the open question.

Problem Statement

Let $w(z)$ be a $\nu$-valued algebroid function in the unit disc, defined by an irreducible equation $A_\nu(z)w^\nu+\cdots+A_0(z)=0$ with analytic coefficients having no common zeros, of order $\rho(w)=\rho$, and let $L(\theta)=\{z:\arg z=\theta,\ 0<|z|<1\}$ be a radius. Suppose the angular characteristic of $w$ has maximal growth along $L(\theta)$: there exists $\varepsilon$ with $0<\varepsilon<\pi$ such that $\limsup_{r\to1^-}\frac{\log T(r,\Delta(\theta,\varepsilon),w)}{\log\frac1{1-r}}=\rho(w),$ where $\Delta(\theta,\varepsilon)=\{z:|\arg z-\theta|<\varepsilon\}$ and $T(r,\Delta(\theta,\varepsilon),w)$ is the sectorial characteristic of $w$ on the part of its Riemann surface over the sector. Must $L(\theta)$ be a Borel radius of $w$ of order $\rho$? That is, must it be true that for every $\varepsilon>0$, $\limsup_{r\to1^-}\frac{\log \overline{N}(r,\Delta(\theta,\varepsilon),w=a)}{\log\frac1{1-r}}=\rho$ holds for every $a\in\widehat{\mathbb{C}}$ with at most $2\nu$ exceptional values?

The question is posed for a $\rho$-order algebroid function without further hypotheses. Settle it: prove the implication, or construct a genuinely algebroid ($\nu\ge2$) function of some order $\rho$ and a radius satisfying the angular-characteristic hypothesis that fails the Borel-radius conclusion. (A proof under this existential-$\varepsilon$ hypothesis also settles the stronger variant in which the hypothesis is required for every sufficiently small $\varepsilon>0$, and a counterexample satisfying that stronger variant also refutes the question as stated here.) The case $\nu=1$ of meromorphic functions is already known to hold and is not part of the open question.

References

1. Wu, Nan; Xuan, Zu-xing. "Common Borel Radii of an Algebroid Function and Its Derivative." Results in Mathematics 62(1-2):89-101, 2012 (online 2011). DOI 10.1007/s00025-011-0132-y. https://doi.org/10.1007/s00025-011-0132-y (preprint: Nan Wu and Zuxing Xuan, "Common Borel radius of an algebroid function and its derivative", arXiv:0906.4409, June 2009, https://arxiv.org/abs/0906.4409).
2. Kong, Yinying. "Borel radius and T-Radius of the algebroidal function in the unit disc." Acta Mathematica Scientia, Series B 32(5):1805-1812, 2012. DOI 10.1016/S0252-9602(12)60141-6. https://doi.org/10.1016/S0252-9602(12)60141-6.
3. Wu, Nan; Xuan, Zu-xing. "On T points of algebroid functions." Mathematica Slovaca 62(1):39-48, 2012. DOI 10.2478/s12175-011-0070-x. https://doi.org/10.2478/s12175-011-0070-x.
4. Zhang, Hongshen. "On T point of algebroid functions in |z|<1." Complex Variables and Elliptic Equations 57(9):929-936, 2012. DOI 10.1080/17476933.2010.534146. https://doi.org/10.1080/17476933.2010.534146.
5. Wu, Zhao-Jun; Zhang, Shao-Hua. "Algebroid functions with given Borel points." Georgian Mathematical Journal 22(2):289-294, 2015. DOI 10.1515/gmj-2015-0018. https://doi.org/10.1515/gmj-2015-0018.
6. Huo, Yingying; Sun, Daochun. "Algebroidal function and its derived function in unit circular disc." Acta Mathematica Scientia, Series B 29(1):129-139, 2009. DOI 10.1016/S0252-9602(09)60013-8. https://doi.org/10.1016/S0252-9602(09)60013-8.
7. Sun, Daochun; Huo, Yingying; Chai, Fujie. "The Growth and Borel Points of Random Algebroid Functions in the Unit Disc." Acta Mathematica Scientia, Series B 41(4):1119-1129, 2021. DOI 10.1007/s10473-021-0406-8. https://doi.org/10.1007/s10473-021-0406-8.
8. Kang, Haigang; Zhang, Qingde. "T-radii and Borel radii of meromorphic functions of finite positive order in the unit disc." Journal of Sichuan University (Natural Science Edition) 46(2):307-314, 2009 (in Chinese, with English summary). zbMATH 5733170 (identifier 1212.30117). https://zbmath.org/?q=an%3A5733170.