Lower-order range of the Barsegian–Yang exceptional-value relations for a-points with bounded derivative

Barsegian and Yang (2004) introduced jumping a-points, a generalization of multiple a-points, and proved that a-points at which $|w'|$ is small (or at which the inverse meromorphic derivative is large) are exceptional in the Nevanlinna sense: the associated counting functions satisfy a deficiency relation summing to at most 4. Their Theorem 2 is proved only for meromorphic functions of lower order $\lambda>2$, and the exponential function ($\lambda=1$) refutes it at $\lambda=1$. The open problem is to determine for which lower orders $0\le\lambda\le2$ these deficiency relations hold.

Background

For a meromorphic function $w$ in the complex plane, Nevanlinna theory measures growth by the characteristic function $T(r,w)$ and measures how often $w$ takes a value $a$ by the integrated counting function $N(r,a,w)$ of the $a$-points (the solutions of $w(z)=a$). The deficiency $\delta(a)=1-\limsup_{r\to\infty}\overline{N}(r,a,w)/T(r,w)$ quantifies how rarely $a$ is taken, and the classical deficiency relation states $\sum_a \delta(a)\le 2$. Multiple $a$-points, i.e. $a$-points where $w'(z)=0$, are also known to be exceptional in this sense: writing $N(r,w=a,w'=0)$ for the counting function of $a$-points at which the derivative vanishes, the classical relation recorded as inequality (1.6) in the source is $\sum_{\nu=1}^q \liminf_{r\to\infty} N(r,w=a_\nu,w'=0)/T(r,w)\le 4$ for any distinct values $a_1,\dots,a_q$.

G. A. Barsegian and C. C. Yang, in a 2004 book chapter, generalized this picture. Given distinct values $a_1,\dots,a_q$ with mutual separation encoded by $\rho^{(w)}=\min_{\nu\ne\mu}|a_\nu-a_\mu|/3$, they call an $a_\nu$-point $z_0$, with Taylor expansion $w(z)=a_\nu+c_1(z-z_0)+c_2(z-z_0)^2+\cdots$, a jumping $a_\nu$-point if for some integer $k\ge 2$ the coefficient $c_k$ dominates a fixed multiple of $|c_1|^k$, the multiple depending only on $k$ and $\rho^{(w)}$. This notion generalizes multiple points (which correspond to $c_1=0$) and captures points where the first derivative is very small relative to the higher Taylor coefficients. Their Theorem 1 proves the same deficiency relation for jumping $a$-points: $\sum_{\nu=1}^q \liminf_{r\to\infty} N(r,a_\nu,\uparrow,w)/T(r,w)\le 4$, where the arrow marks the counting function restricted to jumping $a$-points (counted with multiplicity).

The subject of the present problem is their Theorem 2. The lower order of $w$ is $\lambda=\liminf_{r\to\infty}\log T(r,w)/\log r$. For a fixed $R>0$, let $N(r,w=a,|w'|\le R)$ be the integrated counting function of $a$-points (counted with multiplicity) at which $|w'(z)|\le R$; and for an integer $U\ge 1$ let $F^{(U)}(w(z))$ denote the $U$-th inverse meromorphic derivative, defined as in the second precursor paper (Barsegian–Yang, Complex Variables 44 (2001) 13–27): for an $a$-point $z_i$ of $w$, let $F_i$ be the branch of the local inverse function of $w$ normalized by $F_i(w(z_i))=z_i$, and let $F_i^{(U)}$ be its $U$-th derivative with respect to $w$; the composition $F^{(U)}(w(z))=F_i^{(U)}(w(z))$ is then a meromorphic function of $z$, single-valued near each $a$-point, having at a multiple $a$-point of multiplicity $k$ a pole of order $Uk-1$ (in particular $F^{(1)}(w(z))=1/w'(z)$), so the condition $|F^{(U)}(w(z))|\ge R$ is well defined at every $a$-point independently of any choice of domains in the proofs. Theorem 2 states that if $\lambda>2$, then for every $R>0$: (1) for every value $a$ outside a countable set depending on $w$, $\liminf_{r\to\infty} N(r,w=a,|w'|\le R)/T(r,w)=0$, and $\sum_{a\in\mathbb{C}} \liminf_{r\to\infty} N(r,w=a,|w'|\le R)/T(r,w)\le 4$; and (2) the same pair of conclusions holds for $N(r,w=a,|F^{(U)}|\ge R)$. Thus, for sufficiently fast-growing functions, $a$-points with small derivative (or with large inverse derivative) are as exceptional as multiple points.

The proof rests on the machinery of two earlier Barsegian–Yang papers: Ahlfors-type sets of simple $a$-points grouped into pairwise disjoint simply connected domains $E_i(r,w)$, of diameter at most $\varphi(r) r/A^{1/2}(r,w)$, on which $w$ is univalent, where $A(r,w)$ is the Ahlfors–Shimizu characteristic (the spherical area of the image of $|z|<r$, which is comparable to $T(r,w)$) and $\varphi(r)$ is an auxiliary function growing arbitrarily slowly to infinity. On these domains the derivative at the captured $a$-points is comparable to $A^{1/2}(r,w)/(\varphi(r) r)$. Since $A^{1/2}(r,w)/r$ grows like $r^{\lambda/2-1}$, this lower bound outruns any fixed $R$ only when $\lambda>2$, which is exactly where the method stops. The restriction is not cosmetic: for $w(z)=e^z$, whose lower order is $\lambda=1$, every $a$-point with $a\ne 0$ satisfies $|w'|=|a|$, so for any $R>0$ all values $0<|a|\le R$ have $N(r,w=a,|w'|\le R)/T(r,w)\to 1$, and five such values already violate the sum bound. So Theorem 2 genuinely fails at $\lambda=1$, while the case $0\le\lambda\le 2$ is not covered by the source's theorem.

Problem Statement

Determine the range of validity, in lower order $0\le\lambda\le 2$, of the Barsegian–Yang Theorem 2 for meromorphic functions $w$ in the complex plane. Since $w=e^z$ (lower order $\lambda=1$) verifiably refutes the statement at $\lambda=1$ (see Background), the question is: for which $\lambda\in[0,2]$ is it true that, for every fixed $R>0$, the $a$-points with $|w'|\le R$ are exceptional in the sense that

$$\liminf_{r\to\infty}\frac{N(r,w=a,|w'|\le R)}{T(r,w)}=0$$

for every value $a$ outside a countable set depending on $w$, and

$$\sum_{a\in\mathbb{C}}\liminf_{r\to\infty}\frac{N(r,w=a,|w'|\le R)}{T(r,w)}\le 4,$$

and likewise that the $a$-points with $|F^{(U)}(w(z))|\ge R$ (for each integer $U\ge 1$) are exceptional, with the same individual-liminf and sum statements for $N(r,w=a,|F^{(U)}|\ge R)$? Here $N(r,w=a,|w'|\le R)$ counts $a$-points with multiplicity, $T(r,w)$ is the Nevanlinna characteristic, $F^{(U)}(w(z))$ is the $U$-th inverse meromorphic derivative defined in the Background (the branch of the local inverse normalized by $F_i(w(z_i))=z_i$, differentiated $U$ times with respect to $w$ and composed with $w$; at a multiple $a$-point of multiplicity $k$ it has a pole of order $Uk-1$), and the lower order is $\lambda=\liminf_{r\to\infty}\log T(r,w)/\log r$. A complete answer determines the set $S\subseteq[0,2]$ of lower orders for which both parts of Theorem 2 hold, proves the relations for every $\lambda\in S$, and exhibits meromorphic counterexamples of lower order $\lambda$ for every $\lambda\in[0,2]\setminus S$; no interval structure of $S$ is presumed.

References

1. G. A. Barsegian and C. C. Yang, On Some New Concept of Exceptional Values, in: Value Distribution Theory and Related Topics (Advances in Complex Analysis and Its Applications 3), Kluwer Academic Publishers / Springer, Boston, MA, 2004, pp. 105–116. DOI: 10.1007/1-4020-7951-6_4, https://link.springer.com/chapter/10.1007/1-4020-7951-6_4; zbMATH review Zbl 1090.30033, https://zbmath.org/?q=an:1090.30033
2. G. A. Barsegian and C. C. Yang, Some new generalizations in the theory of meromorphic functions and their applications, I: On the magnitudes of functions on the sets of a-points of derivatives, Complex Variables, Theory and Application 41 (2000), 293–313. DOI: 10.1080/17476930008815256, https://doi.org/10.1080/17476930008815256
3. G. A. Barsegian and C. C. Yang, Some new generalizations in the theory of meromorphic functions and their applications, II: On the derivatives of meromorphic and inverse functions, Complex Variables, Theory and Application 44 (2001), 13–27. DOI: 10.1080/17476930108815340, https://doi.org/10.1080/17476930108815340
4. G. A. Barsegian, I. Laine and C. C. Yang, On a method of estimating derivatives in complex differential equations, Journal of the Mathematical Society of Japan 54 (2002), 923–935. DOI: 10.2969/jmsj/1191591998, https://doi.org/10.2969/jmsj/1191591998
5. J. Lin, W. Xiong and W. Yuan, Meromorphic Solutions of Some Algebraic Differential Equations, Abstract and Applied Analysis 2014 (2014), 796312. DOI: 10.1155/2014/796312, https://doi.org/10.1155/2014/796312
6. Grigor Barsegian, Universal Version of Value Distribution Theory for Functions in a Given Domain, Lobachevskii Journal of Mathematics 45 (2024), 3479–3487. DOI: 10.1134/S1995080224604648, https://doi.org/10.1134/S1995080224604648