For a transcendental meromorphic function ff of order less than one whose Nevanlinna deficiency sum is maximal (∑a∈Cδ(a,f)+δ(∞,f)=2\sum_{a\in\mathbb{C}}\delta(a,f)+\delta(\infty,f)=2), the characteristic-function ratios of its differences are known exactly: lim⁡r→∞T(r,Δcf)/T(r,f)=2−δ(∞,f)\lim_{r\to\infty}T(r,\Delta_c f)/T(r,f)=2-\delta(\infty,f) (Wu 2013) and lim⁡r→∞T(r,∏i=1qΔηif)/T(r,f)=q(2−δ(∞,f))\lim_{r\to\infty}T(r,\prod_{i=1}^{q}\Delta_{\eta_i}f)/T(r,f)=q(2-\delta(\infty,f)) (Shaw 2022). What is not known are the exact zero-and-pole densities. For the combined density K(g)=lim sup⁡r→∞(N(r,g)+N(r,1/g))/T(r,g)K(g)=\limsup_{r\to\infty}(N(r,g)+N(r,1/g))/T(r,g) only upper bounds are proved: K(Δcf)≤2(1−δ(∞,f))/(2−δ(∞,f))K(\Delta_c f)\le 2(1-\delta(\infty,f))/(2-\delta(\infty,f)) (Wu), the same bound for products ∏i=1qΔηif\prod_{i=1}^{q}\Delta_{\eta_i}f (Shaw), and for the difference polynomial L(Δηf)=∑i=1qaiΔηifL(\Delta_\eta f)=\sum_{i=1}^{q}a_i\Delta_\eta^{i}f even the characteristic ratio is confined between two-sided bounds. The open problem, anchored at the Question 1.4 printed in Wu's 2013 paper, is to decide whether the equality K(Δcf)=2(1−δ(∞,f))/(2−δ(∞,f))K(\Delta_c f)=2(1-\delta(\infty,f))/(2-\delta(\infty,f)) holds for every function in this class or whether strict inequality occurs for some member — completing the transfer of the classical derivative theory (Edrei, Weitsman, Singh–Kulkarni, Fang) to difference operators.

Background

Nevanlinna theory describes the value distribution of a meromorphic function ff on the complex plane through its characteristic function T(r,f)=m(r,f)+N(r,f)T(r,f)=m(r,f)+N(r,f), where the proximity function m(r,f)m(r,f) measures the average closeness of ff to ∞\infty on the circle ∣z∣=r|z|=r and the counting function N(r,f)N(r,f) counts poles. The deficiency of a value a∈C∪{∞}a\in\mathbb{C}\cup\{\infty\} is δ(a,f)=lim inf⁡r→∞m(r,1/(f−a))/T(r,f)\delta(a,f)=\liminf_{r\to\infty}m(r,1/(f-a))/T(r,f) (for a=∞a=\infty one uses m(r,f)m(r,f)), a number in [0,1][0,1] measuring how rarely ff takes the value aa; Nevanlinna's second fundamental theorem implies the deficiency relation ∑a∈Cδ(a,f)+δ(∞,f)≤2\sum_{a\in\mathbb{C}}\delta(a,f)+\delta(\infty,f)\le 2. When equality holds, ff is said to have maximal deficiency sum. Throughout, the growth of ff is measured by its order ρ(f)=lim sup⁡r→∞log⁡T(r,f)/log⁡r\rho(f)=\limsup_{r\to\infty}\log T(r,f)/\log r, so that "finite order" means ρ(f)<∞\rho(f)<\infty and "order less than one" means ρ(f)<1\rho(f)<1. This extremal class has been studied since the 1950s because its rigid value distribution forces exact asymptotics for derived functions. Shah and Singh (1956) showed that a finite-order transcendental meromorphic function with ∑a∈Cδ(a,f)=2\sum_{a\in\mathbb{C}}\delta(a,f)=2 satisfies T(r,f′)∼2T(r,f)T(r,f')\sim 2T(r,f); Edrei (1967) and Weitsman (1969) proved that for finite order and maximal deficiency sum one has lim⁡r→∞T(r,f′)/T(r,f)=2−δ(∞,f)\lim_{r\to\infty}T(r,f')/T(r,f)=2-\delta(\infty,f) and lim⁡r→∞N(r,1/f′)/T(r,f′)=0\lim_{r\to\infty}N(r,1/f')/T(r,f')=0; Singh and Gopalakrishna (1971) obtained lim⁡r→∞N(r,a)/T(r,f)=1−δ(a,f)\lim_{r\to\infty}N(r,a)/T(r,f)=1-\delta(a,f) in the same setting. For the combined zero–pole density K(f′)=lim sup⁡r→∞(N(r,f′)+N(r,1/f′))/T(r,f′)K(f')=\limsup_{r\to\infty}(N(r,f')+N(r,1/f'))/T(r,f'), Singh and Kulkarni (1973) proved two-sided bounds at finite order, and Fang (2000) settled the derivative case exactly in the same finite-order setting: K(f′)=2(1−δ(∞,f))/(2−δ(∞,f))K(f')=2(1-\delta(\infty,f))/(2-\delta(\infty,f)).

In parallel, a difference Nevanlinna theory was developed by Halburd and Korhonen (2006) and Chiang and Feng (2008), studying the shift (difference) operator Δηf(z)=f(z+η)−f(z)\Delta_\eta f(z)=f(z+\eta)-f(z), η≠0\eta\ne 0, through difference analogues of the lemma on the logarithmic derivative; Bergweiler and Langley (2007) connected the zeros of differences of low-order meromorphic functions to those of the derivative. Zhaojun Wu (2013) transferred the extremal-deficiency program to the difference setting: for a transcendental meromorphic function of order less than one with maximal deficiency sum, lim⁡r→∞T(r,Δcf)/T(r,f)=2−δ(∞,f)\lim_{r\to\infty}T(r,\Delta_c f)/T(r,f)=2-\delta(\infty,f) and lim⁡r→∞N(r,1/Δcf)/T(r,Δcf)=0\lim_{r\to\infty}N(r,1/\Delta_c f)/T(r,\Delta_c f)=0, so δ(0,Δcf)=1\delta(0,\Delta_c f)=1. For K(Δcf)=lim sup⁡r→∞(N(r,Δcf)+N(r,1/Δcf))/T(r,Δcf)K(\Delta_c f)=\limsup_{r\to\infty}(N(r,\Delta_c f)+N(r,1/\Delta_c f))/T(r,\Delta_c f), however, he proved only the upper bound K(Δcf)≤2(1−δ(∞,f))/(2−δ(∞,f))K(\Delta_c f)\le 2(1-\delta(\infty,f))/(2-\delta(\infty,f)), showed by the example f(z)=1/(ez−1)f(z)=1/(e^{z}-1) that the bound is attained, and printed the open Question 1.4: can the inequality be replaced by equality? Bhoosnurmath, Dyavanal, Barki and Rathod (2018) extended the analysis to the iterated differences Δcnf\Delta_c^{n}f at finite order, adding a lower bound (1−δ(∞,f))/(2−δ(∞,f))≤K(Δcf)(1-\delta(\infty,f))/(2-\delta(\infty,f))\le K(\Delta_c f) and proving K(Δcnf)=0K(\Delta_c^{n}f)=0 in the special configuration ∑a∈Cδ(a,f)=1\sum_{a\in\mathbb{C}}\delta(a,f)=1, δ(∞,f)=1\delta(\infty,f)=1, but left the equality question open. Shaw (2022) treated two further objects: the product ∏i=1qΔηif\prod_{i=1}^{q}\Delta_{\eta_i}f of difference operators with nonzero shifts ηi\eta_i (not necessarily equal), for which he proved the exact characteristic ratio q(2−δ(∞,f))q(2-\delta(\infty,f)) at order less than one but only the bound K(∏i=1qΔηif)≤2(1−δ(∞,f))/(2−δ(∞,f))K(\prod_{i=1}^{q}\Delta_{\eta_i}f)\le 2(1-\delta(\infty,f))/(2-\delta(\infty,f)), attained by an exponential example; and the difference polynomial L(Δηf)=∑i=1qaiΔηifL(\Delta_\eta f)=\sum_{i=1}^{q}a_i\Delta_\eta^{i}f, a fixed linear combination of iterated differences with the same shift and nonzero constants aia_i, for which he obtained only two-sided bounds on lim⁡r→∞T(r,L(Δηf))/T(r,f)\lim_{r\to\infty}T(r,L(\Delta_\eta f))/T(r,f) and limsup bounds on the normalized zero- and pole-counting functions, with constants involving ((q+1)(q+2)−1)/2((q+1)(q+2)-1)/2. Chen and Wu (2021) studied linear difference polynomials of maximal-deficiency-sum functions in the same spirit. Across this literature the exact zero–pole densities remain undetermined: no exact limit formula and no strict-inequality example is known for K(Δcf)K(\Delta_c f), for K(∏i=1qΔηif)K(\prod_{i=1}^{q}\Delta_{\eta_i}f), or for the counting densities of L(Δηf)L(\Delta_\eta f).

Problem Statement

Let ff be a transcendental meromorphic function in the complex plane whose growth order ρ(f)=lim sup⁡r→∞log⁡T(r,f)/log⁡r\rho(f)=\limsup_{r\to\infty}\log T(r,f)/\log r is less than one and whose Nevanlinna deficiency sum is maximal, that is,

∑a∈Cδ(a,f)+δ(∞,f)=2,\sum_{a\in\mathbb{C}}\delta(a,f)+\delta(\infty,f)=2,

let c∈C∖{0}c\in\mathbb{C}\setminus\{0\}, let Δcf(z)=f(z+c)−f(z)\Delta_c f(z)=f(z+c)-f(z), and for a meromorphic function gg write

K(g)=lim sup⁡r→∞N(r,g)+N(r,1/g)T(r,g)K(g)=\limsup_{r\to\infty}\frac{N(r,g)+N(r,1/g)}{T(r,g)}

for the combined normalized density of zeros and poles of gg, where T(r,g)T(r,g) is the Nevanlinna characteristic, N(r,g)N(r,g) counts poles of gg and N(r,1/g)N(r,1/g) counts its zeros. Determine the exact combined zero–pole density of the difference Δcf\Delta_c f:

(Wu's Question 1.4, 2013) Does the equality

K(Δcf)=2(1−δ(∞,f))2−δ(∞,f)K(\Delta_c f)=\frac{2(1-\delta(\infty,f))}{2-\delta(\infty,f)}

hold for every ff in this class, or does there exist such an ff for which Wu's inequality is strict? The analogous two-sided bounds of Bhoosnurmath et al. at finite order leave the same question open there.

References

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