Sharp Deficiency Bounds for Meromorphic Functions in the Unit Disc
Abstract
In 1986, Shea and Sons obtained the following bound for a meromorphic function f of finite order ρ in the unit disc, under the hypothesis 0<λ(f)≤+∞, and for every positive integer n: ∑a≠∞δ(a,f) ≤ δ(0,f(n))(1+n k(f)) +(2n(ρ+1))/λ(f). Under the condition 0<α(f)≤+∞, they also obtained ∑a≠∞δ(a,f) ≤ Δ(0,f(n))(1+n k(f)) +2n/α(f). Shea and Sons asked whether the factor $2$ could be eliminated. We prove that it can. In fact, whenever 0<λ(f)≤+∞, one has ∑a≠∞δ(a,f) ≤ δ(0,f(n))(1+n k(f)) +(n(ρ+1))/λ(f), and under the condition 0<α(f)≤+∞ one has ∑a≠∞δ(a,f) ≤ Δ(0,f(n))(1+n k(f)) +n/α(f). The coefficient n in each of n(ρ+1)/λ(f) and n/α(f) is best possible for every n, and the dependence on ρ and α(f) in these terms is also essential. At the end, we extend both estimates to finite subsets A of an n-dimensional complex vector space V⊂ℂ(z), with f(n) replaced by the monic differential operator DVf of order n whose kernel is V. We prove that both extensions are sharp for every V.
Related
- Sharp unconditional moment bounds for products of L-functions
- Sharp lower bounds for shifted moments of Dedekind zeta functions
- Sharp Volume-Based Bounds for BMO Functions on Compact Domains and Applications
- Sharp lower bounds for the volumes of nodal and positivity sets of harmonic functions
- Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative
BibTeX
@misc{nadi2026sharp,
title = {{Sharp Deficiency Bounds for Meromorphic Functions in the Unit Disc}},
author = {Sina Nadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05835},
eprint = {2609.05835},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05835},
}
RIS
TY - UNPB
TI - Sharp Deficiency Bounds for Meromorphic Functions in the Unit Disc
AU - Sina Nadi
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05835
UR - https://arxiv.org/abs/2609.05835
AB - In 1986, Shea and Sons obtained the following bound for a meromorphic function f of finite order ρ in the unit disc, under the hypothesis 0<λ(f)≤+∞, and for every positive integer n: ∑_a≠∞δ(a,f) ≤ δ(0,f^(n))(1+n k(f)) +(2n(ρ+1))/λ(f). Under the condition 0<α(f)≤+∞, they also obtained ∑_a≠∞δ(a,f) ≤ Δ(0,f^(n))(1+n k(f)) +2n/α(f). Shea and Sons asked whether the factor $2$ could be eliminated. We prove that it can. In fact, whenever 0<λ(f)≤+∞, one has ∑_a≠∞δ(a,f) ≤ δ(0,f^(n))(1+n k(f)) +(n(ρ+1))/λ(f), and under the condition 0<α(f)≤+∞ one has ∑_a≠∞δ(a,f) ≤ Δ(0,f^(n))(1+n k(f)) +n/α(f). The coefficient n in each of n(ρ+1)/λ(f) and n/α(f) is best possible for every n, and the dependence on ρ and α(f) in these terms is also essential. At the end, we extend both estimates to finite subsets A of an n-dimensional complex vector space V⊂ℂ(z), with f^(n) replaced by the monic differential operator D_Vf of order n whose kernel is V. We prove that both extensions are sharp for every V.
ER -