2025/05/27 by Lasse Asikainen, Yu Chen, Asikainen, Lasse +3
Mathematics · #30D35 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2505.21150
openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that for a vanishing period difference operator of a meromorphic function \( f \), there exist the following estimates regarding proximity functions, limη→ 0 mη(r, (Δηf - aη)/(f' - a) ) = 0 and limr → ∞ mη(r, (Δηf - aη)/(f' - a) ) = 0, where \( Δηf = f(z + η) - f(z) \), and \( |η| \) is less than an arbitrarily small quantity \( α(r) \) in the second limit. Then, under certain assumptions on the growth, restrictions on the period tending to infinity, and on the value distribution of a meromorphic function \( f(z) \), we have m(r, (Δωf - aω)/(f' - a) ) = S(r, f'), as \( r → ∞ \), outside an exceptional set of finite logarithmic measure. Additionally, we provide an estimate for the angular shift under certain conditions on the shift and the growth. That is, the following Nevanlinna proximity function satisfies m(r, \fracf(eiω(r)z) - f(z)f' ) = S(r, f), outside an exceptional set of finite logarithmic measure. Furthermore, the above estimates yield additional applications, including deficiency relations between \( Δηf \) (or \( Δωf \)) and \( f' \), as well as connections between \( η/ω\)-separated pair indices and \( δ(0, f') \).