Vanishing, Unbounded and Angular Shifts on the Quotient of the Difference and the Derivative of a Meromorphic Function

TL;DR

This work extends Nevanlinna theory to encompass variable shifts of meromorphic functions, establishing sharp proximity bounds for vanishing, unbounded, and angular shifts between $\Delta_\eta f$ (or $\Delta_\omega f$) and the derivative $f'$. By combining Poisson–Jensen formulas, Green-function oscillation analysis, and shift-difference estimates under precise growth restrictions (notably hyper-order constraints), the authors prove that relevant proximity functions are either vanishing or belong to the small class $S(r,f')$ outside exceptional sets of finite logarithmic measure. The results yield deficiency relations and $\eta/\omega$-separated indices, and culminate in a Second Main Theorem analogue for shifted operators, thereby broadening the linkage between difference and differential value distributions for meromorphic and entire functions. The findings have implications for the distribution of zeros and poles of $f'$, $\Delta_\eta f$, and related difference-operator constructs, enriching the toolbox for difference Nevanlinna theory and its applications.

Abstract

We show that for a vanishing period difference operator of a meromorphic function \( f \), there exist the following estimates regarding proximity functions, \[ \lim_{η\to 0} m_η\left(r, \frac{Δ_ηf - aη}{f' - a} \right) = 0 \] and \[ \lim_{r \to \infty} m_η\left(r, \frac{Δ_ηf - aη}{f' - a} \right) = 0, \] where \( Δ_ηf = f(z + η) - f(z) \), 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 \[ m\left(r, \frac{Δ_ωf - aω}{f' - a} \right) = S(r, f'), \] as \( r \to \infty \), 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 \[ m\left(r, \frac{f(e^{iω(r)}z) - f(z)}{f'} \right) = S(r, f), \] outside an exceptional set of finite logarithmic measure. Furthermore, the above estimates yield additional applications, including deficiency relations between \( Δ_ηf \) (or \( Δ_ωf \)) and \( f' \), as well as connections between \( η/ω\)-separated pair indices and \( δ(0, f') \).

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Lemma 8.3 from MR3206459
• Lemma 2.2
• proof
• Proposition 6.1
• proof
• Proposition 6.2
• proof