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

TL;DR

This paper investigates the asymptotic behavior of shifts and derivatives of meromorphic functions, establishing estimates for proximity functions and angular shifts, with implications for value distribution and deficiency relations.

Contribution

It introduces new estimates for the proximity functions of difference and derivative quotients of meromorphic functions, extending Nevanlinna theory to difference operators and angular shifts.

Findings

Proves limits of proximity functions tend to zero under certain conditions.

Establishes bounds for angular shifts of meromorphic functions.

Links difference operators with value distribution and deficiency relations.

Abstract

We show that for a vanishing period difference operator of a meromorphic function \( f \), there exist the following estimates regarding proximity functions, \[ \lim_{\eta \to 0} m_\eta\left(r, \frac{\Delta_\eta f - a\eta}{f' - a} \right) = 0 \] and \[ \lim_{r \to \infty} m_\eta\left(r, \frac{\Delta_\eta f - a\eta}{f' - a} \right) = 0, \] where \( \Delta_\eta f = f(z + \eta) - f(z) \), and \( |\eta| \) is less than an arbitrarily small quantity \( \alpha(r) \) 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 \[ m\left(r, \frac{\Delta_\omega f - a\omega}{f' - a} \right) = S(r, f'), \] 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 \[ m\left(r, \frac{f(e^{i\omega(r)}z) - f(z)}{f'} \right) = S(r, f), \] outside an exceptional set of finite logarithmic measure. Furthermore, the above estimates yield additional applications, including deficiency relations between \( \Delta_\eta f \) (or \( \Delta_\omega f \)) and \( f' \), as well as connections between \( \eta/\omega \)-separated pair indices and \( \delta(0, f') \).