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