Uniqueness of entire functions concerning their linear differential polynomials in shift
Jeet Sarkar, Debabrata Pramanik
TL;DR
This work addresses the uniqueness problem for entire functions by examining their linear differential polynomials in a shift, \mathscr{L}_k(f). Building on Nevanlinna theory, it generalizes Qi's shift-sharing results by replacing f(z+c) with \mathscr{L}_k(f(z+c)) and allowing the shared small functions a1,a2 with growth constraint ρ_1(f) < 1. The main result (Theorem t1) shows that if f and \mathscr{L}_k(f(z+c)) share a1,a2 IM, then either f ≡ \mathscr{L}_k(f(z+c)) or a specific linear compatibility condition among coefficients, encoded via a recurrence for the psi_i, must hold. Corollaries recover classical special cases where a1,a2 are constants, yielding f ≡ f(z+c) or f ≡ f^{(k)}(z+c); the authors also provide sharpness examples and extend the discussion to meromorphic functions with N(r,f) = S(r,f).
Abstract
In the paper, we investigate the uniqueness problem of entire functions concerning their linear differential polynomial in shift and obtain three results which improve and generalize the recent result due to Qi (Ann. Polon. Math., 102 (2011), 129-142.) in a large extend.