Value Distribution and Uniqueness Results of Zero‐Order Meromorphic Functions to Their q‐Shift

Abstract

We investigate value distribution and uniqueness problems of meromorphic functions with their q‐shift. We obtain that if f is a transcendental meromorphic (or entire) function of zero order, and Q(z) is a polynomial, then afn(qz) + f(z) − Q(z) has infinitely many zeros, where q ∈ ℂ∖{0}, a is nonzero constant, and n ≥ 5 (or n ≥ 3). We also obtain that zero‐order meromorphic function share is three distinct values IM with its q‐difference polynomial P(f), and if limsup r→∞(N(r, f)/T(r, f)) < 1, then f ≡ P(f).