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).