Abstract
We investigate the zeros distributions of difference polynomials of meromorphic functions, which can be viewed as the Hayman conjecture as introduced by (Hayman 1967) for difference. And we also study the uniqueness of difference polynomials of meromorphic functions sharing a common value, and obtain uniqueness theorems for difference.
Highlights
A meromorphic function means meromorphic in the whole complex plane
We say that meromorphic functions f and g share a finite value a IM ignoring multiplicities when f − a and g − a have the same zeros
If f − a and g − a have the same zeros with the same multiplicities, we say that f and g share the value a CM counting multiplicities
Summary
A meromorphic function means meromorphic in the whole complex plane. Given a meromorphic function f, recall that α /≡ 0, ∞ is a small function with respect to f, if T r, α S r, f , where S r, f is used to denote any quantity satisfying S r, f o T r, f , as r → ∞ outside a possible exceptional set of finite logarithmic measure. The restriction of finite order in Theorem 1.2 cannot be deleted This can be seen by taking f z 1/P z eez , ec −n n ≥ 6 , P z is a nonconstant polynomial, and R z is a nonzero rational function. Let f be a transcendental meromorphic function with finite order and c be a nonzero complex constant. Some results about the zeros distributions of difference polynomials of entire functions or meromorphic functions with the condition λ 1/f < ρ f can be found in 9– 12. If n ≥ 12, f z nf z c and g z ng z c share 1 IM, f tg or fg t, where tn 1 1
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