Some results on zeros and uniqueness of difference-differential polynomials

Abstract

We consider the zeros distributions on the derivatives of difference polynomials of meromorphic functions, and present some results which can be seen as the discrete analogues of Hayman conjecture [8], also partly answer the question given in [18, P448]. We also investigate the uniqueness problems of difference-differential polynomials of entire functions sharing one common value. These theorems improve the results of Luo and Lin [18] and some results of present authors [15].

Uniqueness of certain type of differential-difference and difference polyniminals

“…In 2012 Liu-Liu and Cao [9] first considered the value distribution of the differentialdifference counterpart of the above theorems and obtained the following result.…”

Results on the uniqueness of difference polynomials of entire functions

“…Recently, the topic of difference equation and difference product in the complex plane C has attracted many mathematicians, many papers have focused on value distribution of differences and differences operator analogues of Nevanlinna theory (including [3,6,8,7,17,23]), and many people dealt with the uniqueness of differences and difference polynomials of meromorphic function and obtained some interesting results ( [11,12,20,21,22,27,28]).…”

“…. , n. Since g is an entire function and a n ̸ = 0, we can deduce from (20) that T (r, g) = m(r, g) = S(r, g), a contradiction. Then (19) holds.…”

Value distribution of higher order differential-difference polynomial of an entire function

“…In view of Lemma 10, since ρ(h(z)) < 1, the degree of C 1 n q(z) n−1 q(z + c) + q(z) n e αc must be smaller than the degree of q(z) n . Hence, we arrive at the equality e αc = −C 1 n (16) provided that q(z) is not a constant. By using the same arguments as above, we also get D 2 (z) ≡ 0, that is C 2 n q(z) n−2 q(z + c)h(z) 2 + C 1 n q(z) n−1 h(z + c)e αc h(z) ≡ 0.…”