Entire functions sharing sets of small functions with their difference operators or shifts

Abstract

ABSTRACT. We show some interesting results concerning entire functions sharing two sets of small functions CM with their difference operators or shifts.

A note on entire functions sharing a finite set with their difference operators

“…Due to the use of almost the same method, we omit the detailed proof. A similar result can be found in [11]. Before we proceed, we suppose that the reader is familiar with Nevanlinna theory, for example, the first and second main theorems, and the common notations such as T(r, f ), m(r, f ) and N(r, f ).…”

“…In recent years, the Nevanlinna characteristic of f (z + ω), the value distribution theory for difference polynomials, the Nevanlinna theory of the difference operator and the difference analogue of the lemma on the logarithmic derivative had been established; see e.g. [2,10,11,[16][17][18][19][20][21][22][23]. Due to these theories, there has been recent study of whether the derivative f can be replaced by the difference operator c f (z) = f (z + c)f (z) in the above question.…”

Transcendental entire functions of finite order sharing two sets of small functions with their shift differential operators

“…In [7] the author posed an open question: "What happens if in Theorem 1.1, {a, −a} is replaced by {a(z), b(z)} where a(z), b(z) ∈ S(f ) are two non-vanishing periodic functions with period c?" Chen-Chen [2] answered the question of Liu [7] in the following way:…”

“…In view of Theorem 1.3, it will be natural to investigate whether in the same theorem, the set {0} can be replaced by {b(z)}, where b(z) is non-zero periodic small function of f . In this respect, we would first like to mention the following contribution due to Chen-Chen [2].…”

“…) and then automatically z 0 is zero of (e 2γ P (f )) 2 − (P (g)) 2 . Therefore all zeros of (g − a(z))(g + a(z)) are zeros of (e 2γ P (f )) 2 − (P (g)) 2 . Suppose z 0 is a zero of (g − a(z))…”