A note on entire functions sharing a finite set with applications to difference equations

Abstract

Value distribution and uniqueness problems of difference operator of an entire function have been investigated in this article. This research shows that a finite ordered entire function f when sharing a set S = {α(z), β(z)} of two entire functions α and β with max{ρ(α), ρ(β)} < ρ(f ) with its difference L n c (f ) = n j=0 a j f (z + jc), then L n c (f ) ≡ f , and more importantly certain form of the function f has been found. The results in this paper improve those given by k. Liu, X. M. Li, J. Qi, Y. Wang and Y. Gu etc. Some examples have been exhibited to show the condition max{ρ(α), ρ(β)} < ρ(f ) is sharp in our main result. Examples have been also exhibited to show that if CM sharing is replaced by IM sharing, then conclusion of the main results ceases to hold.