Entire function and its linear C-shift operator sharing a set of small functions CM

This paper has been devoted to study the relation between an entire function of hyper order < 1 and its linear c-shift operator sharing a set of small functions. Our first theorem improves a very recent result of the present authors [Meromorphic function of restricted hyper-order sharing one or two sets with its linear C-shift operator. Anal Math. 2021;47(4):747-779.], while the corollary deducted from the same completely provides an answer to an open question posed by Liu [Meromorphic functions sharing a set with applications to difference equations. J Math Anal Appl. 2009;359:384-393]. Most importantly, our another theorem improves and extends a result due to Li [Entire functions sharing a finite set with their difference operators. Comput Methods Funct Theory. 2012;12(1):307-328] which in turn rectifies an erroneous result of Ma-Qi-Zhang [Further results about a special Fermat-type difference equation. J Funct Spaces. 2020. Article ID 3205357]. We have provided a handful number of examples in support of our certain claims. In the last section, as an application of our main result, we extend that of Liu-Ma-Zhai [The generalized Fermat type difference equations. Bull Korean Math Soc. 2018;55(6):1845-1858].