A Note on Entire Functions That Share Two Small Functions

Abstract

This note is to show that if $f$ is a nonconstant entire function that shares two pairs of small functions ignoring multiplicities with its first derivative $f\mathrm{\text{'}}$, then there exists a close linear relationship between $f$ and $f\mathrm{\text{'}}$. This result is a generalization of some results obtained by Rubel and Yang, Mues and Steinmetz, Zheng and Wang, and Qiu. Moreover, examples are provided to show that the conditions in the result are sharp.