On sharing values of meromorphic functions and their differences (Q1936946)

The authors obtain a difference analogue of the Brück conjecture and some results on uniqueness. They prove that \(a=0\) and \[ \frac{f(z+\eta)-f(z)}{f(z)}=A, \] where \(A\) is a nonzero constant, if \(f(z)\) is a finite order transcendental entire function which has a finite Borel exceptional value \(a\) and \(\Delta f(z)=f(z+\eta)-f(z)\not\equiv 0\), and if \(\Delta f\) and \(f\) share the value \(a\) CM. Some other results are also obtained. These contributions are interesting and new, and make the Brück conjecture more popular.