On value distribution of difference polynomials of meromorphic functions (Q638109)

Hayman proved that if \(f(z)\) is a transcendental entire function, \(n\geq 3\) is an integer and \(a\neq 0\) is a constant, then \(f'(z)-af(z)^n\) assumes all finite values infinitely often. The author proves the difference counterpart of Hayman's result in which \(f'(z)\) is replaced by \(f(z+c)-f(z)\), \(f(z)\) is not of period \(c\), with an additional assumption that \(f(z)\) is an entire function of finite order. A simple example shows that the result is not true for all entire functions. The author also obtains some results for the case \(n=2.\)