The periodicity of differential-difference polynomials of entire functions

We present some results on the periodicity of an entire function f ( z ) with its differential-difference polynomials. For instance, we obtain that if f n + L ( f ) is a periodic function with period c , then f ( z ) must be a periodic function with period c , where f ( z ) is a transcendental entire function with hyper-order less than one and N ( r 1 f ) = S ( r f ) L ( f ) is a differential-difference polynomial in f with constant coefficients and degree less than n .