Value Distribution and Uniqueness Results of Zero-Order Meromorphic Functions to Their -Shift

We investigate value distribution and uniqueness problems of meromorphic functions with their -shift. We obtain that if is a transcendental meromorphic (or entire) function of zero order, and is a polynomial, then has infinitely many zeros, where , is nonzero constant, and (or ). We also obtain that zero-order meromorphic function share is three distinct values IM with its -difference polynomial , and if , then . 1. Introduction and Main Results A function is called meromorphic function if it is analytic in the complex plane except at isolated poles. It is assumed that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory such as the characteristic function , proximity function , and the counting function , see [1–3]. Let us recall the definition of the order and the zeros exponent convergence of function . The order of meromorphic function is said by The zeros of exponent convergence of meromorphic function is said by In 1959, Hayman proved the following Theorems. Theorem A (see [4], Theorem 8). Let be a transcendental entire function, and let be an integer and be a nonzero constant. Then assumes all finite values infinitely often. Theorem B (see [4], Theorem 9). Let be a transcendental meromorphic function, and let be an integer and be a nonzero constant. Then assumes all finite values infinitely often. Recently the difference variant of the Nevanlinna theory has been established independently in [5, 6]. Using these theories, value distribution theory uniqueness theory of difference polynomials of finite order transcendental meromorphic functions has been studied as well. We recall the following result by Liu and Laine. Theorem C (see [7], Theorem 1.1). Let be a transcendental entire function of finite order not of period , where is a nonzero constant, and let be a nonzero small function of . Then the difference polynomial has infinitely many zeros in the complex plane provided that . In 2010, Chen considered the difference counterpart of Hayman's theorem and porved an almost direct difference analogue of Hayman's theorem. Theorem D (see [8], Theorem 1.1). Let be a transcendental entire function of finite order not of period , and let be three complex numbers. Then assumes all finite values infinitely often, provided that and for every one has . In this paper, we consider the value distribution of zero-order meromorphic functions with their -shirt and prove the following results. Theorem 1.1. Let be a transcendental meromorphic function of zero order, and let be a polynomial. If is an integer and , then has