The zeros of complex differential-difference polynomials

Advances in Difference Equations, May 2014

This paper is devoted to considering the zeros of complex differential-difference polynomials of different types. Our results can be seen as the differential-difference analogues of Hayman conjecture (Ann. Math. 70:9-42, 1959). MSC: 30D35, 39A05.

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The zeros of complex differential-difference polynomials

Xinling Liu Kai Liu Louchuan Zhou This paper is devoted to considering the zeros of complex differential-difference polynomials of different types. Our results can be seen as the differential-difference analogues of Hayman conjecture (Ann. Math. 70:9-42, 1959). MSC: 30D35; 39A05 1 Introduction and main results Let f (z) be a meromorphic function in the complex domain. Assume that the reader is familiar with standard symbols and fundamental results of Nevanlinna theory [, ]. Recall that a(z) is a small function with respect to f (z), if T (r, a) = S(r, f ), where S(r, f ) is used to denote any quantity satisfying S(r, f ) = o(T (r, f )) as r outside of a possible exceptional set of finite logarithmic measure. Denote by (f ) and (f ) the order and the hyper-order of f . In this paper, c is a non-zero complex constant, n, k are positive integers, unless otherwise specified. Hayman [] conjectured that if f is a transcendental meromorphic function, then f nf takes every finite non-zero value infinitely often. In fact, Hayman [] proved that if f is a transcendental meromorphic function and n , then f nf takes every finite non-zero value infinitely often. Later, the case n = was settled by Mues []. Bergweiler and Eremenko [], Chen and Fang [, Theorem ] proved the case of n = , respectively. In the past years, the topic on the zeros of differential polynomials has always been an important research problem in value distribution of meromorphic functions. With the development of the difference analogues of Nevanlinna theory, some authors paid their attention to the zeros of difference polynomials. Laine and Yang [, Theorem ] firstly considered the zeros distribution of f (z)nf (z + c) - a, where a is a non-zero constant, which can be seen as a difference analogue of Hayman conjecture. Recently, many authors were interested in the zeros distribution of difference polynomials of different types, such as [-]. A polynomial Q(z, f ) can be called a differential-difference polynomial in f whenever Q(z, f ) is a polynomial in f (z), its shifts f (z + c) and their derivatives, with small functions of f (z) as the coefficients. It is interesting to consider the zeros of differential-difference polynomials. The aim of the paper is to explore the differences or analogues among the zeros of differential polynomials, difference polynomials, differential-difference polynomials. Liu et al. [, Theorems . and .] considered this problem and obtained the following result, where cf = f (z + c) - f (z). - Theorem A Let f be a transcendental entire function of finite order and a(z) be a non-zero small function with respect to f (z). If n k + , then [f (z)nf (z + c)](k) a(z) has infinitely many zeros. If f is not a periodic function with period c and n k + , then [f (z)n cf ](k) a(z) has infinitely many zeros. If a(z) in Theorem A, some results can be found in []. In this paper, we will consider the zeros of differential-difference polynomials of f (z)nf (k)(z + c) a(z) and f (z)n( cf )(k) a(z). Theorem . Let f be a transcendental entire function of hyper-order (f ) < . If n , then f (z)nf (k)(z + c) a(z) has infinitely many zeros, where a(z) is a non-zero small function with respect to f (z). Remark () The condition that a(z) is a non-zero small function cannot be removed, which can be seen by f (z) = ez and ec = . Thus we get f (z)nf (k)(z + c) = e(n+)z has no zeros. () The condition (f ) < cannot be deleted, which can be seen by f (z) = eez of (f ) = , thus f (z)nf (z + c) + nez + P(z) = P(z) has finitely many zeros, where ec = n and P(z) is a non-zero polynomial. In fact, for any integer k, we can choose appropriate (z) to make f (z)nf (k)(z + c) + (z) + P(z) = P(z), (z) is a polynomial in ez. If f is a finite order transcendental entire function, we prove the following result. Theorem . Let f be a finite order transcendental entire function. If n , then f (z)nf (k)(z + c) a(z) has infinitely many zeros, where a(z) is an entire function with (a) < (f ). Definition Define that a polynomial p(z) is a Borel exceptional polynomial of f (z) when Theorem . Let f be a finite order transcendental entire function with a Borel exceptional polynomial d(z). If n , then f (z)nf (k)(z + c) b has infinitely many zeros, where b is a nonzero constant. Remark () The condition that b is a non-zero constant cannot be removed, which can be seen by f (z) = ez which has a Borel exceptional value . Thus, we get f (z)nf (k)(z + c) = ece(n+)z has no zeros. () From the above three theorems, we can reduce the value of n with additional conditions. However, we hope that the condition n can be reduced to n in Theorem .. Unfortunately, we have not succeeded in doing that. If f (z) is a transcendental meromorphic function, we obtain the next result. Theorem . Let f be a transcendental meromorphic function of hyper-order (f ) < . If n k + , then f (z)nf (k)(z + c) a(z) has infinitely many zeros, where a(z) is a non-zero small fun (...truncated)


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Xinling Liu, Kai Liu, Louchuan Zhou. The zeros of complex differential-difference polynomials, Advances in Difference Equations, 2014, pp. 157, 2014,