Value distribution of difference and q-difference polynomials
© Li and Yang; licensee Springer. 2013
Received: 20 December 2012
Accepted: 25 March 2013
Published: 10 April 2013
In this paper, we investigate the value distribution of difference polynomial and obtain the following result, which improves a recent result of K. Liu and L.Z. Yang: Let f be a transcendental meromorphic function of finite order σ, c be a nonzero constant, and be a small function of f, and let
be a polynomial with a multiple zero. If , then has infinitely many zeros. We also obtain a result concerning the value distribution of q-difference polynomial.
1 Introduction and main results
possibly outside of a set of finite linear measure in . A meromorphic function is called a small function of provided that . As usual, we denote by the order of a meromorphic function , and denote by () the exponent of convergence of the zeros (poles) of .
Recently, a number of papers concerning the complex difference products and the differences analogues of Nevanlinna’s theory have been published (see [4–12] for example), and many excellent results have been obtained. In 2007, Laine and Yang  investigated the value distribution of difference products of entire functions, and obtained the following result.
Theorem A Let be a transcendental entire function of finite order, and c be a non-zero complex constant. Then for , assumes every non-zero value infinitely often.
Liu and Yang  improved Theorem A, and proved the next result.
Theorem B Let be a transcendental entire function of finite order, and c be a non-zero complex constant. Then for , has infinitely many zeros, where is a polynomial in z.
The purpose of this paper is to investigate the value distribution of difference polynomial and q-difference polynomial , where with constant coefficients , and is a mall function of .
We obtain the following result which improves Theorem A and Theorem B.
be a polynomial with constant coefficients and . If , then has infinitely many zeros, where is a small function of f.
Remark 1 The result of Theorem 1.1 may be false if , for example, , it is obvious that has no zeros. The following example shows that the assumption in Theorem 1.1 cannot be deleted. In fact, let , , , and . Then and has no zeros. Also, let , , , and . Then has no zeros. This shows that the restriction in Theorem 1.1 to the multiple zero case is essential.
Considering the value distribution of q-differences polynomials, we obtain the following result.
Theorem 1.2 Let be a transcendental entire function of zero order, and . Suppose that q is a non-zero complex constant and n is an integer. If , then has infinitely many zeros.
2 Some lemmas
Lemma 2.1 
Lemma 2.2 
The following lemma is a revised form of Lemma 2.4.2 in .
Lemma 2.4 
outside of a possible exceptional set E with finite logarithmic measure.
Lemma 2.5 
on a set of logarithmic density 1.
also holds on a set of logarithmic density 1.
Since () in , we get ().
Following the similar method in the proof of Theorem 1.1 in , we get the result. □
on a set of logarithmic density 1.
The assertion follows from (2.1) and (2.2). □
3 Proof of Theorem 1.1
Now we distinguish two cases.
Since , and is a transcendental, this is impossible.
for and .
This contradicts that f is of order σ. Theorem 1.1 is proved.
4 Proof of Theorem 1.2
Since f is a transcendental entire function with , we deduce that has infinitely many zeros.
This work was supported by the NSF of Shandong Province, P.R. China (No. ZR2010AM030) and the NNSF of China (No. 11171013 and No. 11041005).
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