The zeros of q-shift difference polynomials of meromorphic functions
© Xu and Zhang; licensee Springer 2012
Received: 28 May 2012
Accepted: 6 November 2012
Published: 22 November 2012
In this paper, we investigate the value distribution of difference polynomials and related to two well-known differential polynomials, where is a meromorphic function with finite logarithmic order.
Keywordsdifference equation meromorphic function logarithmic order Nevanlinna theory difference polynomials
In this paper, we shall assume that the reader is familiar with the fundamental results and the standard notation of the Nevanlinna value distribution theory of meromorphic functions (see [1, 2]). The term ‘meromorphic function’ will mean meromorphic in the whole complex plane ℂ. In addition, we will use notations to denote the order of growth of a meromorphic function , to denote the exponents of convergence of the zero-sequence of a meromorphic function , to denote the exponents of convergence of the sequence of distinct poles of .
where the right-hand side is rational in both arguments, has been widely studied during the last decades (see, e.g., [3–7]). There is a variety of methods which can be used to study the value distribution of meromorphic solutions of (1.1).
Recently, the Nevanlinna theory involving q-difference has been developed to study q-difference equations and q-difference polynomials. Many papers have focused on complex difference, giving many difference analogues in value distribution theory of meromorphic functions (see [8–15]).
Hayman  posed the following famous conjecture.
Theorem A If f is a transcendental meromorphic function and , then takes every finite nonzero value infinitely often.
Hayman  also proved the following famous result.
Theorem B If is a transcendental meromorphic function, is an integer, and a (≠0) is a constant, then assumes all finite values infinitely often.
Liu and Qi proved two theorems which considered q-shift difference polynomials (see ), which can be seen as difference versions of the above classical results.
Theorem C Let f be a zero-order transcendental meromorphic function and q be a nonzero complex constant. Then, for , assumes every nonzero value infinitely often.
Theorem D Let f be a zero-order transcendental meromorphic function and a, q be nonzero complex constants. Then, for , assumes every nonzero value infinitely often.
Remark 1 They also conjectured the numbers and can be reduced in Theorems C and D. But they could not deal with it. In fact, Zhang and Korhonen  also proved a result similar to Theorem C under the condition . Obviously, it is an interesting question to reduce the number n. In this paper, our results give some answers in some sense.
2 Main results
is said to be of infinite logarithmic order if the limit superior above is infinite.
Definition 1 If is a function meromorphic in the complex plane ℂ, the logarithmic order of f is the logarithmic order of its characteristic function .
It is clear that the logarithmic order of a non-constant rational function is 1, but there exist infinitely many transcendental entire functions of logarithmic order 1 from Theorem 7.3 . Hence, the transcendental meromorphic function is of the logarithmic order ≥1.
- (2)exists everywhere in except possibly in a countable set where and exist. Moreover, if we use the one-sided derivative or instead of of r in the exceptional set, then(2.2)
- (3)Let , we have for sufficiently large r and(2.3)
The above function is called a logarithmic-type function of . If is a meromorphic function of finite positive logarithmic order λ, then has proximate logarithmic order .
This quantity plays an important role in measuring the value distribution of a-points of .
Throughout this paper, we denote the logarithmic order of by , where is the number of roots of the equation in . It is well known that if a meromorphic function is of finite order, then the order of equals the exponent of convergence of a-points of . The corresponding result for meromorphic functions of finite logarithmic order also holds. That is, if is a non-constant meromorphic function and of finite logarithmic order, then for each , the logarithmic order of equals the logarithmic exponent of convergence of a-points of .
Although for any given meromorphic function with finite positive order and for any , the counting functions and both have the same order, the situation is different for functions of finite logarithmic order. That is, if is a non-constant meromorphic function in ℂ, for each , is of logarithmic order where is the logarithmic order of .
Theorem 2.1 If is a transcendental meromorphic function of finite logarithmic order λ, with the logarithmic exponent of convergence of poles less than and q, c are nonzero complex constants, then for , assumes every value infinitely often.
Remark 2 The following examples show that the hypothesis the logarithmic exponent of convergence of poles is less than is sharp.
Thus, , and the logarithmic exponent of convergence of poles is equal to . Hence, the condition the logarithmic exponent of convergence of poles is less than cannot be omitted.
Example 2 Let , . Then and have no zeros. Note that (see [22–24]), then , and the logarithmic exponent of convergence of poles is equal to . Hence, our condition the logarithmic exponent of convergence of poles is less than cannot be omitted.
Remark 3 We note that the authors claimed b is nonzero in Theorem C. But b can be zero in Theorem 2.1.
Theorem 2.2 If is a transcendental meromorphic function of finite logarithmic order λ, with the logarithmic exponent of convergence of poles less than , and a, q are nonzero complex constants, then for , assumes every value infinitely often.
Remark 4 The authors also claimed b is nonzero in Theorem D. In fact, b can take the zeros in Theorem D from their proof.
In the following, we consider the difference polynomials similar to Theorem 2.2 and Theorem 1.5 in .
Theorem 2.3 If is a transcendental meromorphic function of finite logarithmic order λ, with the logarithmic exponent of convergence of poles less than , and a, q are nonzero constants, then for , assumes every value infinitely often.
3 Proof of Theorem 2.1
We need the following lemmas for the proof of Theorem 2.1.
For a transcendental meromorphic function , is usually dominated by three integrated counting functions. However, when is of finite logarithmic order, can be dominated by two integrated counting functions as the following shows.
Lemma 3.1 ()
on a set of lower logarithmic density 1.
Lemma 3.3 ()
From Lemma 3.2 and (3.6), we can obtain
on a set of lower logarithmic density 1.
Lemma 3.5 ()
on a set of lower logarithmic density 1.
on a set of logarithmic density 1.
Lemma 3.8 If is a transcendental meromorphic function of finite logarithmic order λ and , and is an integer, set , then .
That is, from .
Thus, (3.14) and (3.16) give that . □
Proof of Theorem 2.1 Denote .
We claim that is transcendental if .
Suppose that is a rational function . Then . Therefore, by Lemma 3.5 and (3.8), , which contradicts .
Hence, the claim holds.
where is a logarithmic-type function of .
for sufficiently large r.
That is, has infinitely many zeros, then has infinitely many zeros. This completes the proof of Theorem 2.1. □
4 Proof of Theorems 2.2 and 2.3
We claim that is transcendental. Suppose that is rational, it contradicts (4.2) if . The claim holds.
where is a logarithmic-type function of .
That is, has infinitely many zeros, then has infinitely many zeros.
This completes the proof of Theorem 2.2.
The proof of Theorem 2.3 is similar to the proof of Theorem 2.2, we omit it here.
This work was supported by the National Natural Science Foundation of China (Nos. 11126327, 11171184), the Science Research Foundation of CAUC, China (No. 2011QD10X), NSF of Guangdong Province (No. S2011010000735), STP of Jiangmen, China (No. 133) and The Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. 2012LYM_0126).
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