- Open Access
Some results of certain types of difference and differential equations
Advances in Difference Equations volume 2012, Article number: 127 (2012)
In this article, we shall utilize the value distribution theory and complex oscillation theory to investigate certain types of difference and differential equations. The results we obtain generalize some previous results of Gundersen and Yang.
1 Introduction and main results
In this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna theory (e.g., see [1–3]). In addition, we will use the notation to denote the order of the meromorphic function . We recall the definition of hyper-order (see ), of is defined by
Let f and g be two non-constant meromorphic functions in the complex plane. By , we denote any quantity satisfying as , possibly outside a set of r with finite linear measure. Then the meromorphic function β is called a small function of f, if . If and have the same zeros, counting multiplicity (ignoring multiplicity), then we say f and g share the small function β CM (IM).
Let be a zero of with multiplicity p and a zero of with multiplicity q. We denote by the counting function of the zeros of where , each point counted times. In the same way, we also define .
Let be transcendental meromorphic function in the plane, be a constant such that . The forward differences are defined in the standard way  by
In 1996, Brück raised the following conjecture:
Conjecture Let f be a non-constant entire function such that the hyper order and is not a positive integer. If f and share the finite value a CM, then
where c is a nonzero constant.
The case that had been proved by Brück himself in . From differential equations
we see that when the hyper order of f is a positive integer or infinite, the conjecture of Brück does not hold.
Gundersen and Yang proved the conjecture holds for entire functions of finite order, see , and Yang generalized this finite order for (), instead of , see . Chen and Shon proved the conjecture holds if , see . In terms of sharing a small function α IM, recently Wang has generalized Gundersen and Yang’s results, see .
In this paper, we consider the uniqueness of entire functions sharing a small function with their linear difference and differential polynomial. Now we present the main theorems.
Theorem 1.1 Let be a non-constant entire function, (<∞) is not a positive integer. Set (), where () are entire functions of order less than 1 and . If and share z IM, and
then , where is a meromorphic function of order no greater than s.
Remark 1 Note that the term cannot be contained in , otherwise Theorem 1.1 does not hold. For example: Set and . Then and share z IM, but
Theorem 1.2 Let be a non-constant entire function, (<∞) is not a positive integer. Set (), where () are entire functions and . If and share a IM (a is a constant), then
where is a meromorphic function of order no less than 1.
Theorem 1.3 Let be a non-constant entire function of order less than and be a non-zero small function of . Set , where () are polynomial and . If , then
where is a non-zero polynomial.
2 Some lemmas
In order to prove our theorems, we need the following lemmas and notions.
Following Hayman , pp.75-76], we define an ε-set to be a countable union of open discs not containing the origin and subtending angles at the origin whose sum is finite. If E is an ε-set then the set of for which the circle meets E has finite logarithmic measure, and for almost all real θ the intersection of E with the ray is bounded.
Lemma 2.1 ()
Let . Let be transcendental and meromorphic of order less than 1 in the plane. Then there exists an ε-set such that
Lemma 2.2 ()
Let be an entire function of order , and . If , then
where the lower logarithmic density of subset is defined by
and the upper logarithmic density of subset is defined by
where is the characteristic function of the set H.
Lemma 2.3 ()
Let be an entire function of finite order. Suppose that α is a non-zero small function of . Then there exists a set satisfying , such that
holds for , .
Lemma 2.4 ()
Let be a meromorphic function with . Then for any given , there is a set that has finite logarithmic measure such that
holds for , .
Applying Lemma 2.4 to , it is easy to see that for any given , there is a set of finite logarithmic measure such that
holds for , .
Lemma 2.5 ()
Let be a transcendental meromorphic function, and be a given constant. Then
there exists a set with finite linear measure zero and a constant that depends only on α and , such that if , then there is a constant so that for all z satisfying and , we have(2.1)
for all ;
there exists a set with finite logarithmic measure and a constant that depends only on α and , such that for all z satisfying , we have (2.1) holds.
Lemma 2.6 ()
Let be an entire function of infinite order with , and be the central index of . Then
3 Proof of Theorem 1.1
Under the hypothesis of Theorem 1.1, see , it is easy to get that
where is an entire function, entire functions and satisfy
Therefore, by (3.1), we see that is a meromorphic function of order no greater than s (<1). If is a polynomial, then . Theorem 1.1 holds under this condition. Next we suppose that is transcendental. Set . Then is transcendental, , and .
Substituting into (3.1), we get
If is a constant, then Theorem 1.1 holds. Otherwise, is a polynomial or a transcendental entire function, rewritten (3.2), we have
Set , obviously, has finite linear measure. From Lemma 2.4, for any given , there exists a set that has finite logarithmic measure such that
holds for , , where .
By Lemma 2.5, there exists a set with finite logarithmic measure and a constant such that for all z satisfying , we have
By (3.3)-(3.5), for , , we have
Let , we obtain
From (3.6), we obtain
By Wiman-Valiron theory, there exists a set having finite logarithmic measure, we choose z satisfying , and , then
where is the central index of . By (3.3), (3.5), (3.8), we obtain
By Lemma 2.6, (3.9), , and , we obtain
By combining (3.7) and (3.10), we obtain
If is polynomial, we know ; If is transcendental, we get . Hence this contradicts the hypothesis of Theorem 1.1. Theorem 1.1 is thus proved.
4 Proof of Theorem 1.2
Under the hypothesis of Theorem 1.2, it is easy to get that
where is a meromorphic function. If , by the proof of Theorem 1.1, similarly, we can prove
which contradicts the fact that . This completes the proof of Theorem 1.2.
5 Proof of Theorem 1.3
Proof: By the Hadamard factorization theorem, we have
where is an entire function of order less than . If is polynomial, Theorem 1.3 holds. Next, we consider that is transcendental. Set . Then is transcendental, and . Substituting into (5.1), we have
where . By Lemma 2.1, there exists an ε-set such that
as in . By Wiman-Valiron theory, there is a subset with finite logarithmic measure. We choose z satisfying and , then we have
where is the central index of . By (5.2)-(5.4), we have
By Lemma 2.3, there exists a set satisfying such that
Together with , we get
By (5.5)-(5.7), we have
where C is a constant. By Lemma 2.2, for any α satisfying , there exists a set with , such that
for , where .
Note the characteristic function of and such that the relation
Obviously, . Hence we obtain
Thus, the upper logarithmic density of is also more than . By (5.8) and (5.9), for , we have
If is transcendental, we get , which contradicts our assumption that . So is polynomial. This proves Theorem 1.3.
Hayman W: Meromorphic Functions. Clarendon Press, Oxford; 1964.
Laine I: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin; 1993.
Yang CC, Yi HX: Uniqueness of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.
Whittaker JM Cambridge Tracts in Math. Phys 33. In Interpolatory Function Theory. Cambridge University Press, Cambridge; 1935.
Brück R: On entire functions which share one value CM with their first derivatives. Results Math. 1996, 30: 21–24.
Gundersen GG, Yang LZ: Entire functions that share one value with one or two of their derivatives. J. Math. Anal. Appl. 1998, 223: 88–95. 10.1006/jmaa.1998.5959
Yang LZ: Solution of a differential equation and its application. Kodai Math. J. 1999, 22: 458–464. 10.2996/kmj/1138044097
Chen ZX, Shon K: On conjecture of R. Brück concerning the entire function sharing one value CM with its derivative. Taiwan. J. Math. 2004, 8: 235–244.
Wang J: Uniqueness of entire function sharing a small function with its derivative. J. Math. Anal. Appl. 2010, 362: 387–392. 10.1016/j.jmaa.2009.09.052
Hayman W: The local growth of power series: a survey of the Wiman-Valiron method. Can. Math. Bull. 1974, 17: 317–358. 10.4153/CMB-1974-064-0
Bergweiler W, Langley JK: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 2007, 142: 133–147. 10.1017/S0305004106009777
Barry PD: On a theorem of Besicovitch. Q. J. Math. 1963, 14: 293–302. 10.1093/qmath/14.1.293
Chen ZX: The zero, pole and order meromorphic solutions of differential equations with meromorphic coefficients. Kodai Math. J. 1996, 19: 341–354. 10.2996/kmj/1138043651
Gundersen G: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. Lond. Math. Soc. 1988, 37: 88–104.
Chen ZX, Yang CC: Some furthers on the zeros and growths of entire solutions of second order linear differential equations. Kodai Math. J. 1999, 22: 273–285. 10.2996/kmj/1138044047
The work was supported by the NNSF of China (No. 10771121), the NSF of Shangdong Province, China (No. Z2008A01) and Shandong university graduate student independent innovation fund (yzc11024).
The authors declare that they have no competing interests.
YL and YH completed the main parts of the paper. YL, YH and HX corrected the main theorems. All authors read and approved the final manuscript.