Research Article Uniqueness Theorems on Difference Monomials of Entire Functions

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1 Abstract and Applied Analysis Volume 202, Article ID 40735, 8 pages doi:0.55/202/40735 Research Article Uniqueness Theorems on Difference Monomials of Entire Functions Gang Wang, Deng-li Han, 2 and Zhi-Tao Wen 3 Shandong Transport Vocational College, Weifang, Shandong 26206, China 2 Laiwu Vocational and Technical College, Laiwu, Shandong 2700, China 3 Department of Physics and Mathematics, University of Eastern Finland, P.O. Box, 800 Joensuu, Finland Correspondence should be addressed to Zhi-Tao Wen, zhitaowen@gmail.com Received 5 April 202; Accepted June 202 Academic Editor: Matti Vuorinen Copyright q 202 Gang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aim of this paper is to discuss the uniqueness of the difference monomials f n f z c. It assumed that f and g are transcendental entire functions with finite order and E k,f n f z c E k,g n g z c, wherec is a nonzero complex constant and n, k are integers. It is proved that if one of the following holds i n 6andk 3, ii n 7andk 2, and iii n 0 and k, then fg t or f t 2 g for some constants t 2 and t 3 which satisfy t n 2 andt n 3. It is an improvement of the result of Qi, Yang and Liu.. Introduction and Main Results In this pape a meromorphic respectively entire function always means meromorphic respectively, analytic in the complex plane C. It is also assumed that the reader is familiar with the basic concepts of the Nevanlinna theory. We adopt the standard notations in the Nevanlinna value distribution theory of meromorphic functions as explained in, 2. Let f and g be two nonconstant meromorphic functions, and let a be a value in the extended plane. We say that f and g share the value a CM, provided that f and g have the same a-pints with the same multiplicities. We say that f and g share the value a IM, provided that f and g have the same a-points ignoring multiplicities. The order of f is defined by σ f lim sup r log T f log r..

2 2 Abstract and Applied Analysis Let f be a nonconstant meromorphic function on C, leta C be a finite value, and let k be a positive integer or infinity. We denote by E a, f the set of zeros of f a and count multiplicities, while by E a, f the set of zeros of f a but ignore multiplicities. Also, we denote by E k a, f the set of zeros of f a with multiplicities less than or equal to k and count multiplicities. For a C { }, we denote by N k / f a the counting function corresponding to the set E k a, f while by N k / f a the counting function corresponding to the set E k a, f : E a, f \ E k a, f.ife k a, f E k a, g, we say that f, g share the value a with weight k. The definition implies that if f and g share a value a with weight k, then z 0 is a zero of f a with multiplicity m k if and only if it is a zero of g a with multiplicity m k and z 0 is a zero of f a with multiplicity m > k if and only if it is a zero of g a with multiplicity n > k where m is not necessarily equal to n. Also, we denote by N k / f a and N k / f a the reduced forms of N k / f a and N k / f a, respectively. At last, we set N k f N f N 2 f N k f..2 Hayman proposed the well-known conjecture in 3. Hayman Conjecture If an entire function f satisfies f n f / for all n N, then f is a constant. In fact, Hayman has proved that the conjecture holds in the cases n 2in 4 while Clunie proved the cases n in 5, respectively. In 997, Yang and Hua 6 studied the uniqueness theorem of the differential monomials and obtained the following result. Theorem A. Let f and g be nonconstant entire function, and let n 6 be an integer. If f n f and g n g share CM, then either f z c e cz, g z c 2 e cz where c, c 2 and c are constants satisfying c c 2 n c 2, orf tg for a constant t such that t n. In 200, Qi et al. 7 studied the uniqueness of the difference monomials and obtained the following result. Theorem B. Let f and g be transcendental entire functions with finite orde c a non-zero complex constant, and n 6 an integer. If E,f n f z c E,g n g z c, thenfg t or f t 2 g for some constants t and t 2 which satisfy t n and t n 2. In this pape we will obtain the following results. Theorem.. Let f and g be transcendental entire functions with finite orde c a non-zero complex constant, and n 6 an integer. If E 3,f n f z c E 3,g n g z c, then the assertion of Theorem B holds. Theorem.2. Let f and g be transcendental entire functions with finite orde c a non-zero complex constant, and n 7 an integer. If E 2,f n f z c E 2,g n g z c, then the assertion of Theorem B holds.

3 Abstract and Applied Analysis 3 Theorem.3. Let f and g be transcendental entire functions with finite orde c a non-zero complex constant, and n 0 an integer. If E,f n f z c E,g n g z c, then the assertion of Theorem B holds. 2. Auxiliary Results Lemma 2. see 8, Corollary 2.5. Let f z be a meromorphic function in the complex plane with finite order σ σ f, and let η be a fixed non-zero complex number. Then for each ε>0, one has m f z η f z m f z f z η O r σ ε. 2. Lemma 2.2 see 8, Theorem 2.. Let f z be a meromorphic function in the complex plane with finite order σ σ f, and let η be a fixed non-zero complex number. Then for each ε>0, one has T f z η T f z O r σ ε O log r. 2.2 Lemma 2.3. Let f z be an entire function with finite order σ σ f, c a fixed non-zero complex numbe and P z a n f z n a n f z n a f z a 0, 2.3 where a j j 0,,...,n are constants. If F z P z f z c,then T F n T f O r σ f ε O log r. 2.4 Proof. Since f z is an entire transcendental function with finite orde we can deduce from Lemma 2. and the standard Valiron-Mohon ko theorem that n T f z T f z P z O m f z P z O m f z P z F z f z m m F z O m F z O f z c T F z O r σ ε O. 2.5 Therefore T F n T f O r σ ε O. 2.6

4 4 Abstract and Applied Analysis On the other hand, Lemma 2.2 implies that T F z T P z T f z c nt f z T f z O r σ ε O log r n T f O r σ ε O log r. 2.7 We will obtain the conclusion of Lemma 2.3. Remark 2.4. The condition entire cannot be replaced by meromorphic in Lemma 2.3,as is shown by the following example. Example 2.5. Let f z e z / e z, c πi,andf z f z f z c, we can see T F / 2T f O r σ f ε O log r 2.8 for every set of {r n } with infinite measure. Lemma 2.6 see 9, Lemma 2.. Let f and g be two nonconstant meromorphic functions satisfying E k,f E k,g for some positive integer k N. Define H as follows: f H f 2f f g g 2g. 2.9 g If H / 0, then N H N 2 f N2 N 2 g N2 N 0 f f g N 0 g N k N k f g 2.0 S f S g, where N 0 /f denotes the counting function of zeros of f but not zeros of f f and N 0 /g is similarly defined. Lemma 2.7 see 0. Under the condition of Lemma 2.6, one has N N N H S f S g. f g 2.

5 Abstract and Applied Analysis 5 Lemma 2.8 see 0. Let H be defined as Lemma 2.6. IfH 0, then either f g or fg provided that N f N g N /f N /f lim sup <, 2.2 r,r I T r where T r : max{t f,t g } and I is a set with infinite linear measure. Lemma 2.9 see, Lemma 2.2. Let T : 0, 0, be a nondecreasing continuous function, s>0, 0 <α<, and let F R be the set of all r such that T r αt r s. 2.3 If the logarithmic measure of F is infinite, then log T f lim sup. r log r Proof of Theorem. We define F : f n f z c, G : g n g z c. 3. First of all, suppose that H / 0. We replace f and g by F and G, respectively, in Lemma 2.7 and Lemma 2.8.Thus, N N F N 2 F N 4 N H S f S g G N 2 N 0 F G N 0 N 4 F G S f S g. G 3.2 Applying the second main theorem to F and G jointly implies that T F T G N N F N 0 F N 0 N N F G G S f S g. G 3.3

6 6 Abstract and Applied Analysis Noting that N N F G 2 N 2 N F G N 4 N 4 F G 2 N 2 N T F, F G T G. According to Lemma 2.9 and , we can obtain that T F T G 2N 2 2N 2 S f S g F G 4N 4N 2N f g 2N g z c 6N f z c S f S g 6N S f S g f g 6T 6T S f S g. f g 3.5 Lemma 2.3 shows that T F n T f O r σ f ε O log r, T G n T g O r σ g ε O log r. 3.6 We can deduce that n 5 T f T g O r σ f ε O r σ g ε S f S g, 3.7 which is impossible since n 6. Therefore, we have H 0. Noting that N N 3T f 3T g S f S g T r, F G 3.8 where T r max{t F,T G }. Together with Lemma 2.8, it shows that either F G or FG. We will consider the following two cases. Case. Suppose that F z G z. Therefore f z n f z c g z n g z c. 3.9

7 Abstract and Applied Analysis 7 Let h z f z /g z ; we have h z n h z c. 3.0 If h z is not a constant, then Lemma 2.2 and 3.0 imply that nt h T h z c O T h O r σ h ε O log r, 3. which is a contraction with n 6. Thus, h z t, where t is a constant. From 3.0, we have f z t g z and t n. Case 2. Suppose that F z G z. Therefore f z n f z c g z n g z c. 3.2 Let h 2 z f z g z ; we have h 2 z n h 2 z c. 3.3 By the same way as Case, we can obtain that h 2 is a constant. Therefore, f z g z t 2 and t n Proof of Theorem.2 Noting that N N F G 2 5 N 2 5 N F G 4 5 N N 3 F G 35 N 35 N 3 T F, F G T G. According to 3. and 4., we can obtain the conclusion of Theorem.2 by the same way as Section Proof of Theorem.3 Noting that N N F G 4 N 4 N F G 2 N 2 2 N 2 F G 34 N 34 N 3 T F, F G T G.

8 8 Abstract and Applied Analysis According to 3. and 5., we can obtain the conclusion of Theorem.2 by the same way as Section 3. Acknowledgment The third author was supported in part by China Scholarship Council CSC. References W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, China, W. K. Hayman, Research Problems in Function Theory, The Athlone Press University of London, London, UK, W. K. Hayman, Picard values of meromorphic functions and their derivatives, Annals of Mathematics 2, vol. 70, pp. 9 42, J. Clunie, On a result of Hayman, the London Mathematical Society, vol. 42, pp , C. C. Yang and X. H. Hua, Uniqueness and value-sharing of meromorphic functions, Annales Academiæ Scientiarum Fennicæ Mathematica, vol. 22, no. 2, pp , X.-G. Qi, L. Z. Yang, and K. Liu, Uniqueness and periodicity of meromorphic functions concerning the difference operato Computers & Mathematics with Applications, vol. 60, no. 6, pp , Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of f z η and difference equations in the complex plane, Ramanujan Journal, vol. 6, no., pp , J.-F. Xu, Q. Han, and J.-L. Zhang, Uniqueness theorems of meromorphic functions of a certain form, Bulletin of the Korean Mathematical Society, vol. 46, no. 6, pp , H. X. Yi, Meromorphic functions that share one or two values, Complex Variables and Elliptic Equations, vol. 28, no., pp., 995. R. G. Halburdand and R. J. Korhonen, Nevanlinna theory for the difference operato Annales Academiæ Scientiarium Fennicæ, vol. 3, no. 2, pp , 2006.

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