Research Article Unicity of Entire Functions concerning Shifts and Difference Operators

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1 Abstract and Applied Analysis Volume 204, Article ID 38090, 5 pages Research Article Unicity of Entire Functions concerning Shifts and Difference Operators Dan Liu, Degui Yang, and Mingliang Fang Institute of Applied Mathematics, South China Agricultural University, Guangzhou 50642, China Correspondence should be addressed to Mingliang Fang; mlfang@scau.edu.cn Received 29 October 203; Revised 7 December 203; Accepted 9 December 203; Published 3 February 204 Academic Editor: Zong-Xuan Chen Copyright 204 Dan Liu 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. We prove a unicity theorem of entire functions that share two distinct small functions with their shifts. The corollary of the theorem confirms the conjecture posed by Li and Gao (20).. Introduction Let f be a nonconstant meromorphic function in the complex plane C. We will use the standard notations in Nevanlinna theory of meromorphic functions such as T(r, f), N(r, f), and m(r, f) (see [, 2]). The notation S(r, f) is defined to be any quantity satisfying S(r, f) = o(t(r, f)) as r possibly outside a set of finite linear measures. A meromorphic function a is called a small function related to f provided that T(r, a) = S(r, f). Let f and g be two nonconstant meromorphic functions, and let a be a small function related to both f and g. We say that f and g share a CM if f aand g ahave the same zeros with the same multiplicities. f and g are said to share a IM if f aand g ahave the same zeros ignoring multiplicities. Let N(r, a) be the counting functions of all common zeros with the same multiplicities of f aand g a.if f a ) + ) 2N(r, a) g a =S(r,f)+S(r,g), then we say that f and g share a CM almost. For a nonzero complex constant c C, we define difference operators as Δ c f(z) = f(z+c) f(z) and Δ n c f(z) = Δ c (Δ n c f(z)), n N, n>2. In 977, Rubel and Yang [3] proved the following result. () Theorem A. Let f be a nonconstant entire function. If f(z) and f (z) share two distinct finite values CM, then f(z) f (z). In fact, the conclusion still holds if the two CM values are replaced by two IM values (see Gundersen [4, 5], Mues and Steinmetz [6]). Recently, a number of articles focused on value distribution in shifts or difference operators of meromorphic functions (see [7 ]).Inparticular,somepapersstudiedthe unicity of meromorphic functions sharing values with their shifts or difference operators (see [2 4]). In 2009, Heittokangas et al. [2] proved the following result concerning shifts. Theorem B. Let f be a nonconstant entire function of finite order, c C. If f(z) and f(z+c) share two distinct finite values CM, then f(z) f(z + c). In 20, Li and Gao [4] provedthefollowingresult concerning difference operators. Theorem C. Let f be a nonconstant entire function of finite order, c C,andletn be a positive integer. Suppose that f(z) and Δ n cf(z) share two distinct finite values a, b CM and one of the following cases is satisfied: (i) ab = 0; (ii) ab =0and ρ(f) N. Then f(z) Δ n c f(z).

2 2 Abstract and Applied Analysis In [4], Li and Gao conjectured that the restriction ρ(f) N for the case ab =0 can be removed. In this paper, we confirm their conjecture. In fact, we prove the following more general results. Theorem. Let f be a nonconstant entire function of finite order, let n be a positive integer, let a(z), b(z) be two distinct small functions related to f(z), letm,m 2,...,m n be nonzero complex numbers and c,c 2,...,c n distinct finite values, and let F (z) =m f(z+c )+m 2 f(z+c 2 )+ +m n f(z+c n ). (2) If f(z) and F(z) share a(z), b(z) CM, then f(z) F(z). Corollary 2. Let f be a nonconstant entire function of finite order, let c be a nonzero finite complex number, let n be a positive integer, and let a, b be two distinct finite values. If f(z) and Δ n c f(z) share a, b CM, then f(z) Δn c f(z). Remark 3. Corollary 2 confirms the conjecture of Li and Gao in [4]. Corollary 4. Let f be a nonconstant entire function of finite order, let c be a nonzero finite complex number, and let a(z), b(z) be two distinct small functions related to f. If f(z) and f(z + c) share a(z), b(z) CM, then f(z) f(z + c). 2. Some Lemmas For the proof of Theorem, we require the following results. Lemma 5 (see [5]). Let f and g be two nonconstant meromorphic functions satisfying f )+N(r,f)=S(r,f), g )+N(r,g)=S(r,g). (3) If f(z) and g(z) share CM almost, then either f(z) g(z) or f(z)g(z). Lemma 7 (see [0]). Let f be a nonconstant meromorphic function of finite order, c C.Then m (r, f (z+c) ) =S(r, f), (7) f (z) for all r outsideapossibleexceptionalsete with finite logarithmic measure E dr/r <. In the following, S(r, f) denotes any function satisfying S(r, f) = o(t(r, f)) as r, possibly outside a set with finite logarithmic measure. 3. Proof of Theorem We prove Theorem by contradiction. Suppose that f(z) F(z). Thenitfollowsfromf(z) and F(z) being two distinct entire functions that f(z) and F(z) share a(z), b(z), and CM. By the Nevanlinna second fundamental theorem for three small functions, we have T(r,f) N(r,f)+N(r, + N(r, f a ) f b )+S(r,f) N(r, F a )+N(r, F b )+S(r,f) 2T(r, F) +S(r,f). Similarly, we have T(r, F) 2T(r, f) + S(r, F). Therefore, S(r, f) = S(r, F). Set f (z) = F (z) = f (z) a(z) b (z) a(z), F (z) a(z) b (z) a(z). (8) (9) Lemma 6 (see [5]). Let f and g be two nonconstant meromorphic functions satisfying N(r,f)=S(r,f), N(r,g)=S(r,g). (4) If f(z) and g(z) share 0 and CM almost, and 0) +N(r, ) lim T(r,f)+T(r,g) < 2 3, (5) r r I where I [0, )is a set of infinitely linear measure, then Thus f (z), F (z) share 0,,and CM almost. Obviously, we have T(r,f )=T(r,f)+S(r,f), T(r,F )=T(r, F) +S(r,f), S (r, F) =S(r,F )=S(r,f )=S(r,f). By Nevanlinna s second fundamental theorem, we have (0) f (z) = ag (z) +b cg (z) +d, (6) where a, b, c,andd are constants satisfying ad bc =0. T(r,f ) ) + f f )+N(r,f )+S(r,f ) N(r, 0) +N(r, ) +S(r,f)

3 Abstract and Applied Analysis 3 )+S(r,f) F f T(r,F f )+S(r,f) T(r,F f)+s(r,f) m(r,f f)+s(r,f). () Since F f=m f(z + c )+m 2 f(z + c 2 )+ +m n f(z + c n ) f(z) = f(z)[m (f(z + c )/f(z)) + m 2 (f(z + c 2 )/f(z)) + +m n (f(z + c n )/f(z)) ],thus m(r,f f) m(r,f) +m(r,m f(z+c ) f (z) m(r,f)+s(r,f). + +m n f(z+c n ) f (z) By (), we have T(r,f ) N(r, 0) +N(r, ) +S(r,f) m(r,f)+s(r,f) T(r,f)+S(r,f) ) (2) (3) If a (z) b (z),wecandeduceby(6)that T(r,f ) m (r, If a (z) f a + f b )+S(r,f) m (r, F a f a + F b f b ) + m (r, )+S(r,f) F a T(r, F) +S(r,f) T(r,m f(z+c )+m 2 f(z+c 2 ) =m(r,m f(z+c )+m 2 f(z+c 2 ) m(r,f)+s(r,f) T(r,f)+S(r,f) =T(r,f )+S(r,f). b (z),set (8) =T(r,f )+S(r,f). It follows that 0) +N(r, ) =T(r, f ) +S(r, f). (4) On the other hand, by Nevanlinna first fundamental theorem, we have 2T (r, f ) = T (r, ) + T (r, f f )+S(r,f) So we get T(r,f ) m(r, N(r, 0) +N(r, ) + m (r, + m (r, T(r,f ) + m (r, m(r, + m (r, f )+S(r,f) f ) f )+S(r,f). f ) ) + m (r, f f )+S(r,f) f a ) + m (r, f b )+S(r,f). Set a (z) =m a(z+c )+m 2 a(z+c 2 )+ +m n a(z+c n ), (5) (6) b (z) =m b(z+c )+m 2 b(z+c 2 )+ +m n b(z+c n ). (7) Then we have F a b L (F) = F a b. (9) F a b m (r, F a f a )=m(r,f b f b )=S(r,f), m (r, L (F) L (F) )=m(r, )=S(r,f). F a F b It followed from (6)that T(r,f ) m (r, F a f a ) + m (r, F a ) + m (r, F b f b ) + m (r, F b )+S(r,f) m (r, m (r, ) + m (r, )+S(r,f) F a F b + )+S(r,f) F a F b m (r, L (F) )+S(r,f) T(r, L (F)) +S(r,f) T(r, F) +S(r,f) (20)

4 4 Abstract and Applied Analysis T(r,m f(z+c )+m 2 f(z+c 2 ) =m(r,m f(z+c )+m 2 f(z+c 2 ) m(r,f)+s(r,f) T(r,f)+S(r,f)=T(r,f )+S(r,f). (2) From (28), we have 2T (r, f) T (r, (f a) 2 )+S(r,f) =T(r, (b a) 2 ) + S (r, f) /((F a)/(f a)) T (r, F a f a )+S(r,f) = F a f a )+m(r,f a f a )+S(r,f) By (8)and(2), we can deduce that T (r, f ) =T(r, F) +S(r, f) =T(r, F ) +S(r, f). (22) It follows from (4)and(22)that 0) +N(r, ) lim T(r,f )+T(r,F ) = 2 < 2 3. (23) r r I By Lemma 6,wehave f (z) = AF (z) +B CF (z) +D, (24) where A, B, C, andd are complex numbers satisfying AD BC =0. Now, we consider three cases. Case. Consider N(r, 0) = S(r, f ).Thus Similarly, we have f )+N(r,f )=S(r,f )=S(r,f). (25) F )+N(r,F )=S(r,F )=S(r,f). (26) By Lemma 5, we get that either f F or f F. If f F, we can easily deduce that f F,whichisa contradiction with our assumption. If f F,thatis then we have (f (z) a)(f (z) a) (b a) 2, (27) (f a) 2 = (b a) 2 (F a) /(f a). (28) m(r, (F a )+(a a) )+S(r,f) f a m (r, a a f a )+S(r,f) T(r,f)+S(r,f). (29) It follows that T(r, f) S(r, f), a contradiction. Case 2. Consider N(r, ) = S(r, f ).Usingthesameargument as used in Case, wededucethatt(r, f) S(r, f), a contradiction. Case 3. Consider N(r, 0) =S(r,f ), N(r, ) =S(r,f ).Since f and F share 0, CM almost, we deduce from (24)that f (z) = (C+D) F (z) CF (z) +D. (30) If C=0,thenf F ;thatis,f F, a contradiction. Hence C =0.Thuswehave F + (D/C) )=N(r,f )=S(r,f )=S(r,f). (3) Obviously, D/C =0, D/C =.ThusbyNevanlinna second fundamental theorem and (4), we get 2T (r, f )=2T(r,F )+S(r,f ) )+N(r, F F ) + F + (D/C) )+S(r,f) N(r, 0) +N(r, ) +S(r,f) T(r,f )+S(r,f). (32) It follows that T(r, f ) S(r,f ), a contradiction. Thus we prove that f(z) F(z). This completes the proof of Theorem. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

5 Abstract and Applied Analysis 5 Acknowledgments The authors thank the referees for careful reading of the paper, pointing out a gap in the previous version of this paper, and giving many valuable suggestions. Research is supported by the NNSF of China (Grant no. 3749) and NSF of Guangdong Province, China (Grant no. S ). References [] W. K. Hayman, Meromorphic Function, Clarendon Press, Oxford, UK, 964. [2] L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 993. [3] L.A.RubelandC.C.Yang,Value Shared by an Entire Function and Its Derivative, Lecture Notes in Math, Springer, Berlin, Germany, 977. [4] G. G. Gundersen, Meromorphic functions that share finite values with their derivative, Mathematical Analysis and Applications,vol.75,no.2,pp ,980. [5] G. G. Gundersen, Errata: meromorphic functions that share finite values with their derivative, Mathematical Analysis and Applications,vol.86,no.,p.307,982. [6] E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen, Manuscripta Mathematica, vol. 29,no.2 4,pp ,979. [7] W. Bergweiler and J. K. Langley, Zeros of differences of meromorphic functions, Mathematical Proceedings of the Cambridge Philosophical Society,vol.42,no.,pp.33 47,2007. [8] Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of f(z + η) and difference equations in the complex plane, The Ramanujan Journal,vol.6,no.,pp.05 29,2008. [9] Y.-M. Chiang and S.-J. Feng, On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Transactions of the American Mathematical Society,vol.36,no.7,pp ,2009. [0] R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Annales Academiæ Scientiarum Fennicæ Mathematica,vol.3,no.2,pp ,2006. [] R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, Mathematical Analysis and Applications,vol.34,no.2,pp ,2006. [2] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. Zhang, Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, Mathematical Analysis and Applications,vol.355,no.,pp ,2009. [3] J.Heittokangas,R.Korhonen,I.Laine,andJ.Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Variables and Elliptic Equations,vol.56,no. 4,pp.8 92, 20. [4] S. Li and Z. Gao, Entire functions sharing one or two finite values CM with their shifts or difference operators, Archiv der Mathematik,vol.97,no.5,pp ,20. [5] M. L. Fang, Unicity theorems for meromorphic function and its differential polynomial, Advances in Mathematics, vol.24, no. 3, pp , 995.

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