Journal of Mathematical Analysis and Applications

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1 J. Math. Anal. Appl Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity J. Heittokangas a,, R. Korhonen b,i.laine a, J. Rieppo a, J. Zhang c a University of Joensuu, Mathematics, PO Box, 800 Joensuu, Finland b University of Helsinki, Department of Mathematics and Statistics, PO Box 68 Gustaf Hällströmin katu 2b, FI-0004 University of Helsinki, Finland c Shandong University, School of Mathematics & System Sciences, Jinan, Shandong 25000, PR China article info abstract Article history: Received 7 December 2008 Available online 3 January 2009 Submitted by D. Khavinson Keywords: Meromorphic function Uniqueness theory Nevanlinna theory Shared values Shift Difference Logarithmic difference c-separated This research is a continuation of a recent paper due to the first four authors. Shared value problems related to a meromorphic function f z and its shift f z + c, wherec C, are studied. It is shown, for instance, that if f z is of finite order and shares two values CM and one value IM with its shift f z + c, then f is a periodic function with period c. The assumption on the order of f can be dropped if f shares two shifts in different directions, leading to a new way of characterizing elliptic functions. The research findings also include an analogue for shifts of a well-known conjecture by Brück concerning the value sharing of an entire function f with its derivative f Elsevier Inc. All rights reserved.. Introduction We assume that the reader is familiar with the elementary Nevanlinna theory, see, e.g., [5,2,5,9]. Meromorphic functions are always non-constant, unless otherwise specified. As for the standard notation in the uniqueness theory of meromorphic functions, suppose that f, g are meromorphic and a Ĉ = C { },resp.a is a small meromorphic function in the usual Nevanlinna theory sense. Denoting by Ea, f the set of those points z C where f z = a, resp. f z = az, we say that f, g share a IM ignoring multiplicities, if Ea, f = Ea, g. Provided that Ea, f = Ea, g and the multiplicities of the zeros of f z a and gz a are the same at each z C, then f, g share a CM counting multiplicities. The classical results in the uniqueness theory of meromorphic functions are the five-point, resp. four-point, theorems due to Nevanlinna [7]: If two meromorphic functions f, g share five distinct values in the extended complex plane IM, then f g. Similarly, if two meromorphic functions f, g share four distinct values in the extended complex plane CM, then f T g, where T is a Möbius transformation. The assumption 4 CM in the four-point theorem has been improved to 2CM+ 2 IM by Gundersen [7]. It is well known that 4 CM cannot be improved to 4 IM [6], while CM + 3 IM remains an open problem. This research is partially supported by the Academy of Finland #228 and #834, and the NNSF of China # * Corresponding author. addresses: janne.heittokangas@joensuu.fi J. Heittokangas, risto.korhonen@helsinki.fi R. Korhonen, ilpo.laine@joensuu.fi I. Laine, jarkko.rieppo@jns.fi J. Rieppo, jilongzhang2007@gmail.com J. Zhang X/$ see front matter 2009 Elsevier Inc. All rights reserved. doi:0.06/j.jmaa

2 J. Heittokangas et al. / J. Math. Anal. Appl We recall a well-known conjecture by Brück [2]: Let f be an entire function such that the hyper-order log log T r, f ρ 2 f := r log r is not a positive integer or infinity. If f and f share one finite value a CM, then f a = τ for some non-zero constant τ. Brück s conjecture has been verified in the special cases when a = 0 [2] or when f is of finite order [8]. Examples in [2] show that the conjecture does not hold if ρ 2 f is either a positive integer or infinity. Moreover, an example in [8] shows that the word entire cannot be replaced with the word meromorphic. As for the extensive theory of uniqueness of meromorphic functions, see [9]. In a recent paper [3], the first four authors started to consider the uniqueness of meromorphic functions sharing values with their shifts. The background for these considerations lies in the recent interest of studying Nevanlinna theory with respect to difference operators, see, e.g., the papers [9,0] by Halburd and Korhonen and [3] by Chiang and Feng. Further, shared value problems of meromorphic functions with their shifts naturally appear by looking at shared values of f and f p, asisseennext. Theorem A. See [3, Theorem.5]. Let f and p be entire functions. If f is transcendental and shares two distinct values a, a 2 C IM with f p, then pz = αz + β for some constants α,β C with α =. The same conclusion holds if f is a polynomial and IM is replaced by CM. We specify the notion of small functions as follows: Given a meromorphic function f, the family of all meromorphic functions ω such that T r, ω = ot r, f, where r outside of a possible exceptional set of finite logarithmic measure, is denoted by S f. For convenience, we also include all constant functions in S f. Moreover,letŜ f = S f { }.The two key results in [3] now read as follows. Theorem B. See [3, Theorem 2.a]. Let f be a meromorphic function of finite order, and let c C. If fz and f z + c share three distinct periodic functions a, a 2, a 3 Ŝ f with period c CM, then f z = f z + c for all z C. Theorem C. See [3, Theorem 2.3]. Let f be a meromorphic function, and let c, c 2 C be linearly independent over the real numbers. If f z, fz + c and f z + c 2 share three distinct values a, a 2, a 3 Ĉ CM, then f is an elliptic function with periods c and c 2. The following counterexample from [3] shows that the assumption on the finiteness of the order of growth in Theorem B cannot be dropped. Let c C \{0}, and let f z = expsinπ z/c. Clearly, f is of infinite order of growth, and f z and f z + c share 0, and CM, yet the functions f z and f z + c are not the same. In Section 2 we prove a shifted analogue of Brück s conjecture valid for meromorphic functions, see Theorem. The considerations in Sections 3 5 below are devoted to improving Theorems B and C by relaxing the sharing conditions. In particular, Theorem 2 shows that 3 CM in Theorem B can be replaced with 2 CM + IM. It remains open, whether this could be improved to CM + 2 IM, or even to 3 IM. Moreover, Theorem C, a characterization of elliptic functions, similarly improves from 3 CM to 2 CM + IM, see Theorem 0. In Theorems 6 8 and 2 we proceed to reduce the number of the shared small periodic functions, assuming that the meromorphic function f under consideration, or in fact a simple transformation of f, is close to an entire function in the sense that a certain deficiency condition applies. Moreover, we discuss the generally open 3 IM situation by introducing Theorem 4. In the final Section 6, we prove variants of Theorem A. Theorem 6 is a meromorphic analogue of Theorem A, being a slight improvement at the same time. Theorem 8 presents a special case of Theorem A, assuming that one of the values shared by f and f p is a Picard value. In addition to basic results from Nevanlinna theory, a difference analogue of the lemma on the logarithmic derivative from [3,9,0] takes a key role in the proofs below. For the convenience of the reader, this lemma and a couple of other auxiliary results from the difference variant of value distribution theory will be recalled whenever needed. 2. An analogue of Brück s conjecture The next result is a shifted analogue of Brück s conjecture valid for meromorphic functions. Theorem. Let f be a meromorphic function of order of growth log T r, f ρ f := < 2, r log r and let c C.If fz and f z + c share the values a C and CM, then 2

3 354 J. Heittokangas et al. / J. Math. Anal. Appl f z + c a = τ f z a for some constant τ. 3 To illustrate the necessity of the growth restriction 2, let f z = e z2 + and c C. Then the functions f z and f z + c share the values and CM, and yet f z + c = e 2cz+c2 constant. f z Since ρ f = 2, this counterexample shows that ρ f <2 cannot be relaxed to ρ f 2. We write 3 as a first-order linear difference equation f z + c τ f z = a τ, whose solutions of order < 2 can be written as f z = dz exp Log τ z + a, where Log denotes the principal branch of the c logarithm, and d is a periodic function with period c such that ρd [0, 2. Inparticular,ifτ =, then f is a periodic function with period c. Note that for any σ [, there exists a prime periodic entire function h of order ρh = σ by [8, Theorem ]. Analogously, if is to be considered as a first-order linear differential equation, then its general solution aτ can be written as f z = d expτ z + τ, d C, which is a periodic entire function with period 2π τ i such that ρ f =. To prove Theorem, we need the following result on quotients of shifts. Theorem D. See [9, Theorem 2.], [0, Corollary 2.2]. Let f be a meromorphic function of finite order, and let c C and δ 0,. Then f z + c f z T r, f m r, + m r, = o f z f z + c r δ for all r outside of a possible exceptional set E with finite logarithmic measure. Proof of Theorem. It follows by the assumptions that f z + c a = e Q z, f z a where Q is a polynomial of degree at most one. Theorem D yields T r, e Q z = m r, e Q z = ρ f o r +ε δ for any ε > 0 and δ 0,. Sinceρ f <2, we deduce that Q z must be a constant. Under the assumptions of Theorem D, it is evident that f z + c f z m r, + m r, = Sr, f. f z f z + c This fact will be used later on whenever referring to Theorem D. Chiang and Feng have obtained similar estimates for the logarithmic differences in [3], and this work is independent from [9,0]. 3. Improvements of Theorem B Next we show that 3 CM in Theorem B can be replaced with 2 CM + IM. Theorem 2. Let f be a meromorphic function of finite order, let c C, andleta, a 2, a 3 Ŝ f be three distinct periodic functions with period c. If f z and f z + c share a, a 2 CM and a 3 IM, then f z = f z + c for all z C. Theorem 2 has the following two immediate consequences. Corollary 3. Let f be an entire function of finite order, let c C,andleta, b S f be two distinct periodic functions with period c. If f z and f z + c share a CM and b IM, then f z = f z + c for all z C. Corollary 4. Let c C and f = e Q, where Q is a non-constant polynomial such that f z f z + c 0. Then there does not exist a periodic function a S f \{0} with period c such that f z and f z + c share a IM. To prove Theorem 2, we need the following result.

4 J. Heittokangas et al. / J. Math. Anal. Appl Theorem E. See [4, Theorem 4], [9, Theorem 5.3]. Let f, f 2, f 3, f 4 be meromorphic functions, and let a, a 2, a 3 Ĉ be three distinct points. If the functions f, f 2, f 3, f 4 share a, a 2 CM and a 3 IM, then at least two of f, f 2, f 3, f 4 are the same. Proof of Theorem 2. Suppose first that a, a 2, a 3 S f. Denote gz = f z a z f z a 2 z a3z a 2 z a 3 z a z. 4 Then gz + c = f z + c a z f z + c a 2 z a3z a 2 z a 3 z a z. It suffices to show that gz = gz + c for all z C. Sincenowgz and gz + c share 0, CM, and since g is of finite order, it follows that gz + c gz = e Q z, where Q is a polynomial. Moreover, we conclude that the functions gz, gz + c, gz + 2c, gz + 3c share 0, CM and IM. By Theorem E, at least two of these functions are the same. It suffices to consider the cases gz gz + 2c and gz gz + 3c. Suppose that gz = gz + 2c for all z C. Then = gz + 2c gz + c gz + c gz = e Q z+c e Q z for all z C. ThisgivesQ z + Q z + c = 2nπ i for some n Z and for all z C. BywritingQ z = C k z k + C k z k + + C 0,wehave C k z + c k + z k + C k z + c k + z k + +2C 0 = 2nπ i for all z C. Since the expressions z + c j + z j, j =,...,k, are linearly independent, it follows that C = = C k = 0, and hence Q z nπ i. Ifn is even, then e Q z, and we are done. Suppose then that n is odd. Then e Q z, that is, gz = gz + c for all z C. If there exists a point z 0 C such that gz 0 =, then also gz 0 + c =, which is a contradiction with gz 0 = gz 0 + c. Hence must be a Picard value of gz and of gz + c, and so the functions gz and gz + c share 0,, CM. The assertion now follows by Theorem B. Suppose then that gz = gz + 3c for all z C. Then = gz + 3c gz + 2c gz + c = e Q z+2c e Q z+c e Q z gz + 2c gz + c gz for all z C. ThisgivesQ z + Q z + c + Q z + 2c = 2nπ i for some n Z and for all z C, so that Q z 2 nπ i. We 3 have three possibilities: n 0 mod 3, n mod 3 or n 2 mod 3. Inthefirstcasee Q z, and we are done. In the two remaining cases must be a Picard value of gz and of gz + c, or else we arrive at a contradiction. 2 Suppose then that a =, while a 2, a 3 S f. Letd C \{a 2, a 3 }.Denotehz = / f z d, b 2 z = /a 2 z d and b 3 z = /a 3 z d. Thenb 2, b 3 Sh are two distinct periodic meromorphic functions with period c. Moreover,the functions hz and hz + c share 0, b 2 CM and b 3 IM. Part implies that hz = hz + c for all z C, from which the assertion follows. The cases a 2 = and a 3 = are dealt with analogously. The fact whether the value is shared CM or IM plays no significant role in this reasoning. If 2 CM + IM is replaced with 2 CM in Theorem 2, then an additional deficiency condition needs to be introduced as follows. Theorem F. See [3, Theorem 2.b, c]. Let f be a meromorphic function of finite order, let c C,andleta, a 2, a 3 Ŝ f be three distinct periodic functions with period c. If f z and f z + c share a, a 3 CM, and if Nr, f a 2 <, r T r, f then f z = f z + c for all z C. 5 The following lemma on the growth of non-decreasing real-valued functions will be needed in proving further refinements of Theorem 2.

5 356 J. Heittokangas et al. / J. Math. Anal. Appl Lemma G. See [, Lemma 2.]. Let T : 0, + 0, + be a non-decreasing continuous function, s > 0, α <,andletf R + be the set of all r such that T r αt r + s. If the logarithmic measure of F is infinite, then log T r =. r log r In order to prove a CM + IM version of Theorem F, the following version of the second main theorem is required. Theorem 5. Let f be a meromorphic function, let ε > 0, andleta, a 2, a 3 be pairwise distinct meromorphic functions such that a, a 2 S f,and T r, a 3 νt r, f + Sr, f 6 for some ν [0, /3.Then 3ν εt r, f N r, j= + Sr, f. Proof. We apply the method of proof of the second main theorem for three small target functions [2, Theorem 2.5]. By defining gz as in 4, the usual second main theorem yields T r, g Nr, g + N N r, j= N r, j= r, g + N r, + N r, g a a 3 + Sr, g + N r, a a 2 + N r, a 2 a 3 + Sr, g + 2T r, a 3 + Sr, f. 7 On the other hand, T r, g T r, T r, a 3 a 2 + O = T r, + a 2 a T r, + a a 2 + O 2 a 3 a 2 a 3 a = T r, f T r, a 3 + Sr, f. 8 By combining inequalities 7 and 8, it follows that T r, f N r, j= + 3T r, a 3 + Sr, f, from which the assertion follows by using 6. The next result shows that 2 CM in Theorem F can be replaced with CM + IM by strengthening the deficiency condition 5. Theorem 6. Let f be a meromorphic function of finite order, let c C, andleta, a 2, a 3 Ŝ f be three distinct periodic functions with period c. If f z and f z + c share a CM and a 3 IM, and if Nr, f a 2 < r T r, f 0, then f z = f z + c for all z C. 9 Proof. Suppose first that a, a 2, a 3 S f, and let gz be as in 4. Then gz and gz + c share 0 CM and IM, and there exists an γ [0, /0 such that

6 J. Heittokangas et al. / J. Math. Anal. Appl Nr, g<γ T r, g. It suffices to show that gz = gz + c for all z C. Suppose on the contrary that gz gz + c, and head for a contradiction. We may write gz + c gz = ψz and gz + c gz = φz, where ψ and φ are well-defined meromorphic functions of finite order satisfying mr,ψ= Sr, g and mr,φ= Sr, g by Theorem D. By a simple geometric observation, Lemma G and 0, we conclude that N r, gz + c N r + c, g = Nr, g + Sr, g<γ T r, g + Sr, g. 2 Since gz and gz + c share 0 CM, all poles of ψ are among the poles of gz + c. It now follows by 2 that T r,ψ= Nr,ψ+ Sr, g<γ T r, g + Sr, g. 3 Combining the equations in, we may write φz = ψzgz /gz, from which gz = φz φz φz ψz = ψz +. This gives T r, g T r,φ+ T r,ψ+ O. Using 3, we obtain T r,φ T r, g T r,ψ+ O γ T r, g + Sr, g. 4 Since gz and gz + c share IM, it follows that the -points of gz and of gz + c are among the -points of ψ. Hence, by 3, we have N r, + N r, 2N r, 2T r,ψ+ O 2γ T r, g + Sr, g. 5 g gz + c ψ From, we observe that the zeros and poles of φ are among the -points of either gz or gz + c and the poles of either gz or gz + c. Therefore, by 0, 2 and 5, it follows that Nr,φ+ N r, 4γ T r, g + Sr, g. φ Moreover, T r,ψ γ T r,φ+ Sr,φ γ by 3 and 4. Let ε > 0. Then, by Theorem 5 and 4, we conclude that 3γ γ ε T r,φ Nr,φ+ N r, + N r, + Sr,φ φ φ ψ N r, g + 4γ T r, g + Sr,φ ψ Nr, g + T r,ψ+ 4γ T r, g + Sr,φ 6γ T r,φ+ Sr,φ, γ which contradicts with the fact that γ [0, /0. 2 The cases when exactly one of the functions a, a 2, a 3 is equal to are dealt with as in part 2 of the proof of Theorem 2. Let f be a meromorphic function, and let a S f. Thenn 2 r, is the number of zeros of in z r, where f a the simple zeros are counted once and the multiple zeros twice. Further, n 2 r, f is the number of poles of f, where the simple poles are counted once and the multiple poles twice. The corresponding integrated counting functions N 2 r, f a and N 2 r, f are defined in the usual way. Then N r, N 2 r, 2N r, 6 and 0 Nr, f N 2 r, f 2Nr, f, see [9, p. 365]. 7

7 358 J. Heittokangas et al. / J. Math. Anal. Appl Theorem 7. Let f be a meromorphic function of finite order, let c C, andleta, a 2, a 3 Ŝ f be three distinct periodic functions with period c. If f z and f z + c share a 3 CM, and if r N 2 r, f a + N 2 r, T r, f f a 2 < 2, then f z = f z + c or f z = f z + 2c for all z C. 8 Proof. Similarly as above, we may suppose that a, a 2, a 3 S f. Letgz be as in 4. Then gz and gz + c share CM, and, by 8, there exist constants ε > 0 and r ε > 0 such that N 2 r, g + N 2 r, < g 2 ε T r, g, r r ε. 9 By a simple geometric observation and Lemma G, we have N 2 r, gz + c + N2 r, N 2 r + c, g + N2 r + c, = N 2 r, g + N 2 r, + Sr, g gz + c g g outside of a possible exceptional set E of finite logarithmic measure. Denote F = R + \ E [0, r ε ], where r ε is the constant in 9. Then F is of infinite logarithmic measure, and clearly of infinite linear measure. Hence, by 9, we get N 2 r, gz + c + N2 r, < T r, g, r F. 20 gz + c 2 By combining 9 and 20 with [9, Theorem 7.0], it follows that either gz gz + c or gzgz + c. The latter possibility yields gz gz + 2c. Therefore either f z f z + c or f z f z + 2c. We note that if the deficiency condition 8 is replaced with r Nr, f a + Nr, T r, f f a 2 < 4, and if all the other assumptions of Theorem 7 are valid, then f z = f z + c or f z = f z + 2c for all z C. This follows by 6 and 7. Theorem H. See [, Folgerung 4.]. Let f and f 2 be meromorphic functions such that Nr, f j + N r, f = Sr, f j, j =, 2. j If f and f 2 share IM, then f z = f 2 z or f z f 2 z = for all z C. Finally we introduce a deficiency condition which, together with IM, forces f to be a periodic function. Theorem 8. Let f be a meromorphic function of finite order, let c C, andleta, a 2, a 3 Ŝ f be three distinct periodic functions with period c. If f z and f z + c share a 3 IM, and if N r, + N r, = Sr, f, 2 then f z = f z + c or f z = f z + 2c for all z C. Proof. Similarly as above, we may suppose that a, a 2, a 3 S f. Letgz be as in 4. Then gz and gz + c share IM, and Nr, g + N r, = Sr, g. g Hence, by a simple geometric observation and Lemma G, we have N r, gz + c + N r, N r + c, g + N r + c, gz + c g = Nr, g + N r, + Sr, g g = Sr, g εrt r, g,

8 J. Heittokangas et al. / J. Math. Anal. Appl where lim r εr = 0. The second main theorem and the assumptions give T r, g Nr, g + N r, + N r, + Sr, g g g = N r, gz + c T r, gz + c + Sr, g, + Sr, g that is, + ot r, g T r, gz + c. Wehaveprovedthat N r, gz + c + N r, = S r, gz + c. gz + c Now Theorem H implies that either gz gz + c or gzgz + c. The latter possibility yields gz gz + 2c. Therefore either f z f z + c or f z f z + 2c. Theorem 8 has the following immediate consequence related to Corollary 3. Corollary 9. Let f be an entire function of finite order, let c C,andleta, b S f be two distinct periodic functions with period c. If f z and f z + c share a IM, and if N r, = Sr, f, f b then f z = f z + c or f z = f z + 2c for all z C. 4. An improved criterion for elliptic functions Using Theorem 2, we offer the following improvement of Theorem C. Theorem 0. Let f be a meromorphic function, let c, c 2 C be linearly independent over the real numbers, and let a, a 2, a 3 Ĉ be three distinct values. If f z, fz + c and f z + c 2 share a, a 2 CM and a 3 IM, then f z is an elliptic function with periods c and c 2. Proof. We may assume that a = 0, a 2 = and a 3 =, for otherwise we can replace f with g = f a f a 2 a3 a 2 a 3 a.ifthere exists a point z 0 such that f z 0 = a j, j =, 2, 3, then f z 0 + kc = a j and f z 0 + kc 2 = a j for all k Z. Therefore the parallelogram defined by the vertice points lc + lc 2, l + c + lc 2, lc + l + c 2, l + c + l + c 2 has the same finite number of a j -points of f z for all l Z. Hence there exists a constant C > 0, not depending on r, such that N r, Cr 2, j =, 2, 3, for all r > 0. The second main theorem yields that f is of finite order. The conclusion follows by Theorem Alternative improvements of Theorem B We proceed to find alternative improvements of Theorem B by means of Nevanlinna theory for exact differences [9]. We begin by reviewing some basic definitions and fundamental results of this theory. A more detailed presentation can be found in [9]. Let f be a meromorphic function, and let c C. Ifa C, thenn c r, f a is the number of points z 0, z 0 r, where Ñ c r, f z 0 = a and f z 0 + c = a, counted according to the number of equal terms in the beginning of Taylor series expansions of f z a and f z + c a in a neighborhood of z 0. Such points are called c-separated a-pairs of f in the disc {z: z r}. Further, n c r, f is the number of c-separated pole pairs of f, which are exactly the c-separated 0-pairs of / f. The corresponding integrated counting functions N c r, f a and N cr, f are defined in the usual way. Following [9], we also define := N r, N c r,, which counts the number of those a-points, a Ĉ, of f which are not in c-separated pairs. The point a is an exceptional paired value of f with the separation c if the following property holds for all a-points of f : Whenever f z = a then also f z + c = a with the same or higher multiplicity.

9 360 J. Heittokangas et al. / J. Math. Anal. Appl Theorem I. See [9, Theorem 2.5]. Let c C, and let f be a meromorphic function of finite order such that f z f z + c 0. Let q 2,andleta, a 2,...,a q Ŝ f be distinct periodic functions with period c. Then q T r, f Ñ c r, f + q Ñ c r, + Sr, f. 2 k k= Theorem I has several neat consequences including a c-separated analogue of the classical defect relation [9, Corollary 2.6]. The shortest proof for Picard s theorem is by means of the usual second fundamental theorem by Nevanlinna. The following analogue of Picard s theorem [9, Corollary 2.7] follows immediately by Theorem I: If a finite order meromorphic function f has three exceptional paired values with the separation c, then f is a periodic function with period c. Let f be a meromorphic function of order ρ f 0, ]. We say that a is a Borel exceptional paired value of f with the separation c if r log + Ñ c r, log r f a < ρ f. Note that if a is an exceptional paired value of f with the separation c, thenñ c r, = O, and hence a is also a Borel f a exceptional paired value of f with the separation c. The following analogue of Borel s theorem is an immediate consequence of Theorem I. Corollary. Let c C \{0}. If a meromorphic function f of order ρ f 0, has three Borel exceptional paired values with the separation c, then f is a periodic function with period c. Since Ñ c r, f a Nr, holds for any meromorphic f and any a Ŝ f, we see that the following result is a slight f a improvement of Theorem F. Theorem 2. Let f be a meromorphic function of finite order, let c C, andleta, a 2, a 3 Ŝ f be three distinct periodic functions with period c. If f z and f z + c share a and a 3 CM, and if Ñ c r, f a 2 <, r T r, f then f z = f z + c for all z C. The conclusion of Theorem 2 follows by Theorem I and the following lemma. Lemma 3. Let f be a meromorphic function, let c C, andleta Ŝ f be a periodic function with period c. If f shares a CM with f z + c,then0 is an exceptional paired value of with the separation c, and Ñ c r, 0. Proof. If 0 is a Picard value of f z az, then 0 is also a Picard value of f z +c az, and the conclusions follow trivially. Suppose then that 0 is not a Picard value of. Letz 0 be such that f z 0 az 0 = 0 with multiplicity p. Since f z shares a CM with f z + c, it follows that f z 0 + kc az 0 = 0 with the same multiplicity p for all k Z. The conclusions now follow by definition. We note that the combination of Theorem I and Lemma 3 also yields an alternative proof for Theorem B. Theorem 4. Let f be a meromorphic function of finite order, let c C, andleta, a 2, a 3 Ŝ f be three distinct periodic functions with period c. Given ε 0, 3, if any zero of f z a jz j =, 2, 3, with multiplicity p, is a zero of f z + c a j z, with multiplicity q > 2 εε 3 p + ε, then fz = f z + c for all z C. Theorem 4 is closely related to the generally open 3 IM situation. Note that if f z and f z + c share three values CM, then p = q in Theorem 4, which therefore is an improvement of Theorem B. Also note that 2 εε + ε < forε 0, 3 3, and hence the restriction for the multiplicity q is automatically satisfied in the case p =. Proof of Theorem 4. By the value sharing assumptions, all zeros of are in c-separated pairs for j =, 2, 3. Clearly 2 εε N c r, > N r, + εn r,, j =, 2, 3, 3

10 J. Heittokangas et al. / J. Math. Anal. Appl and so Ñ c r, < + εε N 3 r, εn r,, j =, 2, 3. Suppose on the contrary that f z f z + c. Then, by Theorem I, we have T r, f and hence N r, j= Ñ c r, + Sr, f + εε T r, f ε j= N j= r, + Sr, f, εt r, f + Sr, f. 22 Combining 22 with [2, Theorem 2.5], we obtain + o T r, f εt r, f, which is a contradiction. Hence f z = f z + c for all z C. An analogous reasoning yields the next result for entire functions offering an improvement of [3, Corollary 2.2a]. Theorem 5. Let f be an entire function of finite order, let c C, andleta, a 2 S f be two distinct periodic functions with period c. Given ε 0, 2, if any zero of f z a jz j =, 2, with multiplicity p, is a zero of f z + c a j z, with multiplicity q > εε 2 p + ε,then fz = f z + c for all z C. 6. Functions f and f p share values The next result is a meromorphic analogue of Theorem A. Theorem 6. Let f be a transcendental meromorphic function and p an entire function. If f and f p share three distinct values a, a 2, a 3 Ĉ CM, then pz = αz + β for some constants α,β C with α n = for some n N. The same conclusion holds if p is of finite order and CM is replaced with IM. To prove Theorem 6, we need the following lemma. Lemma 7. Let f be a meromorphic function and pz = αz + β, whereα,β C \{0} and α =. If f and f psharea Ĉ IM, then there exists a positive integer n such that α n =, or f takes the value a at most once. Proof. Suppose that f takes the value a at least twice, and that α. Let z 0 β/ α be such that f z 0 = f αz 0 + β = a. Sincez 0 is not the fixed point of p, thevalue z n = α n z 0 + β αn α is an a-point of f for each positive integer n. Since α = by the assumption, we obtain z n z β α, which means that either the a-point sequence {z n } must accumulate to a finite value or z m = z p for some distinct positive integers m and p. The former is impossible while the latter easily yields α m p =. Proof of Theorem 6. Without loss of generality, we may assume that the shared values are 0, and. By the second main theorem, T r, f 3T r, f p + Sr, f, T r, f p 3T r, f + Sr, f p, T r, f p p 3T r, f + Sr, f p p, 25 and hence S f = S f p = S f p p. By the value sharing assumption, there exist entire functions a and b such that

11 362 J. Heittokangas et al. / J. Math. Anal. Appl Therefore f p = e a and f f p p = e a e a p and f f p = e b. f f p p = e b e b p. 27 f We proceed to show that p must be a polynomial. Assume on the contrary that p is transcendental. We make two observations: If a is a constant, then clearly T r, e a = Sr, f. Suppose then that a is non-constant. Now, by [4, Theorem ii], for an arbitrarily large constant M > there exists a constant r M > 0 such that 26 T r, e a p MT r, e a, r r M. 28 By using 28, 27, 25 and elementary Nevanlinna theory, we obtain MT r, e a T r, f p p + T r, e a + T r, f + O 4T r, f + T r, e a + Sr, f. Hence, for all r outside of a possible exceptional set of finite linear measure, we have the estimate T r, e a T r, f 5 M. Since M > is arbitrarily large, it follows that T r, e a = Sr, f. 2 If b is either a constant or non-constant, it follows similarly as in that T r, e b = Sr, f always holds. We have shown that T r, e a = Sr, f and T r, e b = Sr, f always hold, provided that p is transcendental. Combining the equations in 26 gives e a e b f = e b, which results in e a e b. As a consequence, f p = f. By [4, Theorem 2ii], r T r, f p T r, f =, which is a contradiction. Therefore p must be a polynomial. Similarly, as in the proof of [3, Theorem.5], we see that pz = αz + β, where α,β C and α =. Since f is transcendental, it takes at least one of the shared values a, a 2, a 3 infinitely many times. Hence, by Lemma 7, there exists a positive integer n such that α n =. Finally, we suppose that f shares the values 0, and IM with f p, where p is a non-constant entire function of finite order. We may write f p = ψ and f f p = φ, f where ψ and φ are well-defined meromorphic functions. If ψ is a constant, then clearly T r,ψ= Sr, f. Supposing that p is transcendental yet of finite order, we use [4, Theorem 3i] to conclude that, for an arbitrarily large constant M >, there exists a constant r M > 0 such that T r,ψ p MTr,ψ, r r M. This corresponds to the estimate in 28, and leads to T r,ψ= Sr, f just as in case above. By replacing e a with ψ and e b with φ, the rest of the proof follows that of the CM-case above, word for word. Theorem 8. Let f be an entire function having a Picard value a C, and let p be an entire function. If f and f pshareavalue b C \{a} IM, then one of the following assertions hold: f f pandpz = αz + β for some α,β C such that α n = for some n N. 2 f f p pandpz = αz + β for some α,β C such that α 2n = for some n N. Proof. Define g = f a.theng and g p are both entire functions having zero as a Picard value. Moreover, g and g p b a share IM. By Theorem H we conclude that either g g p or g. The latter possibility yields g g p p. Therefore g p either f f p or f f p p.

12 J. Heittokangas et al. / J. Math. Anal. Appl We proceed to prove the assertions on p. Suppose first that f f p. By Theorem 6 we deduce that pz = αz + β for some α,β C such that α n = for some n N. Suppose then that f f p p. We may apply Theorem 6, with p p in place of p, to deduce that p pz = Az + B for some A, B C such that A n = for some n N. Consequently, pz = αz + β for some α,β C such that α 2n = for some n N. Corollary 9. Let f be a meromorphic function and p an entire function, and let a, a 2, a 3, a 4 Ĉ be pairwise different. If f and f p share a, a 2 CM and a 3, a 4 IM, then f f por f f p p. Proof. By [7, Theorem ], the functions f and f p share all four values a,...,a 4 CM. Then, by the classical 4-point theorem, see [6, p. 8], we may assume that a 3 and a 4 are Picard values of f and of f p. Henceg = f a 3 is an entire f a 4 function and avoids the value zero. Moreover, g and g p share the values a j a 3, j a j a 4 =, 2, CM. Hence, by Theorem 8, we obtain the desired conclusion. References [] G. Brosch, Eindentigkeitssätze für meromorphe Funktionen, Dissertation, Rwth Aachen, 989. [2] R. Brück, On entire functions which share one value CM with their first derivative, Results Math [3] Y.-M. Chiang, S.-J. Feng, On the Nevanlinna characteristic of f z + η and difference equations in the complex plane, Ramanujan J [4] J. Clunie, The Composition of Entire and Meromorphic Functions, Mathematical Essays Dedicated to A.J. Macintyre, OH. Univ. Press, Athens, Ohio, 970, pp [5] A. Goldberg, I. Ostrovskii, Value Distribution of Meromorphic Functions, Transl. Math. Monogr., vol. 236, American Mathematical Society, Providence, RI, 2008, translated from the 970 Russian original by Mikhail Ostrovskii, with an appendix by Alexandre Eremenko and James K. Langley. [6] G.G. Gundersen, Meromorphic functions that share three or four values, J. London Math. Soc [7] G.G. Gundersen, Meromorphic functions that share four values, Trans. Amer. Math. Soc [8] G.G. Gundersen, L.-Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl [9] R.G. Halburd, R.J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math [0] R.G. Halburd, R.J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl [] R.G. Halburd, R.J. Korhonen, Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. London Math. Soc [2] W.K. Hayman, Meromorphic Functions, Oxford Math. Monogr., Clarendon Press, Oxford, 964. [3] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ., in press. [4] G. Jank, N. Terglane, Meromorphic functions sharing three values, Math. Pannon [5] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 993. [6] E. Mues, Shared value problems for meromorphic functions, in: Value Distribution Theory and Complex Differential Equations, Joensuu, 994, in: Joensuun Yliop. Luonnont. Julk., vol. 35, Univ. Joensuu, Joensuu, 995, pp [7] R. Nevanlinna, Le théorème de Picard Borel et la théorie des fonctions méromorphes, Gauthiers Villars, Paris, 929. [8] M. Ozawa, On the existence of prime periodic entire functions, Kodai Math. Sem. Rep [9] C.C. Yang, H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Math. Appl., vol. 557, Kluwer Academic Publishers Group, Dordrecht, 2003.