MEROMORPHIC FUNCTIONS SHARING FOUR VALUES WITH THEIR DIFFERENCE OPERATORS OR SHIFTS

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1 Bull. Korean Math. Soc , No. 4, pp pissn: / eissn: MEROMORPHIC FUNCTIONS SHARING FOUR VALUES WITH THEIR DIFFERENCE OPERATORS OR SHIFTS Xiao-Min Li and Hong-Xun Yi Abstract. We prove a uniqueness theorem of nonconstant meromorphic functions sharing three distinct values IM and a fourth value CM with their shifts, and prove a uniqueness theorem of nonconstant entire functions sharing two distinct small functions IM with their shifts, which respectively improve Corollary 3.3a and Corollary 2.2a from [2], where the meromorphic functions and the entire functions are of hyper order less than. An example is provided to show that the above results are the best possible. We also prove two uniqueness theorems of nonconstant meromorphic functions sharing four distinct values with their difference operators.. Introduction and main results In this pape by meromorphic functions we will always mean meromorphic functions in the complex plane. We use the standard notation of Nevanlinna theory as explained in [, 7, 22]. We will denote by E R + a set of finite logarithmic measure instead of the usual linear measure, not necessarily the same at each occurrence. Then by the errorterm Sf we mean any quantity which is of the growth otf as r tends to infinity outside of E. 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-points 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 cf. [22]. Next we denote by N 0 a,f,g the reduced counting function of common a-points of f and g, and denote N 2 a,f,g by N 2 a,f,g = N f a +N g a 2N 0 a,f,g. Received July 30, 205; Revised December 8, Mathematics Subject Classification. 30D35, 30D30. Key words and phrases. entire functions, meromorphic functions, shift sharing values, difference operators, uniqueness theorems. This work is supported by the National Natural Science Foundation of China No. 784, the National Natural Science Foundation of China No and the National Natural Science Foundation of Shandong Province No. ZR204AM0. 23 c 206 Korean Mathematical Society

2 24 X.-M. LI AND H.-X. YI where and in what follows, N /f means Nf. If N 2 a,f,g = Sf+Sg, we say that f and g share a IM*. Let N E a count those points in N /f a, where each point in N E a is taken by f and g with the same multiplicity, and each such point is counted only once. We say that f and g share the value a CM*, if N f a +N g a 2N E a = Sf+Sg. We say that α is a small function of f, if α is a meromorphic function satisfying Tα = Sf cf. [22]. Suppose that α is a small function of f and g. If f α and g α share 0 CM IM, then we say that f, g share the small function α CM IM. Throughout this pape we denote by ρf the order of f, by ρ 2 f the hyper order of f, and by λf the exponent of convergence of zeros of f cf. [, 7, 22]. We also need the following two definitions: Definition. [2]. Let f be a nonconstant meromorphic function. We define difference operators as η fz = fz +η fz and n ηfz = η n η fz, where η is a nonzero complex numbe n 2 is a positive integer. If η =, we denote η fz = fz. Remark.. Definition. implies n ηfz = n n j=0 j n j fz +jη. Definition.2 [6, Definition]. Letpbeapositiveintegeranda C { }. Then by N p /f a we denote the counting function of those a-points of f counted with proper multiplicities whose multiplicities are not greater than p, by N p /f a we denote the corresponding reduced counting function ignoring multiplicities. By N p /f a we denote the counting function of those a-points of f counted with proper multiplicities whose multiplicities are not less than p, by N p /f a we denote the corresponding reduced counting function ignoring multiplicities, where and what follows, N p /f a, N p /f a, N p /f a and N p /f a mean N p f, N p f, N p f and N p f respectively, if a =. Recently the value distribution theory of difference polynomials, Nevanlinna characteristic of fz + η, Nevanlinna theory for the difference operator and the difference analogue of the lemma on the logarithmic derivative has been established cf. [5, 7, 8, 8, 9]. Using these theories, uniqueness questions of meromorphic functions sharing values with their shifts have been recently treated as well cf. [2, 3, 24]. In this pape we will consider uniqueness questions of meromorphic functions sharing four values with their shifts or difference operators. We first recall two theorems from [2]:

3 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 25 Theorem A [2, Corollary 3.3]. Let f be a nonconstant entire function such that ρf <, and let a, b and c be three distinct finite values. If fz and fz + η share a, b, c IM, where η is a nonzero complex numbe then fz = fz +η for all z C. Theorem B [2, Theorem 2.]. Let f be a nonconstant meromorphic function of finite orde and let a, a 2, a 3 be three distinct values in the extended complex plane. If fz and fz+η share a, a 2, a 3 CM, where η is a nonzero complex numbe then fz = fz +η for all z C. We will prove the following results: Theorem.. Let f be a nonconstantmeromorphic function suchthat ρ 2 f <, and let η be a nonzero complex number. Suppose that f and η f share a, a 2, a 3 IM, and share CM, where a, a 2, a 3 are three distinct finite values. Then 2fz = fz +η for all z C. Theorem.2. Let f be a nonconstant meromorphic function such that ρf <, and let η be a nonzero complex number. Suppose that f and η f share a, a 2, a 3, a 4 IM, where a, a 2, a 3, a 4 are four distinct finite values. Then 2fz = fz +η for all z C. By Theorem. we get the following result: Corollary.. Let f be a nonconstant entire function such that ρ 2 f <, and let η be a nonzero complex number. Suppose that f and η f share a, a 2, a 3 IM, where a, a 2, a 3 are three distinct finite values. Then 2fz = fz+η for all z C. Theorem.3. Let f be a nonconstant meromorphic function of hyper order ρ 2 f <, and let η be a nonzero complex number. Suppose that fz and fz+η share 0,, c IM, and share CM, where c is a finite value such that c 0,. Then fz = fz +η for all z C. We also recall the following result from [2]: Theorem C [2, Corollary 2.2]. Let f be a nonconstant entire function such that ρf <, and let a, a 2 be two distinct finite values. If fz and fz+η share a, a 2 CM, where η is a nonzero complex numbe then fz = fz +η for all z C. We will prove the following result, which improves Theorem C: Theorem.4. Let f be a nonconstant entire function such that ρ 2 f <, and let az and bz be two distinct small functions of fz such that az,bz. Suppose that fz az and fz + η az share 0 IM, fz bz and fz +η bz share 0 IM, where η is a nonzero complex number. Then fz = fz +η for all z C. We give the following example:

4 26 X.-M. LI AND H.-X. YI Example. [2]. Let fz = exp{sinz} and η = π. Then fz and fz+η share 0,,, CM and ρ 2 f =. But fz fz+η. This example shows that the condition ρ 2 f < in Theorems.3 and.4 is best possible. 2. Preliminaries In this section, we will give some lemmas to prove the main results of this pape where Lemmas play an important role in proving Theorems.-.3, Lemmas play an important role in proving Theorem.4. Moreove Lemmas are used to prove Theorems , which can be found in this section. Lemma 2. [6, Lemma]. Let f and g be two distinct nonconstant meromorphic functions, and let a, a 2, a 3 and a 4 be four distinct values in the extended complex plane. If f and g share a, a 2, a 3 and a 4 IM, then i Tf = Tg+OlogrTf as r E and r. ii 2Tf = 4 N f a j + OlogrTf as r E and r, j= where N/f means Nf. Lemma 2.2 [, Theorem 3]. Let f and g be two nonconstant rational functions. If f and g share four distinct values a, a 2, a 3, a 4 IM, then f = g. Lemma 2.3 [9, Theorem 5.]. Let f be a nonconstant meromorphic function and η C. If f is of finite orde then fz +η Tflogr m = O fz r for all r outside of a set E,+ satisfying E [,r lim sup dt/t = 0, r logr i.e., outside of a set E,+ of zero logarithmic density. If ρ 2 f = ρ 2 < and ε > 0, then fz +η Tf m = o fz r ρ2 ε for all r outside of a finite logarithmic measure. Lemma 2.4 [6]. Let f and g be two nonconstant meromorphic functions that share a, a 2, a 3 IM and a 4 CM, where a, a 2, a 3, a 4 are four distinct values in the extended complex plane. Suppose that there exists some real constant µ > 4/5 and some set I R + that has infinite linear measure such that Na 4,f/Tf µ for all r I. Then f and g share all four values CM. Lemma 2.5 [4, Lemma 7] or[22, Theorem4.]. Let F and G be two distinct nonconstant meromorphic functions such that F and G share 0,, c, CM, where c is a finite value such that c 0,.Then F, G satisfy one of the following

5 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 27 six relations: i F +G = 0, ii F +G = 2, iii F /2G /2 = /4, iv FG =, v F G = and vi F+G =, where c {,2,/2}, c = in i and iv, c = 2 in ii and v, c = /2 in iii and vi. Lemma 2.6 [22, Theorem.62]. Let f,f 2,...,f n be nonconstant meromorphic functions, and let f n+ 0 be a meromorphic function such that n+ j= f j =. If there exists a subset I R + satisfying mesi = such that n+ N fi i= Nf i < µ+otf j, j =,2,...,n, n+ +n i= i j as r and r I, where µ <. Then f n+ =. Lemma 2.7 [23, Theorem 5.]. Let f and g be two distinct nonconstant meromorphic functions, and let a, a 2, a 3, a 4, a 5 be five distinct values in the extended complex plane. If f and g share a, a 2, a 3, a 4 IM*, then N 0 a 5,f,g N 2 a 4,f,g+Sf. Lemma 2.8 [5, Theorem 2.2]. Let f be a meromorphic function with exponent of convergence of poles λ/f = λ < +, and let η 0 be a complex number. Then, for each ε > 0, we have j= Nfz +η = Nfz+Or λ +ε +Ologr. We next introduce the term ε-set cf. [4], which is used in the following lemma. We define an ε-set to be a countable union of discs E = Bb j,r j such that lim b r j j = and j b j <. Here Ba,r denotes the open disc of center a and radius and Sa,r will denote the corresponding boundary circle. Note that if E is an ε-set, then the set of r for which the circle S0,r meets E has a finite logarithmic measure and hence a zero logarithmic density. The term ε-set was introduced in the context of the following theorem, which was proved by Hayman [0] for entire functions, and by Anderson-Clunie [2] for meromorphic functions with deficient poles. Lemma 2.9 [4, Lemma 3.3]. Let g be a function transcendental and meromorphic in the plane of order less than, and let h > 0. Then there exists an ε-set E such that j= g z +η gz +η 0 and gz +η as z in C\E, gz uniformly in η for η h. Furthe E may be chosen so that for large z E the function g has no zeros or poles in ζ z h.

6 28 X.-M. LI AND H.-X. YI Next we introduce Wiman-Valiron theory: Let f = n=0 a nz n be an entire function. We define by µr = max{ a n r n : n = 0,,2,...} the maximum term of f, and define by νf = max{m : µr = a m r m } the central index of f cf. [5, pp ]. Lemma 2.0 [5, pp ]. Let g be a transcendental entire function, let 0 < δ < /4 and z be such that z = r and that gz > Mgνg 4 +δ holds. Then there exists a set E 0,+ of finite logarithmic measure, i.e., E dt/t < +, such that gm z = m 0 and r E. νg z m+ogz holds for all Lemma 2. [6, Lemma 3]. Let f and g be distinct nonconstant meromorphic functions that share four values a, a 2, a 3 and a 4 IM, where a 4 =. Then the following statements hold: i N 0,f = OlogrTf and N 0,g = OlogrTf as r E and r, where N 0,f and N 0,g count respectively only those points in N0,f and N0,g which do not occur when fz = gz = a j for some j =,2,3,4. ii For j =,2,3,4, let N 2 a j refer only to those a j -points that are multiple for both f and g and count each such point the number of times of the smaller of the two multiplicities. Then 4 j= N 2a j = OlogrTf as r E and r. Lemma 2.2 [6]. Suppose that f and g are two distinct nonconstant meromorphic functions that share a, a 2, a 3, IM, where a, a 2, a 3 are three distinct finite values. Set φ = f g f g 2 f a f a 2 f a 3 g a g a 2 g a 3. Then φ is an entire function such that Tφ = Sf. Lemma 2.3 [, Theorem]. Let P and P 2 be two nonconstant polynomials, and let a and b be two distinct finite values. If P and P 2 share a and b IM, then P = P 2. Lemma 2.4. Let f be a nonconstant entire function such that ρ 2 f <, and let a and b be two distinct small functions of f such that a and b. Set Θfz{fz fz +η} 2. ϕz = fz azfz bz and 2.2 χz = where Θfz +η{fz fz +η} fz +η azfz +η bz, Θfz = fz az az bz f z a z a z b z

7 2.3 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 29 = fz bz az bz f z b z a z b z = fz az f z a z fz bz f z b z. Suppose that fz az and fz + η az share 0 IM, fz bz and fz +η bz share 0 IM. Then Tϕz+Tχz = Sfz. Proof. By 2.3 we know that 2. can be rewritten respectively as f z b z ϕz = fz bz f z a z 2.4 {fz fz +η} fz az = Θfz{fz az+az} fz +η fz azfz bz fz fz +η 2.5 = ϕ z fz for all z C, where 2.6 fz bz az bz f z b z a z b z az fz az f z a z fz bz f z b z ϕ z = +. fz bz fz azfz bz Similarly, 2.2 can be rewritten respectively as f z +η b z χz = fz +η bz f z +η a z 2.7 {fz fz +η} fz +η az Θfz +ηfz +η fz = fz +η azfz +η bz fz +η fz 2.8 = χ z fz +η for all z C, where fz +η bz az bz f z +η b z a z b z χ z = fz +η bz az fz +η az f z +η a z fz +η bz f z +η b z fz +η azfz +η bz Noting that fz az and fz + η az share 0 IM, fz bz and fz +η bz share 0 IM, we get by 2.4 and 2.7 that 2.0 Nϕz+Nχz 2Naz+2Nbz 2Taz+2Tbz = Sfz.

8 220 X.-M. LI AND H.-X. YI By2.5,2.6,2.8,2.9, Lemma 2.3 and the lemma of logarithmic derivatives cf. [7, Theorem 2.3.3] we deduce 2. mϕz+mχz Sfz+Sfz +η. By Nevanlinna s three small functions theorem cf. [22, Theorem.36] we get Tfz N +N +Sfz fz az fz bz = N +N +Sfz 2.2 Similarly fz +η az 2Tfz +η+sfz. fz +η bz 2.3 Tfz +η 2Tfz+Sfz +η. By 2.2 and 2.3 we deduce 2.4 Sfz = Sfz +η. By 2. and 2.4 we get 2.5 mϕz+mχz = Sfz. By 2.0 and 2.5 we get the conclusion of Lemma 2.4. Lemma 2.5 [8, Proof of Theorem 2.3]. Let f be a transcendental meromorphic solution of a difference equation of the form 2.6 Uz,fPz,f = Qz,f such that ρ 2 f <, where Uz,f, Pz,f, Qz,f are difference polynomials such that the total degree deguz,f = n in fz and its shifts fz +η,..., fz + η k, and degqz,f n. Moreove assume that all coefficients b λ in 2.6 are small in the sense that Tb λ = Sf and that Uz,f contains exactly one term of maximal total degree in fz and its shifts. Then we have mpz,f = Sf. Lemma 2.6 [3, Theorem.]. Let f be a nonconstant zero order meromorphic function, and q C\{0}. Then mfqz/fz = otf on a set of logarithmic density. Lemma 2.7 [3, Theorem 2.] or [8, Theorem 2.5]. Let f be a transcendental meromorphic solution of order zero of a q-difference equation of the form U q z,fp q z,f = Q q z,f, where U q z,f, P q z,f and Q q z,f are q-difference polynomials such that the total degree degu q z,f = n in fz and its q-shifts, where degq q z,f n. Moreove we assume that U q z,f contains just one term of maximal total degree in fz and its q-shifts. Then mp q z,f = otf on a set of logarithmic density.

9 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 22 Lemma 2.8 [25, Theorem.]. Let fz be a nonconstant zero order meromorphic function, and let q C\{0}. Then Tfqz = +otfz on a set of lower logarithmic density. Lemma 2.9 [25, Theorem.]. Let fz be a nonconstant zero order meromorphic function, and let q C\{0}. Then Nfqz = +onfz on a set of lower logarithmic density. Applying Lemmas and then proceeding as in the proof of Lemma 2.4, we can get the following result: Lemma Let fz be a nonconstant zero order entire function, and let q C\{0}, and let az, bz be two distinct small functions of fz such that az and bz. Set and ϕ 2 z = Θfz{fz fqz} fz azfz bz χ 2 z = Θfqz{fz fqz} fqz azfqz bz, where Θfz is defined as in 2.3. Suppose that fz az and fqz az share 0 IM, fz bz and fqz bz share 0 IM. Then on a set of logarithmic density. mϕ 2 z+mχ 2 z = otf Applying Lemmas , we can prove the following uniqueness results of zero order meromorphic functions sharing four values with their q-difference operators or q-shifts, the proofs are just the same as the proofs of Theorems.-.4: Theorem 2.. Let fz be a nonconstant zero order meromorphic function, and let q C \ {0}. Suppose that fz and fqz fz share 0,, c IM, and share CM, where c is a complex number such that c 0,,. Then 2fz = fqz for all z C. Theorem 2.2. Let fz be a nonconstant zero order meromorphic function, and let q C \ {0}. Suppose that fz and fqz fz share a, a 2, a 3, a 4 IM, where a, a 2, a 3, a 4 are four distinct finite values. Then 2fz = fqz for all z C. Theorem 2.3. Let fz be a nonconstant zero order meromorphic function, and let q C \{0}. Suppose that fz and fqz share 0,, c IM, and share CM, where c is a finite value such that c 0,. Then fz = fqz for all z C. Theorem 2.4. Let f be a nonconstant zero order entire function, let q C\{0}, and let a and b be two distinct small functions of f such that a,b. Suppose that fz azand fqz azshare 0 IM, fz bzand fqz bz share 0 IM. Then fz = fqz for all z C.

10 222 X.-M. LI AND H.-X. YI 3. Proof of theorems Proof of Theorem.. We first suppose that f is a transcendental meromorphic function, while η f is a rational function. Then, by Lemma 2. we get Tfz = T η fz+ologrtfz = OlogrTfz as r and r E, this implies that fz is a rational function, which is impossible. Secondly we suppose that fz and η fz are nonconstant rational functions. Then, by Lemma 2.2 we get the conclusion of Theorem.. Finally we suppose that fz and η fz are transcendental meromorphic functions and suppose that f η f. Let z C be a pole of fz. Then, by the condition that fz and η fz share CM we know that either fz + η is analytic at z or z is a pole of fz +η such that the multiplicity of z related to fz +η is not greater than the multiplicity of z related to fz. Combining Lemmas 2. and 2.3, and the condition that fz and η fz share a, a 2, a 3 IM, we get 2Tfz = N i.e., fz a +N +N fz+sfz N fz +η 2fz fz a 2 +N +N fz+sfz Tfz +η 2fz+N fz+sfz fz a 3 = mfz +η 2fz+Nfz +η 2fz+N fz +Sfz mfz+m +Sfz fz +η 2 +Nfz+N fz fz Tfz+Nfz+Sfz, 3. Tfz = Nfz+Sfz. By 3. we know that there exists a subset I R + with logarithmic measure logmesi = + such that Nfz 3.2 lim r Tfz =. r I By 3.2 and Lemma 2.4 we know that fz and η fz share a, a 2, a 3, CM, and so Fz and Gz share 0,, c, CM, where 3.3 Fz = fz a a 2 a, Gz = ηfz a a 2 a

11 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 223 and 3.4 c = a 3 a a 2 a. By Lemma 2.5 we know that Fz, Gz satisfy one of the six relations of Lemma 2.5. We discuss the following six cases: Case. Suppose that Fz and Gz satisfy i of Lemma 2.5. Then Fz+Gz = 0 for all z C, and so we get by 3.3 that fz +η = 2a for all z C, which is impossible. Case 2. Suppose that Fz and Gz satisfy ii of Lemma 2.5. Then Fz+Gz = 2 for all z C, and so we get by 3.3 that fz +η = 2a 2 for all z C, which is impossible. Case 3. Suppose that Fz and Gz satisfy iii of Lemma 2.5. Then /2, are Picard exceptional values of Fz and Gz. Hence 3.5 Fz = +eγz, Gz = +e γz 2 2 for all z C, where γ is a nonconstant entire function. By substituting 3.3 into 3.5 we get 3.6 e γz+η e γz e γz = a 2 +a a 2 a for all z C. Noting that γ is a non-constant entire function, we can get by 3.6 and Lemma 2.6 that a 2 +a /a 2 a = 0, and so 3.6 can be rewritten as 3.7 e γz+η+γz e 2γz = for all z C. By 3.7 and Lemma 2.6 we can get a contradiction. Case 4. Suppose that Fz and Gz satisfy iv of Lemma 2.5. Then 0, are Picard exceptional values of Fz and Gz. Hence 3.8 Fz = e γ2z, Gz = e γ2z, where γ 2 is a nonconstant entire function. By 3.8 we get 3.9 e γ2z+η e γ2z e γ2z = a a 2 a for all z C. Noting that γ 2 is a nonconstant entire function, we can get by 3.9 and Lemma 2.6 that a /a 2 a = 0, and so 3.9 can be rewritten as 3.0 e γ2z+η+γ2z e 2γ2z = for all z C. By 3.0 and Lemma 2.6 we can get a contradiction. Case 5. Suppose that Fz and Gz satisfy v of Lemma 2.5. Then 3. Fz = +e γ3z, Gz = +e γ3z,

12 224 X.-M. LI AND H.-X. YI where γ 3 is a nonconstant entire function. Then, by 3.3 and 3. we get 3.2 e γ3z+η e γ3z e γ3z = a 2 a 2 a for all z C. Proceeding as in Case 3, we can get a contradiction by 3.2. Case 6. Suppose that Fz and Gz satisfy vi of Lemma 2.5. Then Fz+Gz = for all z C. This together with 3.3 gives fz+η = a +a 2, which is impossible. This completes the proof of Theorem.. Proof of Theorem.2. First of all, we suppose that fz is a transcendental meromorphic function, while η fz is a rational function. Then, by Lemma 2. we get Tfz = T η fz+ologrtfz = OlogrTfz, which implies that fz is a rational function, which is impossible. Secondly we suppose that fz and η fz are nonconstant rational functions. Then it follows by Lemma 2.2 that f = η f, which reveals the conclusion of Theorem.2. Finally we suppose that f and η f are transcendental meromorphic functions such that f η f. Combining this with Lemma 2.7 and the condition that fz and η fz share a, a 2, a 3, a 4 IM, we have 3.3 N 0,f, η f N 2 a 4,f, η f+sfz = Sfz. Noting that η fz = fz +η fz, we get by 3.3 that 3.4 N fz =,fz +η +N fz = fz +η = η fz = N 0,f, η f = Sfz, where N fz = fz + η = η fz = denotes the reduced counting function of common poles of fz, fz + η and η fz in {z : z < r}, N fz =,fz+η denotes the reduced counting function of those points in Nfz, which are not poles of fz +η. Therefore, by 3.4 we get 3.5 N η fz = N fz =,fz +η +N fz +η =,fz +N fz = fz +η = η fz = = N fz +η =,fz +Sfz Nfz +η+sfz, where N fz = fz + η = denotes the reduced counting function of common poles of fz and fz +η in z < N fz +η =,fz

13 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 225 denotes the reduced counting function of those points in Nfz+η, which are not poles of fz. Similarly 3.6 Nfz +η 2fz Nfz +η+sfz. By Lemma 2. and the condition that fz, η fz share a, a 2, a 3, a 4 IM we get 3.7 Sfz = S η fz. By Lemma 2. and the second fundamental theorem we get 4 3Tfz Nfz+ N +Sfz fz a j which implies that j= = Nfz+2Tfz+Sfz, 3.8 Tfz = Nfz+Sfz = Nfz+Sfz, which implies that 3.9 N fz = Nfz+Sfz = Tfz+Sfz. Similarly, we can get 3.20 T η fz = N η fz+sfz = N η fz+sfz. and 3.2 N η fz=n η fz+sfz = T η fz+sfz. By 3.9 and 3.2 we get 3.22 By 3.22 we get 3.23 N 2 fz +η = N 2 η fz+sfz N η fz N η fz Sfz. Nfz +η = N fz +η+n 2 fz +η = N fz +η+sfz. By 3.4, 3.5, 3.6, 3.7, 3.9, 3.2, 3.23, Lemma 2.3, Lemma 2.8, the second fundamental theorem and the condition that fz, η fz share a, a 2, a 3, a 4 IM we get 3T η fz N η fz+ 4 N j= Nfz +η+n η fz a j η fz fz +S η fz +Sfz Nfz +η+t fz +η 2fz+Sfz

14 226 X.-M. LI AND H.-X. YI i.e., = Nfz +η+n fz +η 2fz +mfz +η 2fz+Sfz Nfz +η+n fz +η 2fz+mfz fz +η +m 2 +Sfz fz 2Nfz +η+mfz+sfz 2N fz +η+mfz+sfz = 2Nfz+mfz+Or ρf +Ologr+Sfz 2Tfz+Or ρf +Ologr+Sfz, T η fz 2Tfz+Or ρf +ε +Ologr+Sfz. By Lemma 2. and the condition that fz, η fz share a, a 2, a 3, a 4 IM we get 3.25 T η fz = Tfz+Sfz. By 3.24 and 3.25 we get 3.26 Tfz Or ρf +ε +Ologr+Sfz. Noting that Sfz = otfz as r and r E, where E R + is some subset of a finite logarithmic measure, we get 3.27 Tfz = Or ρf +ε +Ologr as r and r E. By 3.27, the supposition that fz is a transcendental meromorphic function we deduce ρf. This together with the standard reasoning of removing exceptional set cf. [7, Lemma..2] gives 3.28 Tfz = O2r ρf +ε +Ologr+log2 as r. By 3.28 we get ρf ρf, which is impossible. This proves Theorem.2. Proof of Theorem.3. First of all, let 3.29 ψz = φ z φ z +η for all z C, where 3.30 φ z = f z fzfz fz c for all z C. Then it follows by 3.29 and 3.30 that 3.3 ψz = fz +ηfz +η fz +η cf z fzfz fz cf z +η

15 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 227 for all z C. By Lemma 2.ii and the assumptions of Theorem.3 we get 3.32 N ψ+n = Sfz. ψ Noting that ρ 2 f = ρ 2 f <, by and Lemma 2.3 we get 3.33 mψ = Sfz. By 3.32 and 3.33 we get 3.34 Tψ = Sfz. By rewriting 3.3 we get 3.35 f z fzfz fz c = ψz f z +η fz +ηfz +η fz +η c for all z C. Next we suppose that fz fz +η and let a = 0, a 2 =, a 3 = c, a 4 =. By Lemma 2.ii, we can see that N n,m a j = Sfz. n 2,m 2 By this equality we discuss the following four cases: Case. Suppose that there exists some a j j 3 such that 3.36 N, a j Sfz, where N, a j denotes the reduced counting function of common simple a j -points of fz and fz+η in z < r. By 3.36 we know that at least one of N, 0, N, and N, c, say N, 0, satisfies N, 0 Sfz. Let z 0 {z : z < r} be a common simple -point of fz and fz+η. Then, by the Laurent expansions of the left side and the right side of 3.35 in a punctured disk about z 0 we deduce ψz 0 =, this together with 3.34 and N, 0 Sfz implies that ψ =, and so 3.35 can be rewritten as f z 3.37 fzfz fz c = f z +η fz +ηfz +η fz +η c for all z C. By 3.37 and the condition that fz and fz + η share 0,, c IM we deduce that fz and fz +η share 0,, c CM. Combining Lemma 2.5, we consider the following two subcases: Subcase.. Suppose that fz andfz+ηsatisfy oneoffz+fz+η = 0, fz+fz +η = 2 and fz+fz +η =, say 3.38 fz+fz +η = 0 for all z C, where c =. Then and are Picard exceptional values of fz and fz +η. Hence 3.39 fz = fz+e γ3z

16 228 X.-M. LI AND H.-X. YI for all z C, where γ 3 is a nonconstant entire function. By 3.39 we have 3.40 fz = +eγ3z e γ3z for all z C. By substituting 3.40 into 3.38 we have e γ3z+γ3z+η = for all z C. Hence 3.4 γ 3 z+γ 3 z +η = c for all z C, where c is some finite complex constant. On the other hand, by 3.40 we have ρ 2 f = ρ 2 e γ3 = ργ 3 <. This implies that γ 3 is a nonconstant entire function of order ργ 3 <. Therefore, by Lemma 2.9 we know that for a positive number h satisfying η < h, there exists an ε-set E, such that as z E and z, we have 3.42 γ 3 z +η γ 3 z, uniformly in η. Next we let E r be the set of r for which the circle S0,r meets E. Then, E r has a finite logarithmic measure cf. [0]. Therefore, by 3.42 and Lemma 2.0 we know that there exist some infinite sequence of points z rk = r k e iθ k, where and in what follows, r k E r E and θ k [0,2π, E,+ is a subset of finite logarithmic measure, i.e., E dt/t < +, such that 3.43 γ 3 z rk = Mr k,γ 3 and γ 3z rk +η γ 3 z rk as r k. By 3.4 and 3.43 we have γ 3 z rk +η c = + lim = lim r k γ 3 z rk r k γ 3 z rk = 0, which is a contradiction. Subcase.2. Supposethat fzandfz+ηsatisfyoneoffzfz+η =, fz /2fz +η /2 = /4 and fz fz +η =, say 3.45 fz /2fz +η /2 = /4 for all z C, where c = /2. Then and /2 are Picard exceptional values of fz and fz +η. Hence fz = e γ4z /2+/2 for all z C, where γ 4 is a nonconstant entire function. This together with 3.45 gives e γ4z+γ4z+η = for all z C. Next in the same manner as in Subcase. we can get a contradiction. Case 2. Suppose that there exists some a j j 3 such that 3.46 N,m a j Sfz, where m 2 is a positive intege N,m a j denotes the reduced counting function of those zeros of fz a j with multiplicity, and of fz + η a j

17 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 229 with multiplicity m. Then, by 3.34, 3.35 and 3.46 we deduce ψ = /m, and so 3.35 can be rewritten as 3.47 mf z fzfz fz c = f z +η fz +ηfz +η fz +η c for all z C. Set 3.48 fz f z +ηfz fz +η 2 φ 2 z = fzfz fz cfz +ηfz +η fz +η c. Then, by 3.48 and Lemma 2.2 we know that φ 2 z 0 is an entire function such that 3.49 Tφ 2 z = Sfz. By 3.47 and 3.48 we get 3.50 fz c 2 = m φ 2 z fz +η fz 2 f 2 z. fz By 3.50, Lemma 2.3 and the lemma of logarithmic derivatives cf. [7, Theorem 2.3.3] we get 2mfz+O = mfz c 2 m m φ 2 z +2m Tφ 2 z+sfz Sfz, fz +η +2m f z fz fz i.e., mfz = Sfz,whichimpliesthatthereexistssomesubsetI R + with its linear measure mesi = + such that Nfz 3.5 lim r Tfz =. r I By 3.5 and Lemma 2.4 we know that fz and fz+η share 0,, c, CM. Next in the same manner as in Case we can get a contradiction. Case 3. Suppose that there exists some a j j 3 such that 3.52 N m, a j Sfz, where m 2 is a positive intege N m, a j denotes the reduced counting function of those zeros of fz a j with multiplicity m, and of fz+η a with multiplicity. Next in the samemannerasin Case2we canget acontradiction. Case 4. Suppose that N, a j + N m, a j + N,m a j = Sfz, j= j= j=

18 230 X.-M. LI AND H.-X. YI where m is any positive integer. Then, by 3.53 we get N = N n,m 0 fz n m n,m 0+ n 5m N N n,m 0 n m 5 + n,m 0+ n 6m N N n,m 0 n m 6 = 2 n,m 0+ n 5m 5N N n,m 0 n 5m 6 + n,m 0+ n 6m 5N N n,m 0 n 6m + N n,m 0 n m 6 2 N +N +Sfz 6 fz fz +η Tfz+ Tfz +η+sfz. 8 Similarly, by 3.53 we get 3.55 N fz 8 Tfz+ 8 Tfz+η+Sfz and 3.56 N fz c 8 Tfz+ Tfz +η+sfz. 8 By Lemma 2. we get 2Tfz = Nfz+N N By we get 3.58 i.e., fz c +N fz +Sfz. fz 2Tfz Tfz+ 3 8 Tfz+ 3 Tfz +η+sfz, Tfz 3Tfz +η+sfz. Similarly 5Tfz+η 3Tfz+Sfz, this together with 3.59 gives 2Tfz+2Tfz+η Sfz,

19 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 23 which is impossible. This complete the proof of Theorem.3. Proof of Theorem.4. Suppose that fz and fz +η are nonconstant polynomials. Then a and b are two distinct complex numbe this together with Lemma 2.3 reveals the conclusion of Theorem.4. Next we suppose that fz and fz+η are transcendental entire functions and fz fz+η. Set 2., 2.2 and 2.3. Then we have 2.4 and 2.7. By Lemma 2.4 we have 3.60 Tϕz = Sfz and 3.6 Tχz = Sfz. Set 3.62 H m,n z = mϕz nχz, where m and n are some two positive integers. We discuss the following two cases: Case. Suppose that there exist some two integers m 0 and n 0 such that H m0,n 0 = 0. This together with 2.4, 2.7 and 3.62 gives f z a z m 0 fz az f z b z fz bz f z +η a z = n 0 fz +η az f z +η b z fz +η bz for all z C, which implies that m0 fz bz fz +η bz 3.63 = A 0 fz az fz +η az for all z C, where A 0 is a nonzero constant. By 3.63 and the standard Valiron-Mokhon ko lemma cf. [20] we deduce n m 0 Tfz = n 0 Tfz +η+sfz. By 3.64 we have 3.65 Sfz = Sfz +η. By Lemma 2.3, the assumptions of Theorem.4 and Nevanlinna s three small functions theorem we get Tfz N +N +Sfz fz az fz bz N +Sfz fz fz +η Tfz fz +η+sfz fz mfz +η+m fz +η +Sfz

20 232 X.-M. LI AND H.-X. YI i.e., Tfz +η+sfz, 3.66 Tfz Tfz +η+sfz. Similarly, 3.67 Tfz +η Tfz+Sfz+Sfz +η. By we get 3.68 Tfz = Tfz +η+sfz. By 3.64 and 3.68 we get m 0 = n 0, and so it follows by 3.63 that 3.69 fz bz fz az = A fz +η bz fz +η az, where A is a nonzero constant satisfying A m0 = A 0. By 3.69 and the supposition fz fz + η we have A, and so 3.69 can be rewritten as 3.70 fza fz +η+az A bz = A az bzfz +η+ A azbz. By 3.70 and Lemma 2.5 we get 3.7 ma fz +η+az A bz = Sfz. By 3.68 and 3.7 we get 3.72 Tfz = Sfz, which is impossible. Case 2. Suppose that for any two positive integers m and n, we have 3.73 H m,n z 0. Let z a be a zero of fz az with multiplicity n, and a zero of fz+η az with multiplicity m, such that az a, bz a and az a bz a 0. Then, by 2.4, 2.7, 3.62 and by a calculating we can get H m,n z a = 0. Similarly, if z b be a zero of fz bz with multiplicity n, and a zero of fz + η bz with multiplicity m such that az b, bz b and az b bz b 0, then H m,n z b = 0. This together with 3.60, 3.6 and 3.73 gives N n,m a+n n,m b N +N H m,n z az bz +Naz+Nbz TH m,n z+sfz Tϕz+Tχz+Sfz

21 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES = Sfz, where and in what follows, N n,m a N n,m b, respectively denotes the reduced counting function of those zeros of fz azfz bz, respectively with multiplicity n, and of fz +η azfz +η bz, respectively with multiplicity m. By 3.74 we can get N fz az = N n,m a n m n,m a+ n 5m N N n,m a n m 5 + n,m a+ n 6m N N n,m a n m 6 = 2 n,m a+ n 5m 5N N n,m a n 5m 6 + n,m a+ n 6m 5N N n,m a n 6m + N n,m a n m 6 2 N +N +Sfz 6 fz fz +η Tfz+ Tfz +η+sfz. 8 Similarly, 3.76 N fz bz 8 Tfz+ Tfz +η+sfz. 8 Noting that f is a nonconstant entire function of hyper-order ρ 2 f <, we can get by 3.75, 3.76, Lemma 2.3 and Nevanlinna s three small functions theorem that Tfz N +N fz az +Sfz fz bz 4 Tfz+ Tfz +η+sfz 4 4 Tfz+ 4 mfz+ 4 m 2 Tfz+o Tf r ρ2 ε 2 Tfz+Sfz, +Sfz fz +η fz +Sfz

22 234 X.-M. LI AND H.-X. YI i.e., Tfz = Sfz, for all r outside of a finite logarithmic measure, which is impossible. This completes the proof of Theorem Concluding remarks Regarding Theorems. and.2, we give the following two conjectures: Conjecture 4.. If the condition ρ 2 f < of Theorem. is replaced with ρf =, then the conclusion of Theorem. still holds. Conjecture 4.2. If the condition ρf < of Theorem.2 is replaced with ρf =, then the conclusion of Theorem.2 still holds. Acknowledgements. The authors wish to express their thanks to the referee for his/her valuable suggestions and comments. References [] W. W. Adams and E. G. Straus, Non-Archimedian analytic functions taking the same values at the same points, Illinois J. Math. 5 97, [2] J. M. Anderson and J.Clunie, Slowly growing meromorphic functions, Comment. Math. Helv , [3] D. Barnett, R. G. Halburd, R. J. Korhonen, and W. Morgan, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edinburgh Sect. A , no. 3, [4] W. Bergweiler and J. K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc , no., [5] Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of fz+η and difference equations in the complex plane, Ramanujan J , no., [6] G. G. Gundersen, Meromorphic functions that share three values IM and a fourth value CM, Complex Variables Theory Appl , no. -4, [7] R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operato Ann. Acad. Sci. Fenn. Math , no. 2, [8], Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl , no. 2, [9] R. Halburd, R. Korhonen, and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc , no. 8, [0] W. K. Hayman, Slowly growing integral and subharmonic functions, Comment. Math. Helv , [], Meromorphic Functions, Clarendon Press, Oxford, 964. [2] J. Heittokangas, R. Korhonen, I. Laine, and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ , no. -4, [3] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. L. Zhang, Value sharing results for shifts of meromorphic function and sufficient conditions for periodicity, J. Math. Anal. Appl , no., [4] X. H. Hua and M. L. Fang, Meromorphic functions sharing four small functions, Indian J. Pure Appl. Math , no. 6, [5] G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuse Basel-Boston, 985. [6] I. Lahiri, Weighted sharing of three values and uniqueness of meromorphic functions, Kodai Math. J , no. 3,

23 MEROMORPHIC FUNCTIONS SHARING FOUR VALUES 235 [7] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyte Berlin/New York, 993. [8] I. Laine and C. C. Yang, Clunie theorems for difference and q-difference polynomials, J. Lond. Math. Soc , no. 3, [9], Value distribution of difference polynomials, Proc. Japan Acad. Ser. A , no. 8, [20] A. Z. Mokhon ko, On the Nevanlinna characteristics of some meromorphic functions, in: Theory of Functions, Functional Analysis and Their Applications vol. 4, 83 87, Izd-vo Khar kovsk, Un-ta, Kharkov, 97. [2] J. M. Whittake Interpolatory Function Theory, Cambridge Tract No. 33, Cambridge University Press, 964. [22] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht/Boston/London, [23] H. X. Yi, Uniqueness theorems for meromorphic functions concerning small functions, Indian J. Pure Appl. Math , no. 6, [24] J. L. Zhang, Value distribution and shared sets of differences of meromorphic functions, J. Math. Anal. Appl , no. 2, [25] J. L. Zhang and R. Korhonen, On the Nevanlinna characteristic of fqz and its applications, J. Math. Anal. Appl , no. 2, Xiao-Min Li Department of Mathematics Ocean University of China Qingdao, Shandong 26600, P. R. China and Department of Physics and Mathematics University of Eastern Finland P.O.Box, FI-800 Joensuu, Finland address: lixiaomin@ouc.edu.cn Hong-Xun Yi Department of Mathematics Shandong University Jinan, Shandong 25099, P. R. China address: hxyi@sdu.edu.cn