On uniqueness of Meromorphic functions with shared-sets in an angular domain

Acta Mathematica Scientia 2011,31B(1):194–206 http://actams.wipm.ac.cn ON UNIQUENESS OF MEROMORPHIC FUNCTIONS WITH SHARED-SETS IN AN ANGULAR DOMAIN∗ Chen Shengjiang (���) Department of Mathematics, Ningde Teachers College, Fujian 352100, China E-mail: [email protected] Lin Weichuan (���) Department of Mathematics, Fujian Normal University, Fuzhou 350007, China E-mail: [email protected] Abstract In this article, we deal with the uniqueness problems on meromorphic func- tions sharing two finite sets in an angular domain instead of the whole plane C. In par- ticular, we investigate the uniqueness for meromorphic functions of infinite order in an angular domain and obtain some results. Moreover, examples show that the conditions in theorems are necessary. Key words Shared-set; uniqueness; meromorphic function; angular domain; infinite or- der 2000 MR Subject Classification 30D35 1 Introduction and Results In this article, the term “meromorphic” will always mean meromorphic in the whole com- plex plane C. As we know, the uniqueness theory of meromorphic functions in the whole plane C is fairly completed as compared to the uniqueness theory of meromorphic functions in an angular domain. However, Zheng in [15] obtained uniqueness theorems of the meromorphic functions which have five shared values in a precise subset of C and in [14] considered the case of dealing with four shared values. Following his idea, we shall investigate the uniqueness problems on meromorphic functions with shared-sets in an angular domain. We assume that the reader is familiar with the basic results and the notations of Nevan- linna’s value distribution theory [6, 12], such as T (r, f), N(r, f), and m(r, f). We define the lower order μ and the order λ of a meromorphic function f as we know as follows: μ := μ (f) = lim inf r→+∞ logT (r, f) log r , λ := λ (f) = lim sup r→+∞ log T (r, f) log r . ∗Received March 4, 2008; revised August 4, 2009. Supported by the NNSFC (10671109), the NSFFC (2008J0190), the Research Fund for Talent Introduction of Ningde Teachers College (2009Y019), and the Scitific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. No.1 Chen & Lin: UNIQUENESS OF MEROMORPHIC FUNCTIONS WITH SHARED-SETS 195 Meanwhile, we define the hyper order σ2 of a meromorphic function f as σ2 := σ2 (f) = lim sup r→∞ log logT (r, f) log r . Let S be a subset of distinct elements in Cˆ and X ⊆ C. We Define EX (S, f) = ⋃ a∈S {z ∈ X | fa (z) = 0, counting multiplicties} , EX (S, f) = ⋃ a∈S {z ∈ X | fa (z) = 0, ignoring multiplicties} , where fa(z) = f (z)− a if a ∈ C and f∞(z) = 1/f (z). Let f and g be two nonconstant meromorphic functions. If EX (S, f) = EX (S, g), we say f and g share the set S CM (counting multiplicities) in X . If EX (S, f) = EX (S, g), we say f and g share the set S IM (ignoring multiplicities) in X . In particular, when S = {a}, where a ∈ Cˆ, we say f and g share the value a CM (or IM ) in X if EX (S, f) = EX (S, g) (or EX (S, f) = EX (S, g)). When X = C, we give the simple notations as before, E (S, f), E (S, f) and so on [12]. In this article, we denote by X an angular domain X (α, β) = {z | α < arg z < β}, where 0 ≤ α < β ≤ 2π, and denote a sector {z ∈ X (α, β) | 1 < |z| < r} by Δ(r). In [8], W. Lin S. Mori and K. Tohge dealt with the uniqueness problem on meromorphic functions sharing three finite sets in an angular domain. They obtained the following theorems. Theorem A Let S1 = {∞}, S2 = {ω | ωn−1 (ω + a) − b = 0}, S3 = {0}, where n (≥ 4) is an integer, and a, b are two nonzero constants, such that the algebraic equation ωn−1 (ω + a)− b = 0 has no multiple roots. Assume that f is a meromorphic function of lower order μ (f) ∈ (1/2,∞) in C and δ (ι, f) > 0 for some ι ∈ Cˆ\ {0,−a}. Then, for each σ < ∞ with μ (f) ≤ σ ≤ λ (f), there exists an angular domain X = X (α, β) with 0 ≤ α < β ≤ 2π and β − α > max { π σ , 2π − 4 σ arcsin √ δ 2 } , (T 1) such that if the conditions E (S3, f) = E (S3, g) and EX (Sj , f) = EX (Sj , g) (j = 1, 2) hold for a meromorphic function g of finite order or, more generally, with the growth satisfying either logT (r, g) = O (logT (r, f)) or lim r→∞ log logT (r, g) min {log r, logT (r, f)} = 0, r �∈ E1, (T 2) where E1 is a set of finite linear measure, then f ≡ g. Theorem B Let S1, S2, and S3 be defined as in Theorem A. Assume that f is a mero- morphic function of infinite order but it grows not so rapidly: σ2 (f) < ∞. (T 3) Assume that δ (ι, f) > 0 for some ι ∈ Cˆ\ {0,−a}. Then, there exists a direction arg z = α (0 ≤ α < 2π) such that for any ε ∈ (0, π/2) , if a meromorphic function g satisfies the growth condition logT (r, g) = O (rτ log rT (r, f)) , r �∈ E2, (T 4) 196 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B for a constant τ > 0 and a set E2 of finite linear measure, and E (S3, f) = E (S3, g) and EX (Sj, f) = EX (Sj , g) for j = 1, 2 in X (α− ε, α + ε), then f ≡ g. Note that, H. Yi and W. Lin recently obtained the following uniqueness theorem of mero- morphic functions sharing two finite sets in the whole plane C [13]. Theorem C [13] Let S1 = {∞} and S2 = {ω | ωn + aωn−1 − b = 0}, where n (≥ 8) is an integer, a and b are two nonzero constants, such that the algebraic equation ωn + aωn−1 − b = 0 has no multiple roots. Assume that f and g are two meromorphic functions satisfying E (Sj , f) = E (Sj , g) for j = 1, 2. If Θ (∞, f) > 2 n−1 , then f ≡ g. Now, it is natural to ask the following question. Question 1 Does there exist an angular domain X = X(α, β) such that f ≡ g always holds in case when f and g are two meromorphic functions satisfying EX (Sj , f) = EX (Sj, g) for two finite sets Sj (j = 1, 2)? In this article, we declare that the answer to Question 1 is affirmative for some class of meromorphic functions. In fact, we obtain the following results. Theorem 1 Let S1 and S2 be defined as in Theorem C. Assume that f is a meromorphic function of lower order μ (f) ∈ (1/2,∞) in C and Θ (∞, f) > 2 n−1 , and that g is a meromorphic function of finite order or, more generally, with the growth satisfying either logT (r, g) = O (logT (r, f)) or condition (T 2) as in Theorem A. Then, for each σ < ∞ with μ (f) ≤ σ ≤ λ (f), there exists an angular domain X = X (α, β) with 0 ≤ α < β ≤ 2π and condition (T 1) as in Theorem A, such that if EX (Sj, f) = EX (Sj , g) (j = 1, 2) then f ≡ g. Remark 1 The following example shows that “>” in (T 1) cannot be replaced by “=”, so condition (T 1) is the best possible. Example 1 Let f (z) = eiz and g (z) = e2iz. It is easy to see that λ (f) = λ (g) = μ (f) = μ (g) = 1. Obviously, δ (∞, f) = 1, σ = 1 and E (S1, f) = E (S1, g) = ∅. Thus, max { π σ , 2π − 4 σ arcsin √ δ 2 } = π. The algebraic equation ω8 +ω7 +3 = 0 has eight distinct roots whose moduli are strictly more that 1. Hence, f8 (z) + f7 (z) + 3 �= 0 and g8 (z) + g7 (z) + 3 �= 0 in {z | �z > 0}, where |f (z) | < 1 and |g (z) | < 1. Therefore, there exists an angular domain X = X1 (0, π), or even X = X1 (0, π), such that EX (S2, f) = EX (S2, g), but f �≡ g. In contrast, for any ε > 0, one can find two different points z1 and z2 in X2 (0, π + ε), such that f8 (z1) + f 7 (z1) + 3 = g 8 (z2) + g 7 (z2) + 3 = 0, because the real axis is a Julia direction for both f and g. Remark 2 The following example [13] shows that the condition of deficiencies in Theo- rem 1 is necessary. Example 2 Let g = − a ( hn−1 − 1 ) hn − 1 , f = h · g = − ah ( hn−1 − 1 ) hn − 1 , where h = u 2ez−u ez−1 and u = exp 2πi n . It is seen that f and g satisfy EX (Sj , f) = EX (Sj , g) for j = 1, 2, and Θ (∞, f) = 2 n−1 . However, f �≡ g. This shows that the assumption “Θ (∞, f) > 2 n−1” in Theorem 1 is necessary. No.1 Chen & Lin: UNIQUENESS OF MEROMORPHIC FUNCTIONS WITH SHARED-SETS 197 Remark 3 As 2π ≥ β − α > π/σ so that condition (T 1) no longer has any meaning when σ ≤ 1/2. In this case, however, by taking X = C regarded as the closure of X (0, 2π), Theorem 1 is reduced to the result of Theorem C. From Theorem 1, one can immediately obtain a corollary as follows. Corollary 1 Suppose that f is a meromorphic function of lower order μ (f) ∈ (1/2,∞) in C and Θ (∞, f) > 2 n−1 . Then, there exists an angular domain X = X(α, β) such that, if any two meromorphic functions f and g of finite order satisfies EX (Sj , f) = EX (Sj , g) for two finite sets Sj (j = 1, 2) as in Theorem C, then, f ≡ g. Especially, considering the case λ (f) =∞, we obtain the following theorem. Theorem 2 Let S1 and S2 be defined as in Theorem C. Assume that f is a meromorphic function of infinite order and Θ (∞, f) > 2 n−1 , and that g is a meromorphic function satisfying the growth condition logT (r, g) = O (rτ log rT (r, f)) , r →∞. (T 5) Then, there exists a direction arg z = α (0 ≤ α < 2π) such that, for any small number ε (0 < ε < π/2), if EX (Sj , f) = EX (Sj , g) (j = 1, 2) in X = X (α− ε, α + ε), then, f ≡ g. Moreover, we denote a set of all the meromorphic functions with a proximity order ρ(r) by E(ρ(r)) (ρ(r) is defined below). Then, from Theorem 2, we have Corollary 2 Suppose that f ,g ∈ E(ρ(r)) and Θ (∞, f) > 2 n−1 . Then, there exists a direction arg z = α (0 ≤ α < 2π), such that for any small number ε (0 < ε < π/2), if EX (Sj, f) = EX (Sj , g) (j = 1, 2) in X = X (α− ε, α + ε), then, f ≡ g. Definition 1 Let f (z) be a meromorphic function of infinite order in C. A real function ρ (r) is called a proximity order of f (z) if ρ (r) has the following properties: (1) ρ (r) is continuous and nondecreasing for r ≥ r0 > 0 and tends to +∞ as r →∞; (2) the function U (r) = rρ(r) (r ≥ r0) satisfies the condition lim r→∞ logU (R) logU (r) = 1, ( R = r + r logU (r) ) ; (3) lim sup r→∞ log T (r,f) log U(r) = 1. Remark This definition is due to Hiong [7]. A simple proof of the existence of ρ (r) was given by Chuang [2]. The function U (r) is called the type function of f (z). 2 Some Lemmas To prove our results, we need the Nevanlinna theory on an angular domain [5, 8, 9] and other results. Let f (z) be a meromorphic function on X. Nevanlinna defined [5, 9] Aα,β (r, f) = ω π ∫ r 1 ( 1 tω − tω r2ω ){ log+ ∣∣f (teiα) ∣∣+ log+ ∣∣f (teiβ) ∣∣} dt t , Bα,β (r, f) = 2ω πrω ∫ β α log+ ∣∣f (reiθ) ∣∣ sinω (θ − α) dθ, Cα,β (r, f) = 2 ∑ bm∈Δ(r) ( 1 |bm|ω − |bm|ω r2ω ) sinω (θm − α) , 198 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B where ω = π/ (β − α), 1 ≤ r < ∞, and the summation Σ is taken over all the poles bm = |bm|eiθm of f (z) in the sector Δ(r), each pole being counted according to its multiplicity. The term Cα,β (r, f) is called the counting function of the poles of f in Δ(r), and C (r, f) is the reduced counting function of poles of f in Δ(r), where each pole is counted only once. Furthermore, for r > 1 we define Sα,β (r, f) = Aα,β (r, f) + Bα,β (r, f) + Cα,β (r, f) . (2.1) Similarly, for a �= ∞, we can define Aα,β (r, fa), Bα,β (r, fa), Cα,β (r, fa), and Sα,β (r, fa) with fa = 1/ (f − a). For sake of simplicity, we omit the subscript in all notations above. For instance, we use S (r, f) instead of Sα,β (r, f). Some properties of S (r, f) are listed as follows. Lemma 1 [5, 9] Let f (z) be a meromorphic function in X (α, β). For any a ∈ C, S (r, a) = S (r, f) + ε (r, a) , where ε (r, a) = O (1) as r → +∞. Lemma 2 [5] Let f (z) be a meromorphic function in X (α, β). For any q ≥ 3 distinct values ai ∈ C ∪ {∞} (i = 1, 2, · · · , q), (q − 2)S(r.f) ≤ q∑ i=1 C ( r, 1 f − ai ) − C0 ( r, 1 f ′ ) + RX(r, f), RX(r, f) = A ( r, f ′ f ) + B ( r, f ′ f ) + q∑ ai �=∞,i=1 { A ( r, f ′ f − ai ) + B ( r, f ′ f − ai )} + O(1), where C0 (r, 1/f ′) is the counting function of the zeros of f ′ but not the zeros of f and f − 1 in Δ(r). Lemma 3 [5] Let f be a meromorphic function in C. Then, A ( r, f ′ f ) ≤ K {( R r )ω ∫ R 1 log+ T (t, f) tω+1 dt + log+ r R− r + log R r + 1 } B ( r, f ′ f ) ≤ 4ω rω m ( r, f ′ f ) , where 1 < r < R < ∞, ω = π β−α and K is a nonzero constant. So that, the error term RX(r, f) in Lemma 2 have two properties as follows. Lemma 4 [5] let f be a meromorphic function in C. Then, Q(r, f) = A ( r, f ′ f ) + B ( r, f ′ f ) = ⎧⎨ ⎩O(1), λ(f) < ∞;O (log rT (r, f)) , λ(f) =∞, r �∈ E ⊂ R+,mesE < ∞. Remark Lemma 4 was first demonstrated by Nevanlinna [9] in the case where the func- tion f (z) is a meromorphic function of finite order in the whole complex plane, and then it No.1 Chen & Lin: UNIQUENESS OF MEROMORPHIC FUNCTIONS WITH SHARED-SETS 199 was generalized to the represented form by Dufresnoy and Ostrovskii. It was an open question whether, for any meromorphic function f (z) defined only on X (α, β), A ( r, f ′ f ) + B ( r, f ′ f ) = o (S (r, f)) (∗) holds as r → +∞ possibly outside a set of finite linear measure. In 1975, Gol’dberg (see [4]) constructed an unexpected counterexample. He showed that, for any function φ (r) → ∞ as r → ∞, there exists an entire function f (z) such that S (r, f) ≡ 0 but A (r, f ′/f) /φ (r) → ∞ as r →∞. Thus, (∗) is not valid in general. Lemma 5 [17] Let f be a meromorphic function in C. Then, RX(r, f) = ⎧⎨ ⎩O(1), λ(f) < ∞;O (logU(r)) , λ(f) = ∞, where U(r) = rρ(r) and ρ(r) is the proximity order of f of infinite order. Therefore, for a meromorphic function f (z) defined in C, by the same argument as in Nevanlinna’s value distribution theory on the whole complex plane, we can obtain some basic results in an angular domain with the error term RX(r, f) [5, 8, 9, 17]. From now on, let f and g be two nonconstant meromorphic functions and S2 = {ω | ωn + aωn−1 − b = 0}. Set F = fn−1 (f + a) b , G = gn−1 (g + a) b . (2.2) Obviously, If EX (S2, f) = EX (S2, g), then, F and G share 1 CM in X . Lemma 6 Let F and G be defined by (2.2). If Θ (∞, f) > 2 n−1 and F ≡ G, then, f ≡ g. Proof Suppose that f �≡ g. As F ≡ G, we know that fn−1 (f + a) = gn−1 (g + a) . (2.3) Put f g = h, (2.4) where h is a meromorphic function. It follows from f �≡ g that h �≡ 1. Then, we have f = − ah ( hn−1 − 1 ) hn − 1 , g = − a ( hn−1 − 1 ) hn − 1 . (2.5) We distinguish the following two cases. Case 1. If h is a constant, then, it follows from (2.5) that f is also a constant. This is a contradiction. Case 2. Suppose that h is nonconstant. By the first main theorem and (2.5), we have T (r, f) = (n− 1)T (r, h) + O (log rT (r, h)) . (2.6) In addition, suppose that ωj ∈ C\{1}, j = 1, 2, · · · , n−1, are the distinct roots of the algebraic equation ωn − 1 = 0. Again by the second main theorem, we obtain N (r, f) = n−1∑ j=1 N ( r, 1 h− ωj ) ≥ (n− 3)T (r, h) + O (log rT (r, h)) . (2.7) 200 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B It follows from (2.6) and (2.7) that Θ (∞, f) ≤ 2 n−1 , which contradicts Θ (∞, f) > 2 n−1 . This completes the proof of Lemma 6. Lemma 7 Let Sj (j = 1, 2) be two sets as in Theorem C, and F and G be defined as (2.2). If EX (Sj , f) = EX (Sj , g) for j = 1, 2 and F �≡ G, then, C (r, f) = C (r, g) ≤ 2 n− 1 {S (r, f) + S (r, g)}+ Q (r, f) + Q (r, g) . (2.8) Proof As EX (Sj , f) = EX (Sj, g) for j = 1, 2, we know that F and G share 1, ∞ CM in X . Set H = ( F ′ F − 1 − G′ G− 1 ) − ( F ′ F − G′ G ) , (2.9) that is, H = F ′ F (F − 1) − G′ G (G− 1) . (2.10) It follows that C (r,H) ≤ C ( r, 1 f ) + C ( r, 1 g ) + C ( r, 1 f + a ) + C ( r, 1 g + a ) . (2.11) Hence, by Lemma 4 and some basic results of Nevanlinna Theory, we deduce from (2.11) that S (r,H) ≤ 2S (r, f) + 2S (r, g) + Q (r, f) + Q (r, g) . (2.12) We discuss the following two cases. Case 1. Suppose that H ≡ 0. By integration, we have from (2.10) F − 1 F = k G− 1 G , (2.13) where k is a nonzero constant. As F �≡ G, we know that k �= 1. So, we deduce from (2.13) that ∞ is a Picard exceptional value in X of f . Therefore, (2.8) holds. Case 2. Suppose that H �≡ 0. Assume that z0 is a pole of f with multiplicity p. Then, an elementary calculation gives that z0 is a zero of H with multiplicity at least np − 1 ≥ n − 1. From this and (2.12), we obtain (n− 1)C (r, f) ≤ C ( r, 1 H ) ≤ S(r,H) ≤ 2S (r, f) + 2S (r, g) + Q (r, f) + Q (r, g) . (2.14) It follows from (2.14) that (2.8) holds. This completes the proof of Lemma 7. Lemma 8 [8] Let F and G be two nonconstant meromorphic functions in C, such that F and G share 1, ∞ CM in X . Then, one of the following three cases holds: (i) S (r) ≤ C2 (r, 1/F ) + C2 (r, 1/G) + 2C (r, F ) + Q (r, F ) + Q (r,G); (ii) F ≡ G; (iii) FG ≡ 1; where S (r) = max{S (r, F ) ,S (r,G)}. For a meromorphic function F (z) defined in X , the term C2 (r, F ) is called the counting function of the poles of f in Δ(r), where a simple pole is counted once and a multiple pole is counted twice. In the same way, we can define C2 (r, 1/F ). Moreover, we need the following important lemmas concerning Po´lya peaks [3, 11]. No.1 Chen & Lin: UNIQUENESS OF MEROMORPHIC FUNCTIONS WITH SHARED-SETS 201 Lemma 9 Let f (z) be a transcendental meromorphic function of finite lower order μ and order λ (0 < λ ≤ ∞) in C. Then, for an arbitrary positive number σ satisfying μ ≤ σ ≤ λ, there exists a set E of finite linear measure and a sequence of positive numbers {rn} such that: (i) rn �∈ E, lim n→∞ (rn/n) =∞; (ii) lim inf n→∞ (logT (rn, f)) / log rn ≥ σ; (iii) T (t, f) < (1 + o (1)) (t/rn) σ T (rn, f) , t ∈ [ rn n , nrn]. A sequence {rn} in Lemma 9 is called a sequence of Po´lya peaks of order σ outside E, which was proved in [11]. Given a positive function Λ = Λ (r) on (0,∞) with Λ → 0 as r →∞, we define, for r > 0 and a ∈ C, DΛ (r, a) = { θ ∈ [−π, π) ∣∣∣∣ log+ 1|f (reiθ)− a| > Λ (r) T (r, f) } and DΛ (r,∞) = { θ ∈ [−π, π) ∣∣∣∣ log+ |f(reiθ)| > Λ (r) T (r, f) } . The following lemma was proved by Baernstein [1]. Lemma 10 Let f (z) be a transcendental meromorphic function of finite lower order μ and order λ (0 < λ ≤ ∞) in C. Suppose that δ = δ (a, f) > 0 for some a ∈ Cˆ. Then, for arbitrary Po´lya peaks {rn} of positive and finite σ (μ ≤ σ ≤ λ) and an arbitrary positive function Λ = Λ (r) with Λ → 0 as r →∞, we have lim inf n→∞ measDΛ (rn, a) ≥ min { 2π, 4 σ arcsin √ δ 2 } . Furthermore, we need the following definitions concerning the Borel direction of a mero- morphic function of infinite order. Definition 2 Let f (z) be a meromorphic function of infinite order and ρ (r) a proximity order of f (z). A direction arg z = θ0 is called a Borel direction of a proximity order ρ (r) of f (z), if for any η (small enough), lim sup r→∞ log+ n (r, θ0, η, f = a) ρ (r) log r = 1 for each value a, except for at most two values, where n (r, θ0, η, f = a) denotes the number of the roots of the equation f (z) = a in the sector | arg z − θ0| < η, 0 < |z| < r, counting with multiplicities. Lemma 11 Let f (z) be a meromorphic function of infinite order and of a proximity order ρ (r). Then, there exists a Borel direction arg z = θ0 of proximity order ρ (r) of f (z). Remark The existence of such a Borel direction was proved using filling circles [2, 7]. Lemma 12 Let f (z) be a meromorphic function of infinite order and of a proximity order ρ(r). Then, for a meromorphic function g with growth condition (T 5), the error term RX(r, g) satisfies RX(r, g) = O(r τ ′ logU(r)), τ < τ ′, r →∞. 202 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B Proof By Lemma 3, we have Rx(r, g) ≤ K {( R r )ω ∫ R 1 log+ T (t, g) tω+1 dt + log+ r R− r + log R r + 1 } + 4ωM rω { log+ T (R, g) + log+ 1 R− r + 1 } for some positive constants K and M . Let R = r + rlog U(r) . By (2), (3) in Definition 1 and assumption (T 5), we deduce Rx(r, g) = O{logT (R, g) + log logU(r)} = O{Rτ logT (R, f) + log logU(r)} = O{rτ ′ logU(r)}, τ ′ > τ, r →∞. (2.15) 3 Proof of Theorem 1 We shall prove Theorem 1 by the method in which idea comes from Zheng [16] and Lin [8]. Define F and G as (2.2), it follows from EX (Sj , f) = EX (Sj , g) for j = 1, 2 that F and G share 1,∞ CM in X . We distinguish the following two cases. Case 1. Suppose that F �≡ G and FG �≡ 1. By Lemma 8, we have S (r) ≤ C2 (r, 1/F ) + C2 (r, 1/G) + 2C (r, F ) + Q (r, F ) + Q (r,G) , (3.1) where S (r) = max {S (r, F ) ,S (r,G)}. From (2.2), we deduce that C2 ( r, 1 F ) + C2 ( r, 1 G ) + 2C (r, F ) ≤ 2C ( r, 1 f ) + 2C ( r, 1 g ) + C ( r, 1 f + a ) + C ( r, 1 g + a ) +2C (r, f) + Q (r, f) + Q (r, g) . (3.2) Set S1 (r) = max{S (r, f) ,S (r, g)}. Then, S (r) = nS1 (r) + Q (r, f) + Q (r, g) . (3.3) From (2.8) obtained in Lemma 7 and (3.1)–(3.3), we obtain nS1 (r) = S (r) + Q (r, f) + Q (r, g) ≤ 3S (r, f) + 3S (r, g) + 4 n− 1 {S (r, f) + S (r, g)}+ Q (r, f) + Q (r, g) ≤ ( 6 + 8 n− 1 ) S1 (r) + Q (r, f) + Q (r, g) . (3.4) Noting that n ≥ 8, it follows immediately from (3.4) that S1 (r) ≤ Q (r, f) + Q (r, g) . (3.5) No.1 Chen & Lin: UNIQUENESS OF MEROMORPHIC FUNCTIONS WITH SHARED-SETS 203 Thus, by Lemma 4, (3.5) gives S (r, f) = O (log (T (r, f)T (r, g))) , r �∈ E3, (3.6) for some set E3 ⊂ [0,∞) of finite linear measure. By assumption (T 1), we choose a real small number ε ∈ (0, (β − α)/4), such that 2π + α− β + 4ε < 4 σ arcsin √ δ 2 . (3.7) Applying Lemma 10 to f (z) gives the existence of the Po´lya peaks {rn} of order σ of f(z) outside the set E := E1 ⋃ E3, where E1 and E3 are sets of finite linear measure appeared in assumption (T 2) and (3.6), respectively. Furthermore, applying Lemma 9 to the Po´lya peaks {rn}, we have either measDΛ (rn,∞) > 4 σ arcsin √ δ 2 − ε, (3.8) or measDΛ (rn,∞) > 2π − ε (3.9) for each sufficiently large n, say n ≥ n0. Note that meas[DΛ (rn,∞) ∩ (α + ε, β − ε)] + meas[DΛ (rn,∞) ∪ (α + ε, β − ε)] = measDΛ(rn,∞) + meas[(α + ε, β − ε)] (3.10) and meas[DΛ (rn,∞) ∪ (α + ε, β − ε)] ≤ 2π. (3.11) We deduce from (3.7)–(3.11) that meas[DΛ (rn,∞) ∩ (α + ε, β − ε)] > ε > 0. Hence, by the definition of DΛ(rn,∞) and setting Λ (r) = 1/ log r, we have∫ β−ε α+ε log+ ∣∣f (rneiθ) ∣∣dθ ≥ ∫ DΛ(rn,∞)∩(α+ε,β−ε) log+ ∣∣f (rneiθ) ∣∣dθ ≥ ε T (rn, f) log rn , n ≥ n0. (3.12) In contrast, from (3.6), we have∫ β−ε α+ε log+ ∣∣f (rneiθ) ∣∣dθ ≤ πrωn 2ω sin (εω) Bα,β (rn, f) = πrωn 2ω sin (εω) O (log (rnT (rn, f)T (rn, g))) , n ≥ n0, (3.13) where ω = π α−β . Combining (3.12) and (3.13), we obtain T (rn, f) ≤ πrωn log rn 2εω sin (εω) O (log (rnT (rn, f)T (rn, g))) , n ≥ n0. (3.14) It follows that logT (rn, f) ≤ ω log rn +K1 log log rn + log logT (rn, f)+ log log T (rn, g)+K2, n ≥ n0, (3.15) 204 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B where K1, K2 are two positive constants. By Lemma 9 and assumptions (T 1) and (T 2), (3.15) implies that σ ≤ lim inf n→∞ logT (rn, f) log rn ≤ ω < σ, which is impossible. Case 2. Suppose that FG ≡ 1. From (2.2), we obtain fn−1 (f + a) gn−1 (g + a) ≡ b2, which implies that f does not take 0, −a and ∞ in X . Using Lemma 2 for q = 3, we obtain S (r, f) = O (log rT (r, f)) , r �∈ E, for some set E ⊂ [0,∞) of finite linear measure. By the same argument as in Case 1, we can also obtain a contradiction. Therefore, we obtain from Lemma 8 F ≡ G. (3.16) It follows from (3.16) and Lemma 6 that f ≡ g . This completes the proof of Theorem 1. 4 Proof of Theorem 2 We shall prove Theorem 2 by the method in which idea comes from Wu and Sun [10]. Suppose that Theorem 2 does not hold. Then, for any α ∈ [0, 2π), we can find a constant ε ∈ (0, π/2), which possibly depends on α, and a meromorphic function g[α] (depending on α) whose growth satisfies condition (T 5), such that EX (Sj , f) = EX ( Sj , g [α] ) holds for j = 1, 2, but f �≡ g[α], where X := X(α) = X (α− ε, α + ε) = {z | | arg z − α| < ε}. Defined F and G[α] as (2.2). Then, F and G[α] share 1, ∞ CM in X (α). We deduce from Lemma 6 that F �≡ G[α]. Let ρ (r) be a proximity order of f (z), and U (r) = rρ(r). Now, we discuss the following two cases. Case 1. Suppose that FG[α] �≡ 1. Proceeding as Case 1 in the proof of Theorem 1, we have SX(α) (r, f) = RX(r, f) + RX(r, g [α]). (4.1) By Lemma 5, Lemma 12, and Definition 1, we have lim sup r→∞ logRX (r, f) logU (r) = 0, (4.2) lim sup r→∞ logRX ( r, g[α] ) logU(r) = 0. (4.3) Thus, lim sup r→∞ logSX(α) (r, f) logU(r) = 0. (4.4) Therefore, there exists a constant λ (0 < λ < 1) such that, for sufficiently large r, say, r ≥ r0, SX(α)(r, f) < U λ (r) , r ≥ r0. (4.5) No.1 Chen & Lin: UNIQUENESS OF MEROMORPHIC FUNCTIONS WITH SHARED-SETS 205 For any a ∈ C, let bm = |bm|eiβm (m = 1, 2, · · ·) be the roots of f (z)− a = 0 in X (α). Set n (r) = n ( r, α, ε 3 , f = a ) . Considering the smaller angular domain X1(α) = {z | | arg z − α| < ε 3}, we have α− ε 3 < βm < α + ε 3 , m = 1, 2, · · · , which lead to ε 6 < βm − α + ε 2 < 5ε 6 , m = 1, 2, · · · . 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