() CUBO A Mathematical Journal Vol.13, No¯ 03, (57–67). October 2011 Some Uniqueness Results On Meromorphic Functions Sharing Three Sets II Abhijit Banerjee 1 Department of Mathematics, West Bengal State University, Barasat, 24 Parganas (North), West Bengal, Kolkata-700126, India. email: abanerjee kal@yahoo.co.in, abanerjee kal@rediffmail.com ABSTRACT With the help of the notion of weighted sharing we investigate the uniqueness of mero- morphic functions concerning three set sharing and significantly improve two results of Zhang [16] and as a corollary of the main result we improve a result of the present author [2] as well. RESUMEN Con la ayuda del concepto de peso repartido, investigamos la unicidad de funciones meromorfas sobre un conjunto compartido y mejoramos significativamente dos resulta- dos de Zhang [16] y como corolario del resultado principal que mejoramos también el resultado de la autora [2]. Mathematics Subject Classification: 30D35. Keywords. Meromorphic functions, uniqueness, weighted sharing, shared set. 1The author dedicates the paper to the memory of his respected teacher Late Prof. B. K. Lahiri who first germinated the inquisition for research work in the author’s mind. 58 Abhijit Banerjee CUBO 13, 3 (2011) 1 Introduction, Definitions and Results In this paper by meromorphic functions we will always mean meromorphic functions in the complex plane. Let f and g be two non-constant meromorphic functions and let a be a finite complex number. We shall use the standard notations of value distribution theory : T (r, f), m(r, f), N(r, ∞; f), N(r, ∞; f), . . . (see [5]). For any constant a, we define Θ(a; f) = 1 − lim sup r−→ ∞ N(r, a; f) T (r, f) . We say that f and g share a CM, provided that f − a and g − a have the same zeros with the same multiplicities. Similarly, we say that f and g share a IM, provided that f − a and g − a have the same zeros ignoring multiplicities. In addition we say that f and g share ∞ CM, if 1/f and 1/g share 0 CM, and we say that f and g share ∞ IM, if 1/f and 1/g share 0 IM. Let S be a set of distinct elements of C ∪ {∞} and Ef(S) = ⋃ a∈S{z : f(z) − a = 0}, where each zero is counted according to its multiplicity. If we do not count the multiplicity the set Ef(S) = ⋃ a∈S{z : f(z) − a = 0} is denoted by Ef(S). If Ef(S) = Eg(S) we say that f and g share the set S CM. On the other hand if Ef(S) = Eg(S), we say that f and g share the set S IM. In [4] Gross posed the following question: Can one find two finite sets Sj (j = 1, 2) such that any two non-constant entire functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1, 2 must be identical ? In the last couple of years or so several attempts have been made in many papers to answer the above question under weaker hypothesis (see [1], [2], [3], [9], [10], [13], [15], [16])). A recent increment to uniqueness theory has been to considering weighted sharing instead of sharing IM/CM which implies a gradual change from sharing IM to sharing CM. This notion of weighted sharing has been introduced by I. Lahiri around 2001 in [7, 8] and since then this notion played a vital role on the uniqueness of meromorphic or entire functions sharing sets concerning the question of Gross. Below we are giving the definition. Definition 1.1. [7, 8] Let k be a nonnegative integer or infinity. For a ∈ C ∪ {∞} we denote by Ek(a; f) the set of all a-points of f, where an a-point of multiplicity m is counted m times if m ≤ k and k + 1 times if m > k. If Ek(a; f) = Ek(a; g), we say that f, g share the value a with weight k. We write f, g share (a, k) to mean that f, g share the value a with weight k. Clearly if f, g share (a, k) then f, g share (a, p) for any integer p, 0 ≤ p < k. Also we note that f, g share a value a IM or CM if and only if f, g share (a, 0) or (a, ∞) respectively. Definition 1.2. [7] Let S be a set of distinct elements of C ∪ {∞} and k be a nonnegative integer or ∞. Let Ef(S, k) = ⋃ a∈S Ek(a; f). CUBO 13, 3 (2011) Some Uniqueness Results On Meromorphic Functions . . . 59 Clearly Ef(S) = Ef(S, ∞) and Ef(S) = Ef(S, 0). Improving the result of Lahiri-Banerjee [10] and Yi-Lin [15] the present author have recently proved the following result. Theorem A. [1] Let S1 = {z : z n + azn−1 + b = 0}, S2 = {0} and S3 = {∞}, where a, b are nonzero constants such that zn + azn−1 + b = 0 has no repeated root and n (≥ 4) is an integer. If for two non-constant meromorphic functions f and g Ef(S1, 4) = Eg(S1, 4), Ef(S2, 0) = Eg(S2, 0) and Ef(S3, ∞) = Eg(S3, ∞) and Θ(∞; f) + Θ(∞; g) > 0 then f ≡ g. In [2] the present author further improved Theorem A as follows. Theorem B. [2] Let Si, i = 1, 2, 3 be defined as in Theorem A. If for two non-constant meromor- phic functions f and g Ef(S1, 4) = Eg(S1, 4), Ef(S2, 0) = Eg(S2, 0) and Ef(S3, 6) = Eg(S3, 6) and Θ(∞; f) + Θ(∞; g) > 0 then f ≡ g. Now it is quite natural to ask the following question. i) What happens in Theorem B if no conditions over the ramification indexes of f and g are imposed ? In the direction of the above question some investigations have already been carried out by Zhang [16] in the following theorems. Theorem C. Let S1 = {z : z n(z + a) − b = 0}, S2 = {0} and S3 = {∞}, where a, b are nonzero constants such that zn(z + a) − b = 0 has no repeated root and n (≥ 3) is an integer. If for two nonconstant meromorphic functions f and g Ef(S1, ∞) = Eg(S1, ∞), Ef(S2, 0) = Eg(S2, 0) and Ef(S3, ∞) = Eg(S3, ∞) then f ≡ g or f = −ae γ (e nγ −1) e(n+1)γ−1 , g = −a(e nγ −1) e(n+1)γ−1 , where γ is a non-constant entire function. Theorem D. Let Si, i = 1, 2, 3 be defined as in Theorem C and n (≥ 4) is an integer. If for two non-constant meromorphic functions f and g Ef(S1, ∞) = Eg(S1, ∞), Ef(S2, 0) = Eg(S2, 0) and Ef(S3, 0) = Eg(S3, 0) then f ≡ g or f = −ae γ (e nγ −1) e(n+1)γ−1 , g = −a(e nγ −1) e(n+1)γ−1 , where γ is a non-constant entire function. The following example shows that in Theorems A-C a 6= 0 is necessary. Example 1.1. Let f(z) = ez and g(z) = e−z and S1 = {z : z 4 − 1 = 0}, S2 = {0}, S3 = {∞}. Since f − ωl = g − ω4−l, where ω = cos 2π 4 + isin 2π 4 , 0 ≤ l ≤ 3, clearly Ef(Si, ∞) = Eg(Si, ∞) for i = 1, 2, 3 but f and g do not satisfy the conclusions of Theorems A-B. Regarding Theorems A-C following example establishes the fact that the set S1 can not be replaced by any arbitrary set containing three distinct elements. However it still remains open for investigations whether the degree of the equation defining S1 in Theorem A-C can be reduced to three or less. 60 Abhijit Banerjee CUBO 13, 3 (2011) Example 1.2. Let f(z) = √ ab e √ abz and g(z) = √ ab e− √ abz and S1 = {a, b, √ ab}, S2 = {0}, S3 = {∞}, where a and b are nonzero complex numbers. Clearly Ef(Si, ∞) = Eg(Si, ∞) for i = 1, 2, 3 but f and g do not satisfy the conclusions of Theorems A-C. In the paper we also concentrate our attention to the above problem as investigated by Zhang [16] and provide a better solution in this direction. We now state the following two theorems which are the main results of the paper. Theorem 1.1. Let S1 = {z : z n(z + a) − b = 0}, S2 = {0} and S3 = {∞}, where a, b are nonzero constants such that zn(z + a) − b = 0 has no repeated root and n (≥ 3) is an integer. If for two non-constant meromorphic functions f and g Ef(S1, 3) = Eg(S1, 3), Ef(S2, 0) = Eg(S2, 0) and Ef(S3, 6) = Eg(S3, 6) then f ≡ g or f = −ae γ (e nγ −1) e(n+1)γ−1 , g = −a(e nγ −1) e(n+1)γ−1 , where γ is a non-constant entire function. Corollary 1.1. Let S1, S2 and S3 be defined as in Theorem 1.1 and n(≥ 3) be an integer. If for two non-constant meromorphic functions f and g Ef(S1, 3) = Eg(S1, 3), Ef(S2, 0) = Eg(S2, 0) and Ef(S3, 6) = Eg(S3, 6) and Θ(∞; f) + Θ(∞; g) > 0 then f ≡ g Theorem 1.2. Let S1, S2 and S3 be defined as in Theorem 1.1 and n(≥ 4) be an integer. If for two non-constant meromorphic functions f and g Ef(S1, 3) = Eg(S1, 3), Ef(S2, 0) = Eg(S2, 0) and Ef(S3, 0) = Eg(S3, 0) then the conclusion of Theorem 1.1 holds . Remark 1. Theorem 1.1, Corollary 1.1 and Theorem 1.2 are respectively the improvements of Theorems C, B and D respectively. We now explain some notations which are used in the paper. Definition 1.3. [6] After a ∈ C∪{∞}, we denote by N(r, a; f |= 1) the counting function of simple a points of f. For a positive integer m we denote by N(r, a; f |≤ m) (N(r, a; f |≥ m)) the counting function of those a points of f whose multiplicities are not greater(less) than m where each a point is counted according to its multiplicity. N(r, a; f |≤ m) (N(r, a; f |≥ m)) are defined similarly, where in counting the a-points of f we ignore the multiplicities. Also N(r, a; f |< m), N(r, a; f |> m), N(r, a; f |< m) and N(r, a; f |> m) are defined analo- gously. Definition 1.4. [2] We denote by N(r, a; f |= k) the reduced counting function of those a-points of f whose multiplicities is exactly k, where k ≥ 2 is an integer. Definition 1.5. [2] Let f and g be two non-constant meromorphic functions such that f and g share (a, k) where a ∈ C ∪ {∞}. Let z0 be a a-point of f with multiplicity p, a a-point of g with multiplicity q. We denote by NL(r, a; f) the counting function of those a-points of f and g where p > q, by N (k+1 E (r, a; f) the counting function of those a-points of f and g where p = q ≥ k+1; each point in these counting functions is counted only once. In the same way we can define NL(r, a; g) and N (k+1 E (r, a; g). CUBO 13, 3 (2011) Some Uniqueness Results On Meromorphic Functions . . . 61 Definition 1.6. [8] We denote by N2(r, a; f) = N(r, a; f) + N(r, a; f |≥ 2) Definition 1.7. [7, 8] Let f, g share a value a IM. We denote by N∗(r, a; f, g) the reduced counting function of those a-points of f whose multiplicities differ from the multiplicities of the corresponding a-points of g. Clearly N∗(r, a; f, g) ≡ N∗(r, a; g, f) and N∗(r, a; f, g) = NL(r, a; f) + NL(r, a; g). Definition 1.8. [11] Let a, b ∈ C ∪ {∞}. We denote by N(r, a; f | g = b) the counting function of those a-points of f, counted according to multiplicity, which are b-points of g. Definition 1.9. [11] Let a, b1, b2, . . . , bq ∈ C ∪{∞}. We denote by N(r, a; f | g 6= b1, b2, . . . , bq) the counting function of those a-points of f, counted according to multiplicity, which are not the bi-points of g for i = 1, 2, . . . , q. 2 Lemmas In this section we present some lemmas which will be needed in the sequel. Let F and G be two non-constant meromorphic functions defined as follows. F = fn(f + a) b , G = gn(g + a) b . (2.1) Henceforth we shall denote by H, Φ and V the following three functions H = ( F ′′ F ′ − 2F ′ F − 1 ) − ( G ′′ G ′ − 2G ′ G − 1 ) , Φ = F ′ F − 1 − G ′ G − 1 and V = ( F ′ F − 1 − F′ F ) − ( G ′ G − 1 − G′ G ) = F′ F(F − 1) − G′ G(G − 1) . Lemma 2.1. Let F, G share (1, 1) and H 6≡ 0. Then N(r, 1; F |= 1) = N(r, 1; G |= 1) ≤ N(r, H) + S(r, F) + S(r, G). Proof. The lemma can be proved in the line of proof of Lemma 1 [8]. Lemma 2.2. Let S1, S2 and S3 be defined as in Theorem 1.1 and F, G be given by (2.1). If for two non-constant meromorphic functions f and g Ef(S1, 0) = Eg(S1, 0), Ef(S2, 0) = Eg(S2, 0), Ef(S3, 0) = Eg(S3, 0) and H 6≡ 0 then N(r, H) ≤ N∗(r, 0, f, g) + N(r, 0; f + a |≥ 2) + N(r, 0; g + a |≥ 2) + N∗(r, 1; F, G) +N∗(r, ∞; f, g) + N0(r, 0; F ′ ) + N0(r, 0; G ′ ), 62 Abhijit Banerjee CUBO 13, 3 (2011) where N0(r, 0; F ′ ) is the reduced counting function of those zeros of F ′ which are not the zeros of F(F − 1) and N0(r, 0; G ′ ) is similarly defined. Proof. The lemma can be proved in the line of proof of Lemma 2.2 [2]. Lemma 2.3. [12] Let f be a nonconstant meromorphic function and let R(f) = n∑ k=0 akf k m∑ j=0 bjf j be an irreducible rational function in f with constant coefficients {ak} and {bj} where an 6= 0 and bm 6= 0. Then T (r, R(f)) = dT (r, f) + S(r, f), where d = max{n, m}. Lemma 2.4. Let F and G be given by (2.1), n ≥ 3 an integer and F 6≡ G. If F, G share (1, m), f, g share (0, p), (∞, k), where 0 ≤ p < ∞ then [np + n − 1] N(r, 0; f |≥ p + 1) ≤ N∗(r, 1; F, G) + N∗(r, ∞; F, G) + S(r, f) + S(r, g). Proof. Suppose 0 is an e.v.P. (Picard exceptional value) of f and g then the lemma follows imme- diately. Next suppose 0 is not an e.v.P. of f and g. If Φ ≡ 0, then by integration we obtain F − 1 ≡ C(G − 1). It is clear that if z0 is a zero of f then it is a zero of g. So it follows that F(z0) = G(z0) = 0. So C = 1 which contradicts F 6≡ G. So Φ 6≡ 0. Since f, g share (0, p) it follows that a common zero of f and g of order r ≤ p is a zero of Φ of order exactly nr − 1 where as a common zero of f and g of order r > p is a zero of Φ of order at least np + n − 1. Let z0 is a zero of f with multiplicity q and a zero of g with multiplicity t. From (2.1) we know that z0 is a zero of F with multiplicity nq and a zero of G with multiplicity nt. So from the definition of Φ it is clear that [np + n − 1]N(r, 0; f |≥ p + 1) = [np + n − 1]N(r, 0; g |≥ p + 1) = [np + n − 1]N (r, 0; F |≥ n(p + 1)) ≤ N(r, 0; Φ) ≤ N(r, ∞; Φ) + S(r, f) + S(r, g) ≤ N∗(r, ∞; F, G) + N∗(r, 1; F, G) + S(r, f) + S(r, g). The lemma follows from above. CUBO 13, 3 (2011) Some Uniqueness Results On Meromorphic Functions . . . 63 Lemma 2.5. Let F, G be given by (2.1), F, G share (1, m), 0 ≤ m < ∞ and ω1, ω2, . . . , ωn are the distinct roots of the equation zn + azn−1 + b = 0 and n ≥ 3. Then N∗(r, 1; F, G) ≤ 1 m [ N(r, 0; f) + N(r, ∞; f) − N⊗(r, 0; f ′ ) ] + S(r, f), where N⊗(r, 0; f ′ ) = N(r, 0; f ′ | f 6= 0, ω1, ω2, . . . , ωn) Proof. We omit the proof since it can be proved in the line of proof of Lemma 2.15 [2]. Lemma 2.6. Let F and G be given by (2.1), n ≥ 3 an integer and F 6≡ G. If F, G share (1, m), f, g share (0, 0), (∞, k) then N(r, 0; f) ≤ m mn − m − 1 N(r, ∞; f |≥ k + 1) + 1 mn − m − 1 N(r, ∞; f) + S(r, f) +S(r, g). Proof. Since using Lemma 2.5 in Lemma 2.4 we get for p = 0 that (n − 1)N(r, 0; f) ≤ N(r, ∞; f |≥ k + 1) + 1 m [N(r, 0; f) + N(r, ∞; f)] +S(r, f) + S(r, g), the lemma follows. Lemma 2.7. Let F, G be given by (2.1), n ≥ 3 an integer and F 6≡ G. If f, g share (0, 0), (∞, k), where 0 ≤ k < ∞, and F, G share (1, m) then the poles of F and G are the zeros of V and (i) nN(r, ∞; f |= 1) + (2n + 1)N(r, ∞; f |= 2) + . . . + [(n + 1)k − 1]N(r, ∞; f |= k) +[(n + 1)k + n]N(r, ∞; f |≥ k + 1) ≤ 1 n − 1 N(r, ∞; f |≥ k + 1) + N(r, 0; f + a) +N(r, 0; g + a) + n n − 1 N∗(r, 1; F, G) + S(r, f) + S(r, g). (ii) N(r, ∞; f |≥ k + 1) ≤ n − 1 (n − 1)[(n + 1)k + n] − 1 [N(r, 0; f + a) + N(r, 0; g + a)] + n (n − 1)[(n + 1)k + n] − 1 N∗(r, 1; F, G) + S(r, f) + S(r, g). Proof. Suppose ∞ is an e.v.P. of f and g then the lemma follows immediately. Next suppose ∞ is not an e.v.P. of f and g. If V ≡ 0, then by integration we obtain 1 − 1 F ≡ A ( 1 − 1 G ) . If z0 is a pole of f then it is a pole of g. Hence from the definition of F and G we have 1 F(z0) = 0 and 1 G(z0) = 0. So A = 1 which contradicts F 6≡ G. So V 6≡ 0. Since f, g share (∞, k), we note that F and G have no pole of multiplicity q where (n + 1)k < q < (n + 1)(k + 1) and so it 64 Abhijit Banerjee CUBO 13, 3 (2011) follows that F, G share (∞, (n + 1)k + n). So using Lemma 2.3 and Lemma 2.4 for p = 0 we get from the definition of V nN(r, ∞; f |= 1) + (2n + 1)N(r, ∞; f |= 2) + . . . + [(n + 1)k − 1]N(r, ∞; f |= k) (2.2) +[(n + 1)k + n]N(r, ∞; f |≥ k + 1) ≤ N(r, 0; V) ≤ N(r, ∞; V) + S(r, f) + S(r, g) ≤ N∗(r, 0; f, g) + N(r, 0; f + a) + N(r, 0; g + a) + N∗(r, 1; F, G) + S(r, f) + S(r, g) ≤ 1 n − 1 N(r, ∞; f |≥ k + 1) + N(r, 0; f + a) + N(r, 0; g + a) + n n − 1 N∗(r, 1; F, G) +S(r, f) + S(r, g), from which (i) follows. Again from (2.2) we note that (n − 1)[(n + 1)k + n] − 1 n − 1 N(r, ∞; f |≥ k + 1) ≤ N(r, 0; f + a) + N(r, 0; g + a) + n n − 1 N∗(r, 1; F, G) + S(r, f) + S(r, g), from which (ii) follows. Lemma 2.8. ([2], Lemma 2.9) Let F, G be given by (2.1) and they share (1, m). If f, g share (0, p), (∞, k) where 2 ≤ m < ∞ and H 6≡ 0. Then T (r, F) ≤ N(r, 0; f) + N(r, 0; g) + N∗(r, 0; f, g) + N2(r, 0; f + a) + N2(r, 0; g + a) +N(r, ∞; f) + N(r, ∞; g) + N∗(r, ∞; f, g) − m(r, 1; G) − N(r, 1; F |= 3) − . . . − (m − 2)N(r, 1; F |= m) − (m − 2) NL(r, 1; F) − (m − 1)NL(r, 1; G) −(m − 1)N (m+1 E (r, 1; F) + S(r, F) + S(r, G) Lemma 2.9. ([14], Lemma 6) If H ≡ 0, then F, G share (1, ∞). If further F, G share (∞, 0) then F, G share (∞, ∞). 3 Proofs of the theorems Proof of Theorem 1.1. Let F, G be given by (2.1). Then F and G share (1, 3), (∞, 7n + 6). We consider the following cases. Case 1. Let H 6≡ 0. Then F 6≡ G. Noting that f, g share (0, 0) and (∞, 6) implies N∗(r, 0; f, g) ≤ N(r, 0; f) = N(r, 0; g) and N∗(r, ∞; f, g) ≤ N(r, ∞; f |≥ 7) = N(r, ∞; g |≥ 7), CUBO 13, 3 (2011) Some Uniqueness Results On Meromorphic Functions . . . 65 using Lemmas 2.3 and 2.6 for m = 3 in Lemma 2.8 we obtain (n + 1){T (r, f) + T (r, g)} (3.1) ≤ 6N(r, 0; f) + 2T (r, f) + 2T (r, g) + 4N(r, ∞; f) + 2N(r, ∞; f |≥ 7) −3N∗(r, 1; F, G) + S(r, f) + S(r, g) ≤ 2T (r, f) + 2T (r, g) + ( 6n + 10 3n − 4 ) N(r, ∞; f |≥ 7) + ( 12n − 10 3n − 4 ) N(r, ∞; f) − 3N∗(r, 1; F, G) + S(r, f) + S(r, g) So using Lemma 2.7 (i) for k = 6 in (3.1) we get (n − 1){T (r, f) + T (r, g)} (3.2) ≤ ( 6n + 10 (3n − 4)(7n + 6) )[ T (r, f) + T (r, g) + 1 n − 1 { N(r, ∞; f |≥ 7) + nN∗(r, 1; F, G) } ] + ( 12n − 10 n(3n − 4) )[ T (r, f) + T (r, g) + 1 n − 1 { N(r, ∞; f |≥ 7) + nN∗(r, 1; F, G) } ] −3N∗(r, 1; F, G) + S(r, f) + S(r, g) ≤ [ 6n + 10 (3n − 4)(7n + 6) + 12n − 10 n(3n − 4) ] {T (r, f) + T (r, g)} + 1 n − 1 [ n(6n + 10) (3n − 4)(7n + 6) + 12n − 10 3n − 4 ] N∗(r, 1; F, G) + 1 (n − 1) [ 6n + 10 (3n − 4)(7n + 6) + 12n − 10 n(3n − 4) ] N(r, ∞; f |≥ 7) −3N∗(r, 1; F, G) + S(r, f) + S(r, g). Now using Lemma 2.7 (ii) for k = 6 in (3.2) we get [ n − 1 − 6n + 10 (3n − 4)(7n + 6) − 12n − 10 n(3n − 4) ] {T (r, f) + T (r, g)} ≤ [ n(6n + 10) (n − 1)(3n − 4)(7n + 6) + 12n − 10 (n − 1)(3n − 4) ] N∗(r, 1; F, G) + [ 6n + 10 (3n − 4)(7n + 6) + 12n − 10 n(3n − 4) ][ 1 7n2 − n − 7 {T (r, f) + T (r, g)} + n (n − 1)(7n2 − n − 7) N∗(r, 1; F, G) ] − 3N∗(r, 1; F, G) + S(r, f) + S(r, g), from which we get a contradiction for n ≥ 3 . Case 2. Let H ≡ 0. Now from Lemma 2.9 we have F and G share (1, ∞) and (∞, ∞). This implies Ef(S1, ∞) = Eg(S1, ∞), Ef(S2, 0) = Eg(S2, 0) and Ef(S3, ∞) = Eg(S3, ∞). Now the theorem follows from Theorem C. Proof of Corollary 1.1. Let F, G be given by (2.1). Then F and G share (1, 3), (∞, 7n + 6). By Theorem 1.1 we get either f ≡ g or f = −ae γ (e nγ −1) e(n+1)γ−1 , g = −a(e nγ −1) e(n+1)γ−1 , where γ is a non- constant entire function. If f 6≡ g then using Lemma 2.3 clearly Θ(∞; f) = Θ(∞; g) = 1 − 66 Abhijit Banerjee CUBO 13, 3 (2011) lim sup r−→ ∞ n∑ k=1 N(r,uk;e γ ) nT (r,eγ) = 0, where uk = exp ( 2kπi n+1 ) for k = 1, 2, . . . , n and hence we deduce a contradiction. This proves the corollary. Proof of Theorem 1.2. Let F, G be given by (2.1). Then F and G share (1, 3), (∞, n). We consider the following cases. Case 1. Let H 6≡ 0. Then F 6≡ G. Noting that f, g share (0, 0) and (∞, 0) implies N∗(r, 0; f, g) ≤ N(r, 0; f) = N(r, 0; g) and N∗(r, ∞; f, g) ≤ N(r, ∞; f |≥ 7) = N(r, ∞; g |≥ 7), using Lemmas 2.3 and 2.6 for m = 3 and k = 0 in Lemma 2.8 we obtain (n + 1){T (r, f) + T (r, g)} (3.3) ≤ 6N(r, 0; f) + 2T (r, f) + 2T (r, g) + 6N(r, ∞; f) − 3N∗(r, 1; F, G) +S(r, f) + S(r, g) ≤ 2T (r, f) + 2T (r, g) + ( 18n 3n − 4 ) N(r, ∞; f) − 3N∗(r, 1; F, G) + S(r, f) + S(r, g) So using Lemma 2.7 (ii) for k = 0 and Lemma 2.5 in (3.3) we get (n − 1){T (r, f) + T (r, g)} (3.4) ≤ ( 18n(n − 1) (3n − 4)(n2 − n − 1) ) [T (r, f) + T (r, g)] + ( 18n2 (3n − 4)(n2 − n − 1) ) N∗(r, 1; F, G) −3N∗(r, 1; F, G) + S(r, f) + S(r, g) ≤ ( 18n(n − 1) (3n − 4)(n2 − n − 1) ) [T (r, f) + T (r, g)] + ( 18n2 6(3n − 4)(n2 − n − 1) − 1 2 ) { N(r, 0; f) + N(r, ∞; f) + N(r, 0; g) + N(r, ∞; g) } + S(r, f) + S(r, g) ≤ ( 18n(n − 1) (3n − 4)(n2 − n − 1) + 18n2 3(3n − 4)(n2 − n − 1) − 1 ) [T (r, f) + T (r, g)] +S(r, f) + S(r, g). Clearly (3.4) implies a contradiction for n ≥ 4 . Case 2. Let H ≡ 0. Now from Lemma 2.9 we have F and G share (1, ∞) and (∞, ∞). This implies Ef(S1, ∞) = Eg(S1, ∞), Ef(S2, 0) = Eg(S2, 0) and Ef(S3, ∞) = Eg(S3, ∞). Now the theorem follows from Theorem C. Received: January 2010. Revised: August 2010. References [1] A.Banerjee, On a question of Gross, J. Math. Anal. Appl. 327(2) (2007) 1273-1283. [2] A.Banerjee, Some uniqueness results on meromorphic functions sharing three sets, Ann. Polon. Math., 92(3)(2007), 261-274. CUBO 13, 3 (2011) Some Uniqueness Results On Meromorphic Functions . . . 67 [3] M.Fang and W. Xu, A note on a problem of Gross, Chinese J. Contemporary Math., 18(4)(1997), 395-402. [4] F.Gross, Factorization of meromorphic functions and some open problems, Proc. Conf. Univ. Kentucky, Leixngton, Ky(1976); Lecture Notes in Math., 599(1977), 51-69, Springer(Berlin). [5] W.K.Hayman, Meromorphic Functions, The Clarendon Press, Oxford (1964). 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[14] H.X.Yi, Meromorphic functions that share one or two values II, Kodai Math. J., 22 (1999), 264-272. [15] H.X.Yi and W.C.Lin, Uniqueness theorems concerning a question of Gross, Proc. Japan Acad., 80, Ser.A(2004), 136-140. [16] Q. Zhang, Meromorphic Functions That Share Three Sets, Northeast Math. J., 23(2)(2007), 103-114. Introduction, Definitions and Results Lemmas Proofs of the theorems