ON THE UNIQUENESS OF MEROMORPHIC FUNCTIONS ... - CiteSeerX

Georgian Mathematical Journal Volume 15 (2008), Number 1, 21–38

ON THE UNIQUENESS OF MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS ABHIJIT BANERJEE

Abstract. Using the notion of weighted and truncated sharings of sets we deal with the problem of uniqueness of meromorphic functions sharing two sets and obtain five theorems which not only improve a recent result of Yi and L¨ u [21], but also improve and supplement the results of Lin and Yi [15], Yi [19]. 2000 Mathematics Subject Classification: 30D35. Key words and phrases: Meromorphic functions, uniqueness, weighted sharing, shared set.

1. Introduction, Definitions and Results Let f and g be two nonconstant meromorphic functions defined on the open complex plane C. The notation S(r, f ) denotes any quantity satisfying S(r, f ) = o(T (r, f )) as r → ∞, outside a possible exceptional set of a finite linear measure. If for some a ∈ C ∪ {∞}, f and g have the same set of a-points with same multiplicities, then we say that f and g share the value a CM (counting multiplicities). If we do not take the multiplicities into account, f and g are said to share the value a IM (ignoring multiplicities). S {z : f (z)−a = Let S be a set of distinct elements of C∪{∞} and Ef (S) = a∈S

0}, where each zero is counted according to its multiplicity. If we do not count S the multiplicity the set {z : f (z) − a = 0} is denoted by E f (S). a∈S

If Ef (S) = Eg (S), we say that f and g share the set S CM. On the other hand, if E f (S) = E g (S), we say that f and g share the set S IM. Let m be a positive integer or infinity and a ∈ C ∪ {∞}. We denote by Em) (a; f ) the set of all a-points of f with multiplicities not exceeding m, where an a-point is counted according to its S multiplicity. For a set S of distinct elements of C we define Em) (S, f ) = Em) (a, f ). If for some a ∈ C ∪ {∞}, a∈S

E∞) (a; f )=E∞) (a; g) we say that f , g share the value a CM. In 1976 F. Gross [7] posed the following question: Question A. Can one find two finite sets Sj (j = 1, 2) such that any two nonconstant entire functions f and g satisfying Ef (Sj ) = Eg (Sj ) for j = 1, 2 must be identical? In [7] Gross wrote that if the answer to Question A is affirmative, it would be interesting to know how large both sets would have to be? c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / °

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A. BANERJEE

Yi [17] and independently Fang and Xu [6] gave the one and same positive answer in this direction. Now it is natural to ask the following question. Question B Can one find two finite sets Sj (j = 1, 2) such that any two nonconstant meromorphic functions f and g satisfying Ef (Sj ) = Eg (Sj ) for j = 1, 2 must be identical ? In 1994 Yi [16] gave an affirmative answer to Question B and proved that there exist two finite sets S1 (with two elements) and S2 (with nine elements) such that any two nonconstant meromorphic functions f and g satisfying Ef (Sj ) = Eg (Sj ) for j = 1, 2 must be identical. In 1996 Li and Yang [14] proved that there exist two finite sets S1 (with one element) and S2 (with 15 elements) such that any two nonconstant meromorphic functions Ef (Sj ) = Eg (Sj ) for j = 1, 2 must be identical. In 1997 Fang and Guo [5] obtained a better result than that of Li and Yang. They [5] proved that there exist two finite sets S1 (with one element) and S2 (with nine elements) such that any two nonconstant meromorphic functions Ef (Sj ) = Eg (Sj ) for j = 1, 2 must be identical. Suppose that the polynomial P (w) is defined by P (w) = awn − n(n − 1)w2 + 2n(n − 2)bw − (n − 1)(n − 2)b2 ,

(1.1)

where n ≥ 3 is an integer and a and b are two nonzero complex numbers satisfying the equality abn−2 6= 2. We claim that the polynomial P (w) has only simple zeros. Indeed, we consider the rational function R(w) =

awn , n(n − 1)(w − α1 )(w − α2 )

(1.2)

where α1 and α2 are two distinct roots of n(n − 1)w2 − 2n(n − 2)bw + (n − 1)(n − 2)b2 = 0. From (1.2) we have 0

R (w) =

(n − 2)awn−1 (w − b)2 . n(n − 1) (w − α1 )2 (w − α2 )2

(1.3)

From (1.3) we know that w = 0 is one root with multiplicity n of the equation R(w) = 0 and w = b is one root with multiplicity 3 of the equation R(w)−c = 0, n−2 where c = ab 2 . Then a(w − b)3 Qn−3 (w) , (1.4) R(w) − c = n(n − 1)(w − α1 )(w − α2 ) where Qn−3 (w) is a polynomial of degree 3. Moreover, from (1.1) and (1.2) we have R(w) − 1 =

P (w) . n(n − 1)(w − α1 )(w − α2 )

(1.5)

UNIQUENESS OF MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS

Noting that c =

abn−2 2 n

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6= 1, from (1.3) and (1.5) we have

P (w) = aw − n(n − 1)w2 − 2n(n − 2)bw + (n − 1)(n − 2)b2 with only simple zeros. In 2002 Yi [19] proved the following results which are the improvements of the earlier one. Theorem A ([19], see also [15], [21]). Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 8. Suppose that f and g are two nonconstant meromorphic functions satisfying Ef (S) = Eg (S) and Ef ({∞}) = Eg ({∞}), then f ≡ g. Theorem B ([19], see also [21]). Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 8. Suppose that f and g are two nonconstant meromorphic functions satisfying Ef (S) = Eg (S) and E f ({∞}) = E g ({∞}), then f ≡ g. In 2003, to deal with the problem of meromorphic functions sharing two sets Lin and Yi [15] proved the following two theorems. Theorem C ([15]). Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 12. Suppose that f and g are two nonconstant meromorphic functions satisfying E1) (S, f ) = E1) (S, g) and E f ({∞}) = E g ({∞}), then f ≡ g. Theorem D ([15]). Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 11. Suppose that f and g are two nonconstant meromorphic functions satisfying E2) (S, f ) = E2) (S, g) and E f ({∞}) = E g ({∞}), then f ≡ g. Recently, to deal with the question of Gross, Yi and L¨ u [21] investigated the problem of the uniqueness of two meromorphic functions f and g when they share the set S as mentioned in Theorem A, IM. They proved the following theorem. Theorem E ([21]). Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 12. Suppose that f and g are two nonconstant meromorphic functions satisfying E f (S) = E g (S) and Ef ({∞}) = Eg ({∞}), then f ≡ g. In 2001 an idea of gradation of sharing known as weighted sharing was introduced in {[10], [11]} which measure how close a shared value is to being shared CM or to being shared IM. In the following definition we explain the notion. Definition 1.1 ([10, 11]). 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 apoint 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.

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Definition 1.2 ([10]). Let S be a set of distinct elements of C S∪ {∞} and k be a nonnegative integer or ∞. We denote by Ef (S, k) the set Ek (a; f ). a∈S

Clearly, Ef (S) = Ef (S, ∞) and E f (S) = Ef (S, 0). Recently, the present author [4] has also provided an affirmative answer to the question of Gross concerning a meromorphic function in a slightly different way in which instead of consideration of the sharing of two sets, three sets have been taken into consideration and as a result a range set having smaller cardinalities has been obtained. But in this paper we shall confine our investigation to the uniqueness of meromorphic functions sharing two sets and thus concentrate our attention on Question B. In the paper we employ the idea of weighted sharing of sets and truncated sharing of sets to investigate the uniqueness of meromorphic functions and improve and supplement Theorems B and E by relaxing the nature of sharing the sets S and {∞}, respectively, and supplement Theorem C and improve Theorem D by replacing the range set S by a smaller one. The following five theorems are the main results of the paper. Theorem 1.1. Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 12. Suppose that f and g are two nonconstant meromorphic functions satisfying Ef (S, 0) = Eg (S, 0) and Ef ({∞}, 3) = Eg ({∞}, 3), then f ≡ g. Theorem 1.2. Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 9. Suppose that f and g are two nonconstant meromorphic functions satisfying Ef (S, 1) = Eg (S, 1) and Ef ({∞}, 0) = Eg ({∞}, 0), then f ≡ g. Theorem 1.3. Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 8. Suppose that f and g are two nonconstant meromorphic functions satisfying Ef (S, 2) = Eg (S, 2) and Ef ({∞}, 0) = Eg ({∞}, 0), then f ≡ g. Theorem 1.4. Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 11. Suppose that f and g are two nonconstant meromorphic functions satisfying E1) (S, f ) = E1) (S, f ) and Ef ({∞}, 1) = Eg ({∞}, 1), then f ≡ g. Theorem 1.5. Let S = {w | P (w) = 0}, where P (w) is given by (1.1) and n ≥ 10. Suppose that f and g are two nonconstant meromorphic functions satisfying E2) (S, f ) = E2) (S, f ) and Ef ({∞}, 0) = Eg ({∞}, 0), then f ≡ g. Though for the standard definitions and notation of the value distribution theory we refer to [8], we now explain some notation which are used in the paper. Definition 1.3 ([9]). For 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.

UNIQUENESS OF MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS

25

Also, N (r, a; f |< m), N (r, a; f |> m), N (r, a; f |< m) and N (r, a; f |> m) are defined analogously. Definition 1.4 ([4]). 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 ([1, 18, 20, 21]). Let f and g be two nonconstant meromorphic functions such that f and g share the value 1 IM. Let z0 be a 1-point of f with multiplicity p, a 1-point of g with multiplicity q. We denote by N L (r, 1; f ) the 1) counting function of those 1-points of f and g where p > q, by NE (r, 1; f ) the counting function of those 1-points of f and g where p = q = 1 and by (2 N E (r, 1; f ) the counting function of those 1-points of f and g where p = q ≥ 2, each point in these counting functions is counted only once. In the same way (2 1) we can define N L (r, 1; g), NE (r, 1; g), N E (r, 1; g). In a similar manner we can define N L (r, a; f ) and N L (r, a; g) for a ∈ C ∪ {∞}. For Ek) (1; f ) = Ek) (1; g) we define N L (r, 1; f ) (N L (r, 1; g)) in a similar manner. Definition 1.6 (cf. [1, 2]). Let k be a positive integer. Let f and g be two nonconstant meromorphic functions such that f and g share the value 1 IM. Let z0 be a 1-point of f with multiplicity p, a 1-point of g with multiplicity q. We denote by N f >k (r, 1; g) the reduced counting function of those 1-points of f and g such that p > q = k. N g>k (r, 1; f ) is defined analogously. For Ek) (1; f ) = Ek) (1; g) we can define N f >k (r, 1; g) (N g>k (r, 1; f )) analogously. Definition 1.7 (cf. [4]). Let f and g be two nonconstant meromorphic functions such that f and g share (1, k) where 1 ≤ k < ∞. Let z0 be a 1point of f with multiplicity p, a 1-point of g with multiplicity q. We denote by (k+1 N E (r, 1; f ) the counting function of those 1-points of f and g where p = q ≥ k + 1, each point in this counting function is counted only once. In the same (k+1 way we can define N E (r, 1; g). (k+1

For Ek) (1; f ) = Ek) (1; g) we can define N E gously.

(k+1

(r, a; f ) (N E

(r, a; g)) analo-

Definition 1.8 ([11]). We denote by N2 (r, a; f ) the sum N (r, a; f ) + N (r, a; f |≥ 2). Definition 1.9. Let m be a positive integer and for a ∈ C, Em) (a; f ) = Em) (a; g). Let z0 be a zero of f (z) − a of multiplicity p and a zero of g(z) − a of multiplicity q. We denote by N f ≥m+1 (r, a; f | g 6= a) (N g≥m+1 (r, a; g | f 6= a)) the reduced counting functions of those a-points of f and g for which p ≥ m + 1 and q = 0 (q ≥ m + 1 and p = 0).

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A. BANERJEE

Definition 1.10 ([10, 11]). Let f , g share (a, 0). 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. For Em) (a; f ) = Em) (a; g) we can define N ∗ (r, a; f, g) in a similar manner. Clearly, N ∗ (r, a; f, g) ≡ N ∗ (r, a; g, f ) and N ∗ (r, a; f, g) = N L (r, a; f ) + N L (r, a; g) when f , g share (a, 0), and N ∗ (r, a; f, g) = N L (r, a; f )+N L (r, a; g)+ N f ≥m+1 (r, 1; f | g 6= 1) + N g≥m+1 (r, 1; g | f 6= 1) when Em) (a; f ) = Em) (a; g). Definition 1.11 ([12]). 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 1.12 ([12]). 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 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 nonconstant meromorphic functions defined in C. Henceforth we shall denote by H and V the following two functions µ 00 µ 00 0 ¶ 0 ¶ F 2F G 2G H= − − − F0 F −1 G0 G−1 and µ ¶ µ ¶ 0 0 F G F0 G0 F0 G0 V = − − − = − . F −1 F G−1 G F (F − 1) G(G − 1) Lemma 2.1 ([18, 20]). If F , G are two nonconstant meromorphic functions such that they share (1,0) and H 6≡ 0, then 1)

NE (r, 1; F ) ≤ N (r, H) + S(r, F ) + S(r, G). Lemma 2.2 ([15]). If F , G are two nonconstant meromorphic functions such that E1) (1; F ) = E1) (1; G) and H 6≡ 0, then N (r, 1; F |= 1) ≤ N (r, H) + S(r, F ) + S(r, G). Lemma 2.3 ([13]). If N (r, 0; f (k) | f 6= 0) denotes the counting function of those zeros of of f (k) which are not the zeros of f , where a zero of f (k) is counted according to its multiplicity, then N (r, 0; f (k) | f 6= 0) ≤ kN (r, ∞; f ) + N (r, 0; f |< k) + kN (r, 0; f |≥ k) + S(r, f ). Lemma 2.4 ([2]). Let f and g be two nonconstant meromorphic functions sharing (1, 0). Then (2

N L (r, 1; f ) + 2N L (r, 1; g) + N E (r, 1; f ) − N f >1 (r, 1; g) − N g>1 (r, 1; f ) ≤ N (r, 1; g) − N (r, 1; g).

UNIQUENESS OF MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS

27

Lemma 2.5 ([1]). Let f and g be two nonconstant meromorphic functions sharing (1, 1). Then (2

2N L (r, 1; f ) + 2N L (r, 1; g) + N E (r, 1; f ) − N f >2 (r, 1; g) ≤ N (r, 1; g) − N (r, 1; g). Lemma 2.6 ([3]). Let f and g be two nonconstant meromorphic functions sharing (1, 2). Then (3

2N L (r, 1; f ) + 3N L (r, 1; g) + 2N E (r, 1; f ) + N (r, 1; f |= 2) ≤ N (r, 1; g) − N (r, 1; g). Lemma 2.7. Let E1) (1; f ) = E1) (1; g). Then (2

2 N L (r, 1; f )+2 N L (r, 1; g)+N E (r, 1; f )+N g≥2 (r, 1; g | f 6= 1)−N f >2 (r, 1; g) ≤ N (r, 1; g) − N (r, 1; g). Proof. Since E1) (1; f ) = E1) (1; g) the simple 1-points of f and g are the same. Let z0 be a 1-point of f with multiplicity p and a 1-point of g with multiplicity q. If q = 2, then the possible values of p are as follows: (i) p = 2 (ii) p ≥ 3 (iii) p = 0. Similarly, when q = 3 the possible values of p are (i) p = 2 (ii) p = 3 (iii) p ≥ 4 (iv) p = 0. If q ≥ 4, we can similarly find possible values of p. Now the lemma follows from the above discussion. This completes the proof. ¤ Lemma 2.8. Let E2) (1; f ) = E2) (1; g). Then (3

2N L (r, 1; f ) + 2N L (r, 1; g) + 2N E (r, 1; f ) + N (r, 1; f |= 2) + 2N g≥3 (r, 1; g | f 6= 1) ≤ N (r, 1; g) − N (r, 1; g). Proof. Since E2) (1; f ) = E2) (1; g), we note that the simple and double 1-points of f and g are the same. Let z0 be a 1-point of f with multiplicity p and an a-point of g with multiplicity q. If q = 3 the possible values of p are as follows: (i) p = 3 (ii) p ≥ 4 (iii) p = 0. Similarly, when q = 4 the possible values of p are (i) p = 3 (ii) p = 4 (iii) p ≥ 5 (iv) p = 0. If q ≥ 5, we can similarly find the possible values of p. Now the lemma follows from above explanation. This completes the proof. ¤ Let f and g be two nonconstant meromorphic function and F = R(f ),

G = R(g),

(2.1)

where R(w) is given by (1.2). From (1.2) and (2.1) it is clear that 1 1 (2.2) T (r, f ) = T (r, F ) + S(r, f ), T (r, g) = T (r, G) + S(r, g) n n Lemma 2.9. Let F , G be given by (2.1) and ω1 , ω2 . . . ωn be the roots of P (w) = 0. Then 1 1 1 0 N F >2 (r, 1; G) ≤ N (r, 0; f ) + N (r, ∞; f ) − N⊗ (r, 0; f ) + S(r, f ), 2 2 2

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A. BANERJEE 0

0

where N⊗ (r, 0; f ) = N (r, 0; f | f 6= 0, ω1 , ω2 . . . ωn ). Proof. Using Lemma 2.3 we get N F >2 (r, 1; G) ≤ N (r, 1; F |≥ 3) ¢ 1¡ ≤ N (r, 1 : F ) − N (r, 1; F ) 2 · n ¸ ¢ 1 X¡ ≤ N (r, ωj ; f ) − N (r, ωj ; f ) 2 j=1 ´ 1³ 0 0 ≤ N (r, 0; f | f 6= 0) − N⊗ (r, 0; f ) 2 1 0 ≤ [N (r, 0; f ) + N (r, ∞; f ) − N⊗ (r, 0; f )] + S(r, f ). 2

¤

Lemma 2.10. Let F and G be given by (2.1). If F , G share (1, m), where 0 ≤ m < ∞, then 1 0 [N (r, 0; f ) + N (r, ∞; f ) − N⊗ (r, 0; f )] + S(r, f ), m+1 1 0 (ii) N L (r, 1; G) ≤ [N (r, 0; g) + N (r, ∞; g) − N⊗ (r, 0; g )] + S(r, g), m+1 (i) N L (r, 1; F ) ≤

0

0

where N⊗ (r, 0; g ) is defined similarly to N⊗ (r, 0; f )in Lemma 2.9. Proof. We prove (i), since (ii) can be proved in a similar way. We get from (1.5) and (2.1) that N L (r, 1; F ) ≤ N (r, 1; F |≥ m + 2) ¢ 1 ¡ ≤ N (r, 1; F ) − N (r, 1; F ) . m+1 By Lemma 2.3 the proof can be carried out in the line of the proof of Lemma 2.9. ¤ Lemma 2.11. Let F and G be given by (2.1). If Em) (1; F ) = Em) (1; G), then N (r, 1; F |≥ m + 1) + N (r, 1; G |≥ m + 1) ¢ 1 ¡ N (r, 0; f ) + N (r, 0; g) + N (r, ∞; f ) + N (r, ∞; g) . ≤ m Proof. The proof can be carried out in the line of the proof of Lemma 2.9.

¤

Lemma 2.12. Let F and G be given by (2.1) and E1) (1; F ) = E1) (1; G). Then N F >2 (r, 1; G) + N F ≥2 (r, 1; F | g 6= 1) 0

≤ N (r, 0; f ) + N (r, ∞; f ) − N⊗ (r, 0; f ) + S(r, f ).

UNIQUENESS OF MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS

29

Proof. We note that a 1-point of F with multiplicity 2 is counted at most once only for the counting function N F ≥2 (r, 1; F | G 6= 1). Also, since a 1-point of F with multiplicity ≥ 3 may or may not be a 1-point of G, those 1-points of F are counted only once either in N F >2 (r, 1; G) or in N F ≥2 (r, 1; F | G 6= 1). So, using Lemma 2.3 we get N F >2 (r, 1; G) + N F ≥2 (r, 1; F | G 6= 1) ≤ N (r, 1; F |≥ 2) ≤ N (r, 1; F ) − N (r, 1; F ). Now the lemma can be proved in the line of the proof of Lemma 2.9.

¤

Lemma 2.13. Let F and G be given by (2.1) and E1) (1; F ) = E1) (1; G). Then 0

N F ≥2 (r, 1; F | G 6= 1) ≤ N (r, 0; f ) + N (r, ∞; f ) − N⊗ (r, 0; f ) + S(r, f ). Proof. The proof is obvious.

¤

Lemma 2.14. Let F and G be given by (2.1) and E2) (1; F ) = E2) (1; G). Then 1 1 1 0 N F ≥3 (r, 1; F | G 6= 1) ≤ N (r, 0; f ) + N (r, ∞; f ) − N⊗ (r, 0; f ) + S(r, f ). 2 2 2 Proof. Since N F ≥3 (r, 1; F | G 6= 1) ≤ N (r, 1; F | ≥ 3) 1 ≤ [N (r, 1; F ) − N (r, 1; F )], 2 the proof can be carried out in the line of the proof of Lemma 2.9.

¤

Lemma 2.15. [15, 21] Let F , G be given by (2.1). If V ≡ 0 and n ≥ 8, then F ≡ G. Lemma 2.16. Let F , G be given by (2.1) and H 6≡ 0. If F , G share (1, m) and f , g share (∞, k), where 0 ≤ m < ∞, 0 ≤ k < ∞, then [(n − 2)k + n − 3)] N (r, ∞; f |≥ k + 1) = [(n − 2)k + n − 3)]N (r, ∞; g |≥ k + 1) m+2 [N (r, 0; f ) + N (r, 0; g)] ≤ m+1 2 + N (r, ∞; f ) + S(r, f ) + S(r, g). m+1 Proof. Since H 6≡ 0, from Lemma 2.15 we have V 6≡ 0. We suppose that z0 is a pole of f with multiplicity p and a pole of g with multiplicity q. From (1.2) and (2.1) we know that z0 is a pole of f with multiplicity (n − 2)p and a pole of g with multiplicity (n − 2)q. Noting that f , g share (∞; k), from the definition of

30

A. BANERJEE

V it is clear that z0 is a zero of V with multiplicity at least (n − 2)(k + 1) − 1. Thus, using Lemma 2.10, from the definition of V we obtain [(n − 2)k + n − 3]N (r, ∞; f |≥ k + 1) = [(n − 2)k + n − 3]N (r, ∞; g |≥ k + 1) ≤ N (r, 0; V ) ≤ N (r, ∞; V ) + S(r, f ) + S(r, g) ≤ N (r, 0; F ) + N (r, 0; G) + N ∗ (r, 1; F, G) + S(r, f ) + S(r, g) ≤ N (r, 0; f ) + N (r, 0; g) + N L (r, 1; F ) + N L (r, 1; G) + S(r, f ) + S(r, g) m+2 2 ≤ [N (r, 0; f ) + N (r.0; g)] + N (r, ∞; f ) + S(r, f ) + S(r, g). ¤ m+1 m+1 Lemma 2.17. Let F , G be given by (2.1) and H 6≡ 0. If Em) (1; F ) = Em) (1; G), where 1 ≤ m < ∞ and f , g share (∞, k), where 0 ≤ ∞ < k, then [(n − 2)k + n − 3)] N (r, ∞; f |≥ k+1) = [(n − 2)k + n − 3)]N (r, ∞; g |≥ k+1) m+1 ≤ [N (r, 0; f ) + N (r, 0; g)] m 2 + N (r, ∞; f ) + S(r, f ) + S(r, g). m Proof. We omit the proof since using Lemma 2.11 and noting that N ∗ (r, 1; F, G) ≤ N (r, 1; F |≥ m + 1) + N (r, 1; G |≥ m + 1) the proof of the lemma can be carried out in the line of the proof of Lemma 2.16. ¤ Lemma 2.18. Let F , G be given by (2.1) and H 6≡ 0. If F , G share (1, m) and f , g share (∞, k), where 0 ≤ m ≤ ∞ and 0 ≤ k ≤ ∞, then 1)

NE (r, 1; F ) ≤ N L (r, 1; F ) + N L (r, 1; G) + N (r, 0; f ) + N (r, b; f ) 0

0

+ N ∗ (r, ∞; f, g) + N (r, 0; g) + N (r, b; g) + N 0 (r, 0; f ) + N 0 (r, 0; g ), 0

where N 0 (r, 0; f ) denotes the reduced counting function corresponding to the 0 0 zeros of f which are not the zeros of f (f − b) and F − 1, N 0 (r, 0; g ) is defined similarly. Proof. We have from (1.2) and (2.1) 0

0

F =

(n − 2)af n−1 (f − b)2 f , n(n − 1)(f − α1 )2 (f − α2 )2 0

(2.3)

(n − 2)ag n−1 (g − b)2 g G = . (2.4) n(n − 1)(g − α1 )2 (g − α2 )2 It is obvious that the simple zeros of f − α1 and f − α2 are the simple poles of F , the simple zeros of g − α1 and g − α2 are the simple poles of G. It can be easily verified that the simple zeros of f − α1 , f − α2 , g − α1 and g − α2 are not the poles of H. 0

UNIQUENESS OF MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS

31

We note that the multiple zeros of f − α1 , f − α2 , g − α1 and g − α2 are 0 0 the zeros of f and g , respectively. Also, the poles of H come from those poles of f and g whose multiplicities are different and those 1-points of F which are different in multiplicities from the corresponding 1-points of G. Since all poles of H are simple, using Lemma 2.1 we get the conclusion of the lemma from (1.2), (2.3) and (2.4). ¤ Lemma 2.19. Let F , G be given by (2.1) and H 6≡ 0. If Em) (1; F ) = Em) (1; G) and f , g share (∞, 0), where 1 ≤ m < ∞, then 1)

NE (r, 1; F ) ≤ N L (r, 1; F ) + N L (r, 1; G) + N F ≥m+1 (r, 1; F | G 6= 1) + N G≥m+1 (r, 1; G | F 6= 1) + N (r, 0; f ) + N (r, b; f ) + N ∗ (r, ∞; f, g) + N (r, 0; g) + N (r.b; g) 0

0

+ N 0 (r, 0; f ) + N 0 (r, 0; g ). Proof. We omit the proof since using Lemma 2.2 the proof can be carried out in the line of the proof of Lemma 2.18. ¤ Lemma 2.20. Let F and G be given by (2.1). If F , G share (1, 0), f , g share (∞, k), where 0 ≤ k < ∞ and H 6≡ 0, then (n + 1) T (r, f ) + T (r, g) ≤ 2N (r, 0; f ) + 2N (r, b; f ) + 2N (r, ∞; f ) + N (r, ∞; f |≥ k + 1) + 2N L (r, 1; F ) + 2N (r, 0; g) + 2N (r, b; g) + N L (r, 1; G) + S(r, f ) + S(r, g). Proof. By the second fundamental theorem we get (n + 1) T (r, f ) + (n + 1)T (r, g) ≤ N (r, 1; F ) + N (r, 0; f ) + N (r, b; f ) + N (r, ∞; f ) + N (r, 1; G) + N (r, 0; g) + N (r, b; g) + N (r, ∞; g) 0

0

− N0 (r, 0; f ) − N0 (r, 0; g ) + S(r, f ) + S(r, g).

(2.5)

Since f and g share (∞, k), N ∗ (r, ∞; f, g) ≤ N (r, ∞; f |≥ k + 1). By Lemmas 2.4, 2.18, and (2.2) we get 1)

N (r, 1; F ) + N (r, 1; G) = NE (r, 1; F ) + N L (r, 1; F ) + N L (r, 1; G) (2

+ N E (r, 1; F ) + N (r, 1; G) 1)

≤ NE (r, 1; F ) + N (r, 1; G) + N F >1 (r, 1; G) + N G>1 (r, 1; F ) − N L (r, 1; G) ≤ N (r, 0; f ) + N (r, b; f ) + N (r, 0; g) + N (r, b; g) + N (r, ∞; f |≥ k + 1) + nT (r, g) − m(r, 1; G) 0

+ O(1) + 2N L (r, 1; F ) + N L (r, 1; G) + N 0 (r, 0; f )

32

A. BANERJEE 0

+ N 0 (r, 0; g ) + S(r, f ) + S(r, g).

(2.6)

Using (2.6) in (2.5) and noting that N (r, ∞; f ) = N (r, ∞; g), we obtain the conclusion of the lemma. ¤ Lemma 2.21. Let F and G be given by (2.1). If F , G share (1, 1), f , g share (∞, 0) and H 6≡ 0, then (n + 1)T (r, f ) + T (r, g) ≤ 2N (r, 0; f ) + 2N (r, b; f ) + 3N (r, ∞; f ) + N F >2 (r, 1; G) + 2N (r, 0; g) + 2N (r, b; g) + S(r, f ) + S(r, g). Proof. We omit the proof since using Lemmas 2.5, 2.18 and (2.2) the proof of the lemma can be carried out in the line of the proof of Lemma 2.20. ¤ Lemma 2.22. Let F and G be given by (2.1). If F , G share (1, 2), f , g share (∞, 0) and H 6≡ 0, then (n + 1)T (r, f ) + T (r, g) ≤ 2N (r, 0; f ) + 2N (r, b; f ) + 3N (r, ∞; f ) + 2N (r, 0; g) + 2N (r, b; g) + S(r, f ) + S(r, g). 1)

Proof. Since F , G share (1, 2) implies NE (r, 1; F ) = N (r, 1; F |= 1), by Lemmas 2.6, 2.18 and (2.2) we see that (3

N (r, 1; F )+N (r, 1; G) = N (r, 1; F |= 1) + N (r, 1; F |= 2) + N E (r, 1; F ) + N L (r, 1; F ) + N L (r, 1; G) + N (r, 1; G) (3

≤ N (r, 1; F |= 1) + N (r, 1; F |= 2) + N E (r, 1; F ) + N L (r, 1; F )+N L (r, 1; G)+N (r, 1; G)−2N L (r, 1; F ) (3

− 3N L (r, 1; G) − 2N E (r, 1; F ) − N (r, 1; F |= 2) ≤ N (r, 0; f ) + N (r, b; f ) + N (r, ∞; f ) + N (r, 0; g) + N (r, b; g) + nT (r, g) − m(r, 1; G) + O(1) 0

0

+ N 0 (r, 0; f ) + N 0 (r, 0; g ) + S(r, f ) + S(r, g). (2.7) Using (2.7) in (2.5) the lemma follows. This proves the lemma.

¤

Lemma 2.23. Let F and G be given by (2.1) and E1) (1; F ) = E1) (1; G), f , g share (∞, k), where 0 ≤ k < ∞ and H 6≡ 0. Then (n + 1) T (r, f ) + T (r, g) ≤ 2N (r, 0; f ) + 2N (r, b; f ) + 2N (r, ∞; f ) + N (r, ∞; f |≥ k + 1) + 2N (r, 0; g) + 2N (r, b; g) + N F >2 (r, 1; G) + 2N F ≥2 (r, 1; F | G 6= 1) − m(r, 1; g) + S(r, f ).

UNIQUENESS OF MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS

33

Proof. By Lemmas 2.2, 2.7 and 2.19 for m = 1 and (2.2) we get N (r, 1; F ) + N (r, 1; G) (2

≤ N (r, 1; F |= 1) + N E (r, 1; F ) + N L (r, 1; F ) + N L (r, 1; G) + N F ≥2 (r, 1; F | G 6= 1) + N (r, 1; G) ≤ N (r, 0; f ) + N (r, b; f ) + N (r, ∞; f |≥ k + 1) + N (r, 0; g) + N (r, b; g) + N F ≥2 (r, 1; F | G 6= 1) + N G≥2 (r, 1; G | F 6= 1) + N L (r, 1; F ) + N L (r, 1; G) 0

0

(2

+ N 0 (r, 0; f ) + N 0 (r, 0; g ) + N E (r, 1; F ) + N L (r, 1; F ) + N L (r, 1; G) + N F ≥2 (r, 1; F | G 6= 1) (2

+ N (r, 1; g) − N E (r, 1; F ) − 2N L (r, 1; F ) − 2N L (r, 1; G) − N G≥2 (r, 1; G | F 6= 1) + N F >2 (r, 1; G) + S(r, f ) + S(r, g) ≤ N (r, 0; f ) + N (r, b; f ) + N (r, ∞; f |≥ k + 1) + N (r, 0; g) + N (r, b; f ) + N F ≥2 (r, 1; F | G 6= 1) + nT (r, g) 0

− m(r, 1; g) + O(1) + N F >2 (r, 1; g) + N 0 (r, 0; f ) 0

+ N 0 (r, 0; g ) + S(r, f ) + S(r, g).

(2.8)

Using (2.8) in (2.5), the lemma follows. This completes the proof.

¤

Lemma 2.24. Let F and G be given by (2.1) and E2) (1; F ) = E2) (1; G), f , g share (∞, 0) and H 6≡ 0. Then (n + 1) T (r, f ) + T (r, g) ≤ 2N (r, 0; f ) + 2N (r, b; f ) + 3N (r, ∞; f ) + 2N (r, 0; g) + 2N (r, b; g) + 2N F ≥3 (r, 1; F | G 6= 1) − m(r, 1; g) + S(r, f ). Proof. Using Lemma 2.2, 2.8 and 2.19 for m = 2 and (2.2), we can prove the lemma in the line of the proof of Lemma 2.23. ¤ Lemma 2.25 ([18]). If H ≡ 0, then F , G share (1, ∞). 3. Proofs of the Theorems Proof of Theorem 1.1. Let F and G be given by (2.1). Since Ef (S, 0) = Eg (S, 0), from (1.5) and (2.1) it follows that F and G share (1, 0). Suppose H 6≡ 0. Then F 6≡ G and so by Lemma 2.15 we have V 6≡ 0. So using Lemmas 2.10, 2.16 for m = 0 and k = 3 and from Lemma 2.20 we get (n + 1)T (r, f ) + T (r, g) ≤ 2N (r, 0; f ) + 2N (r, b; f ) + 2N (r, ∞; f ) + N (r, ∞; f |≥ 4) + 2N (r, 0; g) + 2N (r, b; g) + 2[N (r, 0; f ) + N (r, ∞; f )] + N (r, 0; g)

34

A. BANERJEE

+ N (r, ∞; g) + S(r, f ) + S(r, g) ≤ 4N (r, 0; f ) + 2N (r, b; f ) + 5N (r, ∞; f ) + N (r, ∞; f |≥ 4) + 3N (r, 0; g) + 2N (r, b; g) + S(r, f ) + S(r, g) ≤ 6T (r, f ) + 5T (r, g) + 5N (r, ∞; f ) 2 + [T (r, f ) + T (r, g) + N (r, ∞; f )] 4n − 9 + S(r, f ) + S(r, g).

(3.1)

Now, again using Lemma 2.16 for m = 0 and k = 0 we get from (3.1) that µ ¶ µ ¶ 2 2 (n + 1)T (r, f ) + T (r, g) ≤ 6 + T (r, f ) + 5 + T (r, g) 4n − 9 4n − 9 µ ¶ 2 2 + 5+ [T (r, f ) + T (r, g)] 4n − 9 n − 5 + S(r, f ) + S(r, g) µ ¶ 42n − 96 ≤ 6+ T (r, f ) (4n − 9)(n − 5) ¶ µ 42n − 96 T (r, g) + 5+ (4n − 9)(n − 5) + S(r, f ) + S(r, g). (3.2) In a similar manner we obtain

µ

¶ 42n − 96 T (r, f ) + (n + 1)T (r, g) ≤ 5 + T (r, f ) (4n − 9)(n − 5) ¶ µ 42n − 96 T (r, g) + 6+ (4n − 9)(n − 5) + S(r, f ) + S(r, g).

(3.3)

Adding (3.2) and (3.3), we get µ ¶ 84n − 192 T (r, f ) (n + 2)T (r, f ) + (n + 2)T (r, g) ≤ 11 + (4n − 9)(n − 5) µ ¶ 84n − 192 + 11 + T (r, g) + S(r, f ) + S(r, g), (4n − 9)(n − 5) i.e., µ

84n − 192 n−9− (4n − 9)(n − 5)

¶

µ

¶ 84n − 192 T (r, f ) + n − 9 − T (r, g) (4n − 9)(n − 5) ≤ S(r, f ) + S(r, g),

UNIQUENESS OF MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS

35

which is a contradiction for n ≥ 12. So H ≡ 0. Thus by Lemma 2.25 we get that F and G share (1, ∞). Hence Ef (S, ∞) = Eg (S, ∞) and the theorem follows from Theorem B. ¤ Proof of Theorem 1.2. Let F and G be given by (2.1). Since Ef (S, 1) = Eg (S, 1), from (1.5) and (2.1) it follows that F and G share (1, 1). Suppose H 6≡ 0. Then F 6≡ G and so by Lemma 2.15 we have V 6≡ 0. Using Lemmas 2.9, 2.16 for m = 1 and k = 0 and from Lemma 2.21 we get (n + 1)T (r, f )+T (r, g) ≤ 2N (r, 0; f ) + 2N (r, b; f ) + 2N (r, 0; g) + 2N (r, b; g) 7 1 + N (r, ∞; f ) + N (r, 0; f ) 2 2 5 ≤ N (r, 0; f ) + 2N (r, b; f ) + 2N (r, 0; g) + 2N (r, b; g) 2 21 + [T (r, f ) + T (r, g)] + S(r, f ) + S(r, g) 4(n − 4) µ ¶ µ ¶ 9 21 21 ≤ + T (r, f ) + 4 + T (r; g) 2 4(n − 4) 4(n − 4) + S(r, f ) + S(r, g). (3.4) Similarly, we obtain µ

¶ µ ¶ 9 21 21 T (r, f ) + (n + 1)T (r, g) ≤ 4 + T (r, f ) + + T (r, g) 4(n − 4) 2 4(n − 4) + S(r, f ) + S(r, g). (3.5) Adding (3.4) and (3.5) we get µ

¶ 17 21 (n + 2)T (r, f ) + (n + 2)T (r, g) ≤ + T (r, f ) 2 2(n − 4) µ ¶ 17 21 + + T (r, g) + S(r, f ) + S(r, g), 2 2(n − 4) i.e., µ

21 13 − n− 2 2(n − 4)

¶

µ ¶ 21 13 − T (r, f ) + n − T (r, g) 2 2(n − 4) ≤ S(r, f ) + S(r, g),

which is a contradiction for n ≥ 9. So H ≡ 0. Thus by Lemma 2.25 we get that F and G share (1, ∞). Now the theorem follows from Theorem B. ¤ Proof of Theorem 1.3. Let F and G be defined as in the proof of Theorem 1.1. Then F and G share (1, 2). Suppose H 6≡ 0. Then F 6≡ G and so by Lemma 2.15 we have V 6≡ 0. Now from Lemma 2.16 for m = 2 and k = 0 and from

36

A. BANERJEE

Lemma 2.22 we get (n + 1)T (r, f ) + T (r, g) ≤ 2N (r, 0; f ) + 2N (r, b; f ) + 2N (r, 0; g) + 2N (r, b; g) 12 + [T (r, f ) + T (r, g)] + S(r, f ) + S(r, g) 3n − 11 µ ¶ µ ¶ 12 12 ≤ 4+ T (r, f ) + 4 + T (r; g) 3n − 11 3n − 11 + S(r, f ) + S(r, g). (3.6) Similarly, we obtain

µ ¶ ¶ 12 12 T (r, f ) + 4 + T (r, g) T (r, f ) + (n + 1)T (r, g) ≤ 4 + 3n − 11 3n − 11 + S(r, f ) + S(r, g). (3.7) µ

Adding (3.6) and (3.7) we get µ ¶ 24 (n + 2)T (r, f ) + (n + 2)T (r, g) ≤ 8 + T (r, f ) 3n − 11 µ ¶ 24 + 8+ T (r, g) + S(r, f ) + S(r, g) 3n − 11 i.e., µ n−6−

24 3n − 11

¶

µ

24 T (r, f ) + n − 6 − 3n − 11

¶ T (r, g) ≤ S(r, f ) + S(r, g),

which is a contradiction for n ≥ 8. So H ≡ 0. Thus by Lemma 2.25 we get that F and G share (1, ∞). Now the theorem follows from Theorem B. ¤ Proof of Theorem 1.4. Let F and G be defined as in the proof of Theorem 1.1. Since E1) (S, f ) = E1) (S, g), from (1.5) and (2.1) it follows that E1) (1, F ) = E1) (1, G). Suppose H 6≡ 0. Then F 6≡ G and so by Lemma 2.15 we have V 6≡ 0. Then from Lemmas 2.12, 2.13, 2.17 for m = 1, k = 1, and again for m = 1, k = 0 and 2.23 we obtain (n + 1) T (r, f ) + T (r, g) ≤ 4N (r, 0; f ) + 2N (r, b; f ) + 2N (r, 0; g) + 2N (r, b; g) 2 [T (r, f ) + T (r, g) + N (r, ∞; f )] + 4N (r, ∞; f ) + 2n − 5 + S(r, f ) + S(r, g) µ ¶ µ ¶ 2 2 ≤ 6+ T (r, f ) + 4 + T (r; g) 2n − 5 2n − 5 µ ¶ 2 2 + 4+ [T (r, f ) + T (r, g)] 2n − 5 n − 5 + S(r, f ) + S(r, g)

UNIQUENESS OF MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS

µ ≤

37

¶

18n − 46 6+ T (r, f ) (2n − 5)(n − 5) µ ¶ 18n − 46 + 4+ T (r, g) (2n − 5)(n − 5) + S(r, f ) + S(r, g).

(3.8)

In a similar way we obtain ¶ 18n − 46 T (r, f ) T (r, f ) + (n + 1)T (r, g) ≤ 4 + (2n − 5)(n − 5) µ ¶ 18n − 46 + 6+ T (r, g) (2n − 5)(n − 5) + S(r, f ) + S(r, g). µ

(3.9)

Adding (3.8) and (3.9) we get µ ¶ µ ¶ 36n − 92 36n − 92 n−8− T (r, f ) + n − 8 − T (r, g) (2n − 5)(n − 5) (2n − 5)(n − 5) ≤ S(r, f ) + S(r, g), which is a contradiction for n ≥ 11. So H ≡ 0. Therefore F and G share (1, ∞). Now, following the proof of Theorem 1.1 the proof of the theorem can be carried out. ¤ Proof of Theorem 1.5. Let F and G be defined as in the proof of Theorem 1.1. Since E2) (S, f ) = E2) (S, g), from (1.5) and (2.1) it follows that E2) (1, F ) = E2) (1, G). Suppose H 6≡ 0. Then F 6≡ G and so by Lemma 2.15 we have V 6≡ 0. Now using Lemmas 2.14, 2.17 for m = 2, k = 0 and 2.24 the proof can be carried out in the line of the proof of Theorem 1.4. ¤ Acknowledgement The author is thankful to Prof. H. X. Yi for supplying him the electronic file of the paper [21]. References 1. Th. C. Alzahary and H. X. Yi, Weighted value sharing and a question of I. Lahiri. Complex Var. Theory Appl. 49(2004), No. 15, 1063–1078. 2. A. Banerjee, Meromorphic functions sharing one value. Int. J. Math. Math. Sci. 2005, No. 22, 3587–3598. 3. A. Banerjee, Uniqueness of meromorphic functions that share three values. Aust. J. Math. Anal. Appl. 3(2006), No. 2, Art. 1, 10 pp. (electronic). 4. A. Banerjee, On a question of Gross. J. Math. Anal. Appl. 327(2007), No. 2, 1273–1283. 5. M. Fang and H. Guo, On meromorphic functions sharing two values. Analysis 17(1997), No. 4, 355–366.

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6. M. Fang and W. Xu, A note on a problem of Gross. (Chinese) Chinese Ann. Math. Ser. A 18(1997), No. 5, 563–568; English transl.: Chinese J. Contemp. Math. 18(1997), No. 4, 395–402 (1998). 7. F. Gross, Factorization of meromorphic functions and some open problems. Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), pp. 51–67. Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977. 8. W. K. Hayman, Meromorphic functions. Oxford Mathematical Monographs Clarendon Press, Oxford, 1964. 9. I. Lahiri, Value distribution of certain differential polynomials. Int. J. Math. Math. Sci. 28(2001), No. 2, 83–91. 10. I. Lahiri, Weighted sharing and uniqueness of meromorphic functions. Nagoya Math. J. 161(2001), 193–206. 11. I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions. Complex Variables Theory Appl. 46(2001), No. 3, 241–253. 12. I. Lahiri and A. Banerjee, Weighted sharing of two sets. Kyungpook Math. J. 46(2006), No. 1, 79–87. 13. I. Lahiri and S. Dewan, Value distribution of the product of a meromorphic function and its derivative. Kodai Math. J. 26(2003), No. 1, 95–100. 14. P. Li and C. C. Yang, On the unique range set of meromorphic functions. Proc. Amer. Math. Soc. 124(1996), No. 1, 177–185. 15. W. C. Lin and H. X. Yi, Some further results on meromorphic functions that share two sets. Kyungpook Math. J. 43(2003), No. 1, 73–85. 16. H. X. Yi, Uniqueness of meromorphic functions and question of Gross. Sci. China Ser. A 37(1994), No. 7, 802–813. 17. H. X. Yi, On a question of Gross concerning uniqueness of entire functions. Bull. Austral. Math. Soc. 57(1998), No. 2, 343–349. 18. H. X. Yi, Meromorphic functions that share one or two values. II. Kodai Math. J. 22(1999), No. 2, 264–272. 19. H. X. Yi, Meromorphic functions that share two sets. (Chinese) Acta Math. Sinica (Chin. Ser.) 45(2002), No. 1, 75–82. 20. H. X. Yi and L. Z. Yang, Meromorphic functions that share two sets. Kodai Math. J. 20(1997), No. 2, 127–134. ¨ , Meromorphic functions that share two sets. II. Acta Math. Sci. 21. H. X. Yi and W. R. Lu Ser. B Engl. Ed. 24(2004), No. 1, 83–90.

(Received 13.07.2006) Author’s address: Department of Mathematics Kalyani Government Engineering College West Bengal 741235 India E-mails: abanerjee [email protected], abanerjee [email protected], [email protected]