Meromorphic functions sharing three values

J. Math. Soc. Japan Vol. 56, No., 2004 Meromorphic functions sharing three values By Xiao-Min Li and Hong-Xun Yi* (Received Feb. 7, 2002) (Revised Aug. 5, 2002) Abstract. In this paper, we prove a result on uniqueness of meromorphic functions sharing three values counting multiplicity. As applications of this, many known results can be improved. Examples are provided to show that the results in this paper are best possible.. Introduction and main results. In this paper, by meromorphic function we shall always mean a meromorphic function in the complex plane. We adopt the standard notations in the Nevanlinna theory of meromorphic functions as explained in []. For any nonconstant meromorphic function f ðzþ, we denote by Sðr; f Þ any quantity satisfying Sðr; f Þ¼oðTðr; f ÞÞ for r! y except possibly a set of r of finite linear measure. Let k be a positive integer, we denote by N kþ ðr; f Þ the counting function of poles of f with multiplicitya k. We further define (see [2]) N ð2 ðr; f Þ¼Nðr; f Þ N Þ ðr; f Þ; N ð2 ðr; f Þ¼Nðr; f Þ N Þ ðr; f Þ; N ð3 ðr; f Þ¼Nðr; f Þ N 2Þ ðr; f Þ: Let f and g be two nonconstant meromorphic functions and let a be a finite complex number. If f and g have the same a-points with the same multiplicities, we say that f and g share the value a CM (counting multiplicity) (see [2]). If =f and =g share the value 0 CM, we say that f and g share y CM. M. Ozawa [3], H. Ueda [4], G. Brosch [5], H. Yi [6], [7], [8], S. Ye [9], P. Li [0], Q. Zhang [] and other authors (see [2]) dealt with the problem of uniqueness of meromorphic functions that share three distinct values. In 995, H. Yi proved the following result, which is an improvement of some theorems given by H. Ueda [4], H. Yi [6] and S. Ye [9]. Theorem A (see [8, Theorem 4]). Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM, and let a ð00; Þ be a finite complex number. If N r; 0 Tðr; f ÞþSðr; f Þ; ð:þ 2000 Mathematics Subject Classification. 30D35, 30D30. Key Words and Phrases. Nevanlinna Theory, Meromorphic Function, Shared Value. Project supported by the NSF of Shandong (No. Z2002A0) and the NSF of China (No. 037065). *Corresponding author.

48 X.-M. Li and H.-X. Yi then a is a Picard value of f, f is a fractional linear transformation of g and one of the following three cases will hold: (i) y is a Picard value of f, aand y are Picard values of g, and ð Þ ðg þ a Þ að aþ; (ii) 0 is a Picard value of f, a=ða Þ and 0 are Picard values of g, and f þ ða Þg a; (iii) is a Picard value of f,=a and are Picard values of g, and f ag. In this paper, we improve the above theorem and obtain the following result. Theorem.. Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. If there exists a finite complex number a ð00; Þ such that a is not a Picard value of f, and N Þ r; 0 Tðr; f ÞþSðr; f Þ; ð:2þ then N Þ r; ¼ k 2 Tðr; f ÞþSðr; f Þ; k ð:3þ and one of the following cases will hold: (i) f ¼ eðkþþg e sg, g ¼ e ðkþþg ða Þkþ s e sg, with a kþ ¼ ss ðk þ sþ kþ s ðk þ Þ kþ and a 0 k þ ; s (ii) f ¼ esg e ðkþþg, g ¼ e sg e ðkþþg, with as ð aþ kþ s ¼ ss ðk þ sþ kþ s ðk þ Þ kþ and a 0 s k þ ; e sg (iii) f ¼ e ðkþ sþg, g ¼ e sg e ðkþ sþg, with ð aþ s ð aþ kþ ¼ ss kþ s ðk þ sþ ðk þ Þ kþ and s a 0 k þ s ; (iv) f ¼ ekg le sg, g ¼ e kg ð=lþe sg, with lk 0 0; and ða Þk s l k a k ¼ ss ðk sþ k s k k ; (v) f ¼ esg le kg, g ¼ e sg ð=lþe kg, with ls 00; and l s a s ð aþ k s ¼ ss ðk sþ k s k k ; (vi) f ¼ esg le ðk sþg, g ¼ e sg ð=lþe ðk sþg, with ls 00; and ð laþs ð aþ k ¼ ss ðk sþ k k ; where g is a nonconstant entire function, s and k ðb2þ are positive integers such that s and k þ are mutually prime and asak in (i), (ii) and (iii), s and k are mutually prime and asak in (iv), (v) and (vi). From Theorem., we immediately obtain the following corollary: Corollary.. Let f and g be two nonconstant meromorphic functions sharing 0; and y CM, and let a ð00; Þ be a finite complex number such that a is not a Picard value of f, and N Þ ðr; =ð ÞÞ 0 Tðr; f ÞþSðr; f Þ. If for any positive integer k ðb2þ, k s

Meromorphic functions sharing three values 49 then f g. N Þ r; 0 k 2 Tðr; f ÞþSðr; f Þ; k 2. Some lemmas. Lemma 2.. Let f be a nonconstant meromorphic function, and let a and a 2 be two distinct values in the extended complex plane, and a 3 be a meromorphic function satisfying Tðr; a 3 Þ¼Sðr; f Þ and a 3 D a j for j ¼ ; 2. If N r; þ N r; ¼ Sðr; f Þ; ð2:þ 2 then N Þ r; ¼ Tðr; f ÞþSðr; f Þ: 3 ð2:2þ Proof. have Using (2.), by the second fundamental theorem for small functions we Tðr; f ÞaN r; þ Sðr; f Þ: 3 ð2:3þ Thus, N r; ¼ Tðr; f ÞþSðr; f Þ: 3 ð2:4þ Obviously, N r; þ 3 2 N ð2 r; a N r; a Tðr; f ÞþSðr; f Þ: 3 3 From (2.4) and (2.5) we obtain N ð2 r; ¼ Sðr; f Þ: 3 Again from (2.4) and (2.6) we get (2.2). ð2:5þ ð2:6þ r Lemma 2.2. Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. If f is a fractional linear transformation of g, then for any finite complex number a ð00; Þ, either a is a Picard value of f, or N Þ r; ¼ Tðr; f ÞþSðr; f Þ: Proof. By assumption, there is a fractional linear transformation w ¼ LðuÞ such that f ¼ LðgÞ. Assume that a is not a Picard value of f. By virtue of Lemma 2., it su ces to show that f have two distinct Picard values. Assume that f has at most one Picard values. Then, two of the values 0; and y are not Picard values of f, and the

50 X.-M. Li and H.-X. Yi rest b of them is a Picard value, because the values among 0; ; y which are not Picard values of f are fixed points of LðuÞ and LðuÞ ðduþ has at most two fixed points. Set c :¼ LðbÞ. Obviously, c 0 b and c is a Picard value of f, because b is a Picard value of g too. This is a contradiction. This completes the proof of Lemma 2.2. r Lemma 2.3 (see [2, Lemma 4.5] or [3, Lemma 5]). Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM, and let a ð00; Þ be a finite complex constant. Then N ð3 r; þ N ð3 r; ¼ Sðr; f Þ: g a Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. We use N 0 ðrþ to denote the counting function of the zeros of f g that are not zeros of f, f and =f (see [8] or[]). The following lemma is essentially due to Q. Zhang. Lemma 2.4 (see [, Proof of Theorem and Theorem 2]). Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM, and let N 0 ðrþ 0 Sðr; f Þ. If f is a fractional linear transformation of g, then N 0 ðrþ ¼Tðr; f ÞþSðr; f Þ: If f is not any fractional linear transformation of g, then N 0 ðrþa Tðr; f ÞþSðr; f Þ; 2 and f and g assume one of the following relations: (i) f eðkþþg e sg, g e ðkþþg e sg ; (ii) f esg e ðkþþg, g e sg e ðkþþg ; (iii) f esg e ðkþ sþg, g e sg e ðkþ sþg ; where g is a nonconstant entire function, s and k ðb2þ are positive integers such that s and k þ are mutually prime and asak. Remark. Let f be a nonconstant meromorphic function. By the definition of Sðr; f Þ, there is a set E of r of finite linear measure such that Sðr; f Þ¼oðTðr; f ÞÞ ðr! y; r B EÞ: ð2:7þ In [], Q. Zhang first proved the conclusion of Lemma 2.4. Using the conclusion of Lemma 2.4, Q. Zhang proved the following theorems: Theorem in []. Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. If N 0 ðrþ lim Tðr; f Þ > 2 ; r!y r B E

Meromorphic functions sharing three values 5 where E is a set of r of finite linear measure with (2.7), then f is a fractional linear transformation of g. Theorem 2 in []. Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. If N 0 ðrþ 0 < lim Tðr; f Þ a 2 ; r!y r B E where E is a set of r of finite linear measure with (2.7), then f is not any fractional linear transformation of g, and f and g assume one of the three relations in Lemma 2.4. Lemma 2.5 (see [4]). Let s ð> 0Þ and t are mutually prime integers, and let c be a finite complex number such that c s ¼, then there exists one and only one common zero of o s and o t c. Lemma 2.6 (see [5]). Let f be a nonconstant meromorphic function, and let F ¼ Xp k¼0 a k f k X q be an irreducible rational function in f with constant coe cients fa k g and fb j g, where a p 0 0 and b q 0 0. Then where d ¼ maxfp; qg. j¼0 b j f j Tðr; FÞ ¼dTðr; f ÞþSðr; f Þ; Lemma 2.7 (see [6]). Let PðoÞ ¼o n þ ao m þ b; ð2:8þ where m and n are positive integers such that n > m, a and b are finite nonzero complex numbers. (i) The algebraic equation PðoÞ ¼0 has no roots with multiplicityb3; (ii) If b n m a n 0 ð Þn m m ðn mþ n m n n ; ð2:9þ the algebraic equation PðoÞ ¼0 has n distinct simple roots, no multiple roots; (iii) If n and m are mutually prime and b n m a n ¼ ð Þn m m ðn mþ n m n n ; ð2:0þ the algebraic equation PðoÞ ¼0 has n distinct roots, where n 2 roots are simple, one is double. Proof. (i) The conclusion is obvious, we now omit it. (ii) Let PðoÞ ¼o n þ ao m þ b; ð2:þ

52 X.-M. Li and H.-X. Yi then P 0 ðoþ ¼no n þ amo m : ð2:2þ If o 0 is a double root of PðoÞ ¼0, then if and only if PðoÞ ¼P 0 ðoþ ¼0: Combining (2.) and (2.2) we can easily get o m 0 ¼ nb aðn mþ ; on 0 ¼ bm n m : ð2:3þ Since ðo0 mþn ¼ðo0 nþm, from (2.3) we have nb n ¼ bm m ; ð2:4þ aðn mþ n m which can be rewritten as b n m a n ¼ ð Þn m m ðn mþ n m n n ; ð2:5þ accordingly, if PðoÞ ¼0 has n distinct simple roots, then PðoÞ ¼0 has no any multiple root, if and only if (2.9) holds. (iii) Let o 0 be a double root of PðoÞ ¼0, using proceeding as in (ii), we can get (2.3) and (2.4). On the other hand, since n and m are mutually prime, there exist one and only one pair of integers s and t such that ns mt ¼ ð0 < s < m; 0 < t < nþ: ð2:6þ From (2.3) and (2.6) we can easily have o 0 ¼ o0 ns mt ¼ bm s nb t ; n m aðn mþ which implies that PðoÞ ¼0 has one and only one double root. r Lemma 2.8 (see [8, Lemma ]). Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM, then there exist two entire functions a and b such that where e b D, e a D and e b a D, and f ea e b ; g e a e b ; ð2:7þ Tðr; gþþtðr; e a ÞþTðr; e b Þ¼OðTðr; f ÞÞ ðr B EÞ; ð2:8þ where E is a set of r of finite linear measure. Lemma 2.9 (see [8, Lemma 3]). Let a be a nonconstant entire function, then Tðr; a 0 Þ¼Sðr; e a Þ: ð2:9þ

Meromorphic functions sharing three values 53 Lemma 2.0 (see [, Lemma 6]). Let f and f 2 be two nonconstant meromorphic functions satisfying Nðr; f j ÞþN r; ¼ SðrÞ ðj ¼ ; 2Þ: f j Then either N 0 ðr; ; f ; f 2 Þ¼SðrÞ or there exist two integers p and q ðjpjþjqj > 0Þ such that f p f q 2 ; where N 0 ðr; ; f ; f 2 Þ denotes the reduced counting function of the common -points of f and f 2, and TðrÞ ¼Tðr; f ÞþTðr; f 2 Þ, SðrÞ ¼oðTðrÞÞ ðr! y; r B EÞ, E is a set of r of finite linear measure. Lemma 2. (see [8, Lemma 4]). Let f and g be two nonconstant meromorphic functions sharing 0; and y CM. If f D g, then N ð2 ðr; f ÞþN ð2 r; þ N ð2 r; ¼ Sðr; f Þ: f f 3. Proof of Theorem.. If f is a fractional linear transformation of g, by Lemma 2.2 we have that either a is a Picard value of f,orn Þ ðr; =ð ÞÞ ¼ Tðr; f ÞþSðr; f Þ; which contradicts the assumption of Theorem.. Thus, f is not a fractional linear transformation of g. By Theorem A we have N r; ¼ Tðr; f ÞþSðr; f Þ: ð3:þ From (.2) and (3.) we obtain N ð2 r; 0 Sðr; f Þ: ð3:2þ By Lemma 2.3, N ð3 r; ¼ Sðr; f Þ: ð3:3þ Combining (3.2) and (3.3) we get N ð2 r; 0 Sðr; f Þ: ð3:4þ We discuss the following two cases. Case. Suppose that N 0 ðrþ 0 Sðr; f Þ:

54 X.-M. Li and H.-X. Yi By Lemma 2.4 we know that f and g assume one of the three relations in Lemma 2.4. We discuss the following three subcases. Subcase.. Suppose that f and g assume the form (i) in Lemma 2.4. Thus, f ¼ eðkþþg e sg ; g ¼ e ðkþþg e sg ; ð3:5þ which assume the form (i) in Theorem.. By Lemma 2.5, we know that there exists one and only one common zero of o kþ and o s. By Lemma 2.6, we have from (3.5) Tðr; f Þ¼kTðr; e g ÞþSðr; f Þ: ð3:6þ From (3.5) we have ¼ eðkþþg ae sg þða Þ e sg : ð3:7þ Let PðoÞ ¼o kþ ao s þða Þ; ð3:8þ QðoÞ ¼ okþ ao s þða Þ o s : ð3:9þ If a ¼ðkþÞ=s, from (3.8) we know that o ¼ is a double root of PðoÞ ¼0. Again by Lemma 2.7, the equation PðoÞ ¼0 has k distinct simple roots. From (3.9) we know that QðoÞ ¼0 has k distinct simple roots. From (3.6) and (3.7), N Þ r; ¼ ktðr; e g ÞþSðr; f Þ¼Tðr; f ÞþSðr; f Þ; which contradicts (.2). Thus, a 0 ðk þ Þ=s and o ¼ is a simple root of PðoÞ ¼0. If ða Þ kþ s a kþ 0 ss ðk þ sþ kþ s ðk þ Þ kþ ; by Lemma 2.7, we know that QðoÞ ¼0 has k distinct simple roots, which is also a contradiction. Thus, ða Þ kþ s a kþ ¼ ss ðk þ sþ kþ s ðk þ Þ kþ : By Lemma 2.7, we know that QðoÞ ¼0 has k 2 distinct simple roots and one double root. From (3.6), (3.7) and (3.9) we obtain N ð2 r; ¼ 2 Tðr; f ÞþSðr; f Þ: k ð3:0þ Combining (3.) and (3.0) we get (.3).

Meromorphic functions sharing three values 55 Subcase.2. Suppose that f and g assume the form (ii) in Lemma 2.4. Thus, f ¼ esg e ðkþþg ; g ¼ e sg e ðkþþg ; ð3:þ which assume the form (ii) in Theorem.. By Lemma 2.6, we have from (3.) Tðr; f Þ¼kTðr; e g ÞþSðr; f Þ: ð3:2þ From (3.) we have ¼ aðeðkþþg ð=aþe sg ða Þ=aÞ : ð3:3þ e ðkþþg In the same manner as Subcase., we have a 0 s=ðk þ Þ and a s ð aþ kþ s ¼ s s ðk þ sþ kþ s =ðk þ Þ kþ, and can obtain (.3). Subcase.3. Suppose that f and g assume the form (iii) in Lemma 2.4. Thus, f ¼ e sg e ðkþ sþg ; g ¼ e sg e ðkþ sþg ; ð3:4þ which assume the form (iii) in Theorem.. From (3.4) we have ¼ eðkþþg þða Þe ðkþ sþg a e ðkþ sþg : ð3:5þ In the same manner as Subcase., we have a 0 s=ðk þ sþ and ð aþ s =ð aþ kþ ¼ s s ðk þ sþ kþ s =ðk þ Þ kþ, and can obtain (.3). Case 2. Suppose that N 0 ðrþ ¼Sðr; f Þ: Noting f and g share 0; and y CM, by Lemma 2.8 we have (2.7) and (2.8). (2.7) we have Tðr; f ÞaTðr; e a ÞþTðr; e b ÞþOðÞ: ð3:6þ From ð3:7þ From (2.8) and Lemma 2.9 we have Tðr; a 0 ÞþTðr; b 0 Þ¼Sðr; f Þ: ð3:8þ Again from (2.7) we get ¼ ea ae b þða Þ e b : ð3:9þ Assume that Tðr; e b Þ¼Sðr; f Þ. Noting 0 and y are Picard values of e a, by Lemma 2. we have from (2.7) and (3.8) N Þ r; ¼ Tðr; f ÞþSðr; f Þ;

56 X.-M. Li and H.-X. Yi which is a contradiction. Thus, Tðr; e b Þ 0 Sðr; f Þ. Similarly, we have Tðr; e a Þ 0 Sðr; f Þ and Tðr; e a b Þ 0 Sðr; f Þ. Particularly, none of e a ; e b and e a b are constants. From (2.7) we obtain f g ¼ ðea Þð e b a Þ e b : ð3:20þ We use N0 ðrþ to denote the counting function of the common zeros of ea and e b. From (3.20), the following formula is obviously From this and (3.6), N 0 ðrþ ¼N0 ðrþþsðr; f Þ: N0 ðrþ ¼Sðr; f Þ: ð3:2þ Let z 0 be a multiple zero of, but not a zero of a 0 ;b 0 and b 0 a 0. From (3.9) we obtain and e aðz 0Þ ae bðz 0Þ þ a ¼ 0 a 0 ðz 0 Þe aðz 0Þ ab 0 ðz 0 Þe bðz 0Þ ¼ 0: ð3:22þ ð3:23þ From (3.22) and (3.23) we have e aðz 0Þ ¼ ð aþb 0 ðz 0 Þ b 0 ðz 0 Þ a 0 ðz 0 Þ ; ebðz 0Þ ¼ ð aþa0 ðz 0 Þ aðb 0 ðz 0 Þ a 0 ðz 0 ÞÞ : ð3:24þ Let f ¼ ðb 0 a 0 Þe a ð aþb 0 ; f 2 ¼ aðb 0 a 0 Þe b ð aþa 0 : ð3:25þ Set TðrÞ ¼Tðr; f ÞþTðr; f 2 Þ; SðrÞ ¼oðTðrÞÞ ðr! y; r B EÞ; ð3:26þ E is a set of r of finite linear measure. From (3.7), (3.8), (3.25) and (3.26) we get Sðr; f Þ¼SðrÞ: ð3:27þ From (2.9), (3.8), (3.25) and (3.27) we have Nðr; f j ÞþN r; ¼ SðrÞ f j ðj ¼ ; 2Þ: ð3:28þ From (3.24) and (3.25), we have f ðz 0 Þ¼, f 2 ðz 0 Þ¼. Thus, N ð2 r; a N 0 ðr; ; f ; f 2 ÞþSðr; f Þ; ð3:29þ where N 0 ðr; ; f ; f 2 Þ denotes the reduced counting function of the common -points of f and f 2. From (3.4), (3.27) and (3.29) we obtain

Meromorphic functions sharing three values 57 N 0 ðr; ; f ; f 2 Þ 0 SðrÞ: ð3:30þ Noting (3.28) and (3.30) and using Lemma 2.0, we know that there exist two integers p and q ðjpjþjqj > 0Þ such that f p f q 2 : ð3:3þ Noting Tðr; e a Þ 0 Sðr; f Þ and Tðr; e b Þ 0 Sðr; f Þ, from (3.25) and (3.3) we have p 0 0 and q 0 0. From (3.25) and (3.3), we obtain e paþqb 0 p ð aþb ð aþa 0 q b 0 a 0 aðb 0 ; ð3:32þ a 0 Þ by logarithmic di erentiation, we can get pa 0 þ qb 0 q þ pa 0 =b 0 a 0 0 ¼ ða 0 =b 0 Þð a 0 =b 0 Þ b 0 : ð3:33þ If a 0 =b 0 D q=p, from (3.33) we have a 0 ða0 =b 0 Þ 0 ða 0 =b 0 Þ : By integration, we obtain e a a0 b 0 c ; ð3:34þ where c is a nonzero constant. From (3.34) we get Again by integration, we have b 0 a0 e a e a c : e b c 2 ðe a c Þ; ð3:35þ where c 2 is also a nonzero constant. e a, which is impossible. Thus, and hence By integration, we obtain From (3.35) we know that c is a Picard value of a 0 b 0 q p pa 0 þ qb 0 0: pa þ qb c 0 ; ð3:36þ ð3:37þ where c 0 is a finite constant. Noting e a b is not a constant, from (3.37) we know that p 0 q. Without loss of generality, from (3.36) we may assume that p and q are two integers such that p and q are mutually prime and q > 0. Let g ¼ a=q. From this, (2.7) and (3.37) we have

58 X.-M. Li and H.-X. Yi f ¼ eqg le pg ; g ¼ e qg ð=lþe pg ; ð3:38þ where l ¼ e c0=q is a nonzero constant. lo p ¼ 0 have a common root. Obviously, if and only if l q ¼, o q ¼ 0 and Noting N0 ðrþ ¼Sðr; f Þ, from (3.38) we get l q 0 0; : ð3:39þ Noting q > 0 and p 0 q, we discuss the following three subcases. get Subcase 2.. Suppose that q > p > 0. Setting k ¼ q and s ¼ p, from (3.38) we f ¼ ekg le sg ; g ¼ e kg ð=lþe sg ; ð3:40þ which assume the form (iv) in Theorem.. From (3.39) we have l k 0 0; : ð3:4þ By Lemma 2.6, we have from (3.40) From (3.40) we have Tðr; f Þ¼kTðr; e g ÞþSðr; f Þ: ð3:42þ Let If ¼ ekg ale sg þða Þ le sg : ð3:43þ RðoÞ ¼ ok alo s þða Þ lo s : ð3:44þ ða Þ k s l k a k 0 ss ðk sþ k s k k ; by Lemma 2.7, we know that RðoÞ ¼0 has k distinct simple roots, which is a contradiction. Thus, ða Þ k s l k a k ¼ ss ðk sþ k s k k : ð3:45þ By Lemma 2.7, we know that QðoÞ ¼0 has k 2 distinct simple roots and one double root. From (3.42), (3.43) and (3.44) we obtain N ð2 r; ¼ 2 Tðr; f ÞþSðr; f Þ: k ð3:46þ Combining (3.) and (3.46) we get (.3). get Subcase 2.2. Suppose that p > q > 0. Setting k ¼ p and s ¼ q, from (3.38) we

Meromorphic functions sharing three values 59 f ¼ esg le kg ; g ¼ e sg ð=lþe kg ; ð3:47þ which assume the form (v) in Theorem.. From (3.39) we have l s 0 0; : ð3:48þ By Lemma 2.6, we have from (3.47) Tðr; f Þ¼kTðr; e g ÞþSðr; f Þ: ð3:49þ From (3.47) we have ¼ alðekg ð=ðalþþe sg þð aþ=ðalþþ le kg : ð3:50þ In the same manner as Subcase 2., we have and can obtain (.3). l s a s ð aþ k s ¼ ss ðk sþ k s k k ; ð3:5þ get Subcase 2.3. Suppose that p > 0. Setting k ¼ p þ q and s ¼ q, from (3.38) we f ¼ esg le ðk sþg ; g ¼ e sg ð=lþe ðk sþg ; ð3:52þ which assume the form (vi) in Theorem.. From (3.39) we have l s 0 0; : ð3:53þ From (3.52) we have In the same manner as Subcase 2., we have and can obtain (.3). Theorem. is thus completely proved. 4. On two results of P. Li. In 998, P. Li proved the following result: ¼ ekg ð aþe ðk sþg al l e ðk sþg : ð3:54þ ð laþ s ð aþ k ¼ ss k s ðk sþ k k ; ð3:55þ Theorem B (see [0, Theorem ]). Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. Suppose additionally that f is not a fractional linear transformation of g and that there exists a finite complex number a ð00; Þ such that

60 X.-M. Li and H.-X. Yi Tðr; f ÞacN ð2 r; þ Sðr; f Þ; ð4:þ here c ð> 0Þ is a constant, then there exist a nonconstant entire function g, a nonzero constant l and two integers t ð> 0Þ, s which are mutually prime, such that f ¼ etg le sg ; g ¼ e tg ð=lþe sg ; ð4:2þ with y ¼ t=s 0 ; a. ð aþ sþt a t ¼ l t ð yþ sþt y t ; ð4:3þ From Theorem., we can obtain the following result, which is an improvement and supplement of Theorem B. Theorem 4.. Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. If there exists a finite complex number a ð00; Þ such that N ð2 r; 0 Sðr; f Þ; ð4:4þ then the conclusions of Theorem. hold, and N ð2 r; ¼ Tðr; f ÞþSðr; f Þ: k ð4:5þ Proof. From (4.4) we know that a is not a Picard value of f, and N Þ ðr; =ð ÞÞ 0 Tðr; f ÞþSðr; f Þ. By Theorem., we immediately obtain the conclusion of Theorem 4.. r In 998, P. Li proved the following result: Theorem C (see [0, Theorem 2]). Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. Suppose additionally that f is not a fractional linear transformation of g and that there exists a finite complex number a ð00; Þ such that N Þ r; ¼ Sðr; f Þ; ð4:6þ then f and g assume one of the following forms: (i) f ¼ e3g e g, g ¼ e 3g e g, with a ¼ 3 4 ; (ii) f ¼ e3g le 2g, g ¼ e 3g ð=lþe 2g, with a ¼ 3 and l3 ¼ ; (iii) f ¼ eg e 3g, g ¼ e g e 3g, with a ¼ 4 3 ; (iv) f ¼ e2g le 3g, g ¼ e 2g ð=lþe 3g, with a ¼ 3 and l2 ¼ ;

Meromorphic functions sharing three values 6 (v) f ¼ e2g e g, g ¼ e 2g e g, with a ¼ 4 ; (vi) f ¼ eg e 2g, g ¼ e g e 2g, with a ¼ 4; (vii) f ¼ e2g le g, g ¼ e 2g ð=lþe g, with l2 0 and a 2 l 2 ¼ 4ða Þ; (viii) f ¼ eg le 2g, g ¼ e g ð=lþe 2g, with l 0 and 4að aþl ¼ ; (ix) f ¼ eg le g, g ¼ e g a ð=lþe g, with l 0 and ð aþ 2 þ 4al ¼ 0; 2 where g is a nonconstant entire function. From Theorem., we can obtain the following result. Theorem 4.2. Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. If there exists a finite complex number a ð00; Þ such that a is not a Picard value of f, and N Þ r; a utðr; f ÞþSðr; f Þ; ð4:7þ where u < =3, then N Þ r; ¼ 0; ð4:8þ and f and g assume one of the following forms: (i) f ¼ e3g e g, g ¼ e 3g e g, with a ¼ 3 4 ; (ii) f ¼ e3g e 2g, g ¼ e 3g e 2g, with a ¼ 3; (iii) f ¼ eg e 3g, g ¼ e g e 3g, with a ¼ 4 3 ; (iv) f ¼ e2g e 3g, g ¼ e 2g e 3g, with a ¼ 3 ; (v) f ¼ e2g e g, g ¼ e 2g e g, with a ¼ 4 ; (vi) f ¼ eg e 2g, g ¼ e g e 2g, with a ¼ 4; (vii) f ¼ e2g le g, g ¼ e 2g ð=lþe g, with l2 0 and a 2 l 2 ¼ 4ða Þ; (viii) f ¼ eg le 2g, g ¼ e g ð=lþe 2g, with l 0 and 4að aþl ¼ ; (ix) f ¼ eg le g, g ¼ e g ð=lþe g, with l 0 and ð aþ2 þ 4al ¼ 0; where g is a nonconstant entire function.

62 X.-M. Li and H.-X. Yi Proof. By Theorem. and (4.7), we know that the conclusions of Theorem. hold, where k ¼ 2. From this, we immediately obtain the conclusion of Theorem 4.2. r Remark. Obviously, Theorem 4.2 is an improvement of Theorem C. It is easy to show that (ix) in Theorem C and (ix) in Theorem 4.2 are equivalent to each other. We next prove that (iv) in Theorem C and (iv) in Theorem 4.2 are equivalent to each other. In fact, in (iv) of Theorem C, l 2 ¼. From this we obtain l ¼ orl¼. When l ¼, from (iv) in Theorem C we obtain (iv) in Theorem 4.2. When l ¼, using g þ pi in place of g in (iv) of Theorem C, we obtain (iv) in Theorem 4.2. Similarly, we can prove that (ii) in Theorem C and (ii) in Theorem 4.2 are equivalent to each other. Example p 4.. Let f ðzþ ¼e 3z þ e 2z þ e z þ, gðzþ ¼e 3z þ e 2z þ e z þ and a ¼ ð20 þ 4 ffiffi 2 iþ=27. Then it is easily verified that f and g share 0; andy CM, and N Þ r; ¼ Tðr; f ÞþSðr; f Þ: 3 Moreover, f and g do not assume one of the forms in Theorem 4.2. that the assumption u < =3 in Theorem 4.2 is best possible. This illustrates 5. Some result of entire functions. In 995, H. Yi proved the following result. Theorem D (see [8, Theorem ]). Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM, and let a ð00; Þ be a finite complex number. If N r; 0 Tðr; f ÞþSðr; f Þ ð5:þ and Nðr; f Þ 0 Tðr; f ÞþSðr; f Þ; then a and a are Picard values of f and g respectively, and also y is so, and ð5:2þ ð Þðg þ a Þ að aþ: ð5:3þ From Theorem., we immediately obtain the following result. Theorem 5.. If, in addition to the assumptions of Theorem., then N Þ r; Nðr; f Þ¼Sðr; f Þ; ¼ k 2 Tðr; f ÞþSðr; f Þ; k ð5:4þ ð5:5þ and one of the following two cases will hold:

Meromorphic functions sharing three values 63 (i) f ¼ e kg þ e ðk Þg þþ, g ¼ e kg þ e ðk Þg þþ, with ða Þ k =a kþ ¼ k k =ðk þ Þ kþ and a 0 k þ ; (ii) f ¼ e kg e ðk Þg e g, g ¼ e kg e ðk Þg e g, with ð aþ k = ð aþ kþ ¼ k k =ðk þ Þ kþ and a 0 k; where g is a nonconstant entire function, k ðb2þ is a positive integer. From Theorem 4. we immediately obtain the following result. Theorem 5.2. If, in addition to the assumptions of Theorem 4., Nðr; f Þ¼Sðr; f Þ; ð5:6þ then the conclusions of Theorem 5. hold. From Theorem 4.2 we immediately obtain the following result. Theorem 5.3. If, in addition to the assumptions of Theorem 4.2, Nðr; f ÞavTðr; f ÞþSðr; f Þ; ð5:7þ where v < =2, then N Þ r; ¼ 0; Nðr; f Þ¼0; ð5:8þ and one of the following two cases will hold: (i) f ¼ e 2g þ e g þ, g ¼ e 2g þ e g þ, with a ¼ 3=4; (ii) f ¼ e 2g e g, g ¼ e 2g e g, with a ¼ =4; where g is a nonconstant entire function. Example 5.. Let f ðzþ ¼ðe 3z Þ=ðe 2z Þ, gðzþ ¼ðe 3z Þ=ðe 2z Þ and a ¼ 3. Then it is easily verified that f and g share 0; and y CM, N Þ ðr; =ð ÞÞ ¼ 0 and Nðr; f Þ¼ Tðr; f ÞþSðr; f Þ: 2 Moreover, f and g do not assume one of the forms in Theorem 5.3. that the assumption v < =2 in Theorem 5.3 is best possible. This illustrates Theorem 5.4. Let f and g be two distinct nonconstant meromorphic functions sharing 0; and y CM. If there exists a finite complex number a ð00; Þ such that N Þ r; a utðr; f ÞþSðr; f Þ; ð5:9þ and Nðr; f ÞavTðr; f ÞþSðr; f Þ; ð5:0þ N Þ r; 0 Tðr; gþþsðr; gþ; g a ð5:þ where u < =3 and v < =2, then

64 X.-M. Li and H.-X. Yi N Þ r; ¼ 0; ð5:2þ and one of the following three cases will hold: (i) f g 2 2 4, with a ¼ 2 ; (ii) f ¼ e 2g þ e g þ, g ¼ e 2g þ e g þ, with a ¼ 3 4 ; (iii) f ¼ e 2g e g, g ¼ e 2g e g, with a ¼ 4 ; where g is a nonconstant entire function. Proof. We discuss the following two cases. Case. Suppose that a is a Picard value of f. By Theorem A, we know that f and g assume one of the three relations in Theorem A. We discuss the following three subcases. Subcase.. Suppose that f and g assume the relation (i) in Theorem A. From this we obtain, ð Þðg þ a Þ að aþ; ð5:3þ and a and y are Picard values of g. N Þ r; ¼ Tðr; gþþsðr; gþ; g a If a 0 a, by Lemma 2.2 we have which contradicts (5.). Thus a ¼ a, and hence a ¼ =2. From this we obtain the form (i) in Theorem 5.4. Subcase.2. Suppose that f and g assume the relations (ii) in Theorem A. From this we obtain, 0 and a are Picard values of f. By Lemma 2.2 we have which contradicts (5.0). N Þ ðr; f Þ¼Tðr; f ÞþSðr; f Þ; Subcase.3. Suppose that f and g assume the relations (iii) in Theorem A. From this we obtain, and a are Picard values of f. By Lemma 2.2 we have which contradicts (5.0). N Þ ðr; f Þ¼Tðr; f ÞþSðr; f Þ; Case 2. Suppose that a is not a Picard value of f. Using Theorem 5.3, we obtain the forms (ii) and (iii) in Theorem 5.4. r Remark 5.. It is clear that the conclusions of Theorem 5., Theorem 5.2, Theorem 5.3 and Theorem 5.4 hold when f and g be two distinct nonconstant entire functions. Example 5.2. Let f ðzþ ¼2ðe z þ Þ, gðzþ ¼ ðe z þ Þ and a ¼ 2. Then it is

Meromorphic functions sharing three values 65 easily verified that f and g share 0; and y CM, N Þ ðr; =ð ÞÞ ¼ 0, Nðr; f Þ¼0, and N Þ r; ¼ Tðr; gþþsðr; gþ: g a Moreover, f and g do not assume one of the forms in Theorem 5.4. that the assumption (5.) in Theorem 5.4 is best possible. This illustrates 6. An application of the results in this paper. Let h be a nonconstant meromorphic function, and let S is a subset of distinct elements in extended complex plane. Define E h ðsþ ¼6 fz j hðzþ a ¼ 0g; a A S where each zero of hðzþ a ¼ 0 with multiplicity m is repeated m times in E h ðsþ (see [7]). In 982, F. Gross and C. Yang [8] asked whether there exist two sets S ¼fa ; a 2 g and S 2 ¼fb ; b 2 g such that for any two nonconstant entire functions f and g the conditions E f ðs j Þ¼E g ðs j Þ ðj ¼ ; 2Þ imply f g or not. F. Gross and C. Yang (see [8]) studied the question for the case a þ a 2 ¼ b þ b 2. In 990, H. Yi (see [9]) proved the following Theorem which is an extension and correction of the result of Gross and Yang. Theorem E (see [9]). Let S ¼fa ; a 2 g and S 2 ¼fb ; b 2 g be two pairs of distinct elements with a þ a 2 ¼ b þ b 2 ¼ c but a a 2 0 b b 2. Suppose that there are two nonconstant entire functions f and g of finite order such that E f ðs j Þ¼E g ðs j Þ for j ¼ ; 2. Then f and g must satisfy exactly one of the following relations: (i) f g, (ii) f þ g a þ a 2, (iii) ð f c=2þðg c=2þ Gðða a 2 Þ=2Þ 2, where c ¼ a þ a 2. This occurs only for ða a 2 Þ 2 þðb b 2 Þ 2 ¼ 0. (iv) ð j Þðg a k Þ ð Þ jþk ða a 2 Þ 2 for j; k ¼ ; 2. This occurs only for 3ða a 2 Þ 2 þðb b 2 Þ 2 ¼ 0. (v) ð f b j Þðg b k Þ ð Þ jþk ðb b 2 Þ 2 for j; k ¼ ; 2. This occurs only for ða a 2 Þ 2 þ 3ðb b 2 Þ 2 ¼ 0. In 998, Y. H. Li and C. T. Zhou [20] and independently P. Li [] proved the following theorem, which is an improvement and extension of Theorem E. Theorem F. Let S ¼fa ; a 2 g and S 2 ¼fb ; b 2 g be two pairs of distinct elements with a þ a 2 ¼ b þ b 2 but a a 2 0 b b 2, and let S 3 ¼fyg. Suppose that f and g are two nonconstant meromorphic functions satisfying E f ðs j Þ¼E g ðs j Þ for j ¼ ; 2; 3. Then the conclusions of Theorem E hold. The proofs of Theorem F are long in [] and [20]. Theorem F. Now we give a simple proof of

66 X.-M. Li and H.-X. Yi Let F ¼ ð f c=2þ2 ðða a 2 Þ=2Þ 2 ðg c=2þ 2 ðða a 2 Þ=2Þ 2 ððb b 2 Þ=2Þ 2 ; G ¼ 2 ðða a 2 Þ=2Þ ððb b 2 Þ=2Þ 2 ðða a 2 Þ=2Þ 2 ; ð6:þ where c ¼ a þ a 2 ¼ b þ b 2. If F G, from (6.) we have f g or f þ g a þ a 2 ; ð6:2þ which assume the forms (i) and (ii) in Theorem F. Next, suppose that F D G. From E f ðs j Þ¼E g ðs j Þðj¼; 2; 3Þ we know that F and G share 0; and y CM. From (6.), we have N Þ ðr; FÞ ¼0. Again by Lemma 2.3 we obtain Nðr; FÞ ¼Sðr; FÞ: ð6:3þ Set ðða a 2 Þ=2Þ 2 a ¼ ððb b 2 Þ=2Þ 2 ðða a 2 Þ=2Þ 2 : ð6:4þ From (6.) we have ð f c=2þ 2 2 F a ¼ ððb b 2 Þ=2Þ 2 ðða a 2 Þ=2Þ 2 ; G a ¼ ðg c=2þ ððb b 2 Þ=2Þ 2 ðða a 2 Þ=2Þ 2 : ð6:5þ From (6.5) we obtain N Þ r; ¼ 0; N Þ r; ¼ 0: F a G a ð6:6þ Noting (6.3) and (6.6), by Theorem 5.4 we know that one of the three cases in Theorem 5.4 holds. From this we obtain the form (iii), (iv) and (v) in Theorem F. This completes the proof of Theorem F. Acknowledgement. The authors want to express their thanks to the anonymous referee for his or her valuable suggestions. References [ ] W. K. Hayman, Meromorphic Functions, Oxford Math. Monogr., Clarendon Press, Oxford, 964. [ 2 ] H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, In: Pure Appl. Math. Monographs, No. 32, Science Press, Beijing, 995. [ 3 ] M. Ozawa, Unicity theorems for entire functions, J. d Anal. Math. J., 30 (976), 4 420. [ 4 ] H. Ueda, Unicity theorems for meromorphic or entire functions II, Kodai Math. J., 6 (983), 26 36. [ 5 ] G. Brosch, Eindeutigkeissätze für meromorphe Funktionen (in German), Thesis, Technical University of Aachen, February 989, Supervisor: Prof. Dr. L. Volkmann. [ 6 ] H. X. Yi, Meromorphic functions sharing three values, Chinese Ann. Math. Ser. A, 9 (988), 434 439. [ 7 ] H. X. Yi, Meromorphic functions that share two or three values, Kodai Math. J., 3 (990), 363 372. [ 8 ] H. X. Yi, Unicity theorems for meromorphic functions that share three values, Kodai Math. J., 8 (995), 300 34. [ 9 ] S. Z. Ye, Uniqueness of meromorphic functions that share three values, Kodai Math. J., 5 (992), 236 243.

Meromorphic functions sharing three values 67 [0] P. Li, Meromorphic functions sharing three values or sets CM, Kodai Math. J., 2 (998), 38 52. [] Q.-C. Zhang, Meromorphic functions sharing three values, Indian J. Pure Appl. Math., 30 (999), 667 682. [2] Amer H. H. Al-Khaladi, Unicity theorems for meromorphic functions, Kodai Math. J., 23 (2000), 27 34. [3] X. Hua and M. Fang, Meromorphic functions sharing four small functions, Indian J. Pure Appl. Math., 38 (997), 797 8. [4] Q.-C. Zhang, Meromorphic functions sharing values or sets (in Chinese), Doctoral Thesis, Shandong University, March 999, Supervisor: Prof. Hong-Xun Yi. [5] A. Z. Mokhon ko, On the Nevanlinna characteristic of some meromorphic functions, In: Theory of Function, Funct. Anal. Appl., 4 (97), 83 87. [6] H. X. Yi, On a question of Gross, Sci. China Ser. A, 38 (995), 8 6. [7] F. Gross, On the distribution of values of meromorphic functions, Trans. Amer. Math. Soc., 3 (968), 99 24. [8] F. Gross and C. Yang, Meromorphic functions covering certain finite sets at the same points, Illinois J. Math., 26 (982), 432 44. [9] H. X. Yi, On a result of Gross and Yang, Tohoku Math. J. (2), 42 (990), 49 428. [20] Y. H. Li and C. T. Zhou, Meromorphic functions sharing three finite sets CM, J. Shandong Univ. Nat. Sci. Ed., 33 (998), 40 45. Xiao-Min Li Department of Mathematics Shandong University Jinan, Shandong 25000 People s Republic of China E-mail: li-xiaomin@63.com Hong-Xun Yi Department of Mathematics Shandong University Jinan, Shandong 25000 People s Republic of China E-mail: hxyi@sdu.edu.cn