Uniqueness of meromorphic functions sharing two finite sets

Abstract We prove uniqueness theorems of meromorphic functions, which show how two meromorphic functions are uniquely determined by their two finite shared sets. This answers a question posed by Gross. Moreover, some examples are provided to demonstrate that all the conditions are necessary.


Introduction and main results
Throughout this paper, for a meromorphic function, the word "meromorphic" means meromorphic in the whole complex plane C. Let M.C/ (resp. E.C/) be the field of meromorphic (resp. holomorphic) functions in C. The order .f / and the lower order .f / of f 2 M.C/ are defined in turn as follows: .f / D lim sup where each zero of h.z/ a of multiplicity l appears l times in E.S; h/. The notation E.S; h/ expresses the set containing the same points as E.S; h/ but without counting multiplicities. Let f; g 2 M.C/. If E.S; f / D E.S; g/, then f and g share the set S CM (counting multiplicity). If E.S; f / D E.S; g/, then f and g share the set S IM (ignoring multiplicity). For fundamental concepts and results from Nevanlinna theory and further details related to M.C/, see [1,2]. In the sequel, we mainly consider a subset M 1 .C/ of M.C/ defined by M 1 .C/ D ff 2 M.C/jf has only finitely many poles in Cg: In 1976, Gross (see [3]) posed the following interesting question. ; q/ is as small as possible and minimise the number q such that any two elements f and g of G are algebraically dependent if E.S i ; f / D E.S i ; g/ for every i .i D 1; 2; ; q/, that is, if f and g share every S i .i D 1; 2; ; q/ CM (counting multiplicity).
In [4], Yi proved that there exist two finite sets S 1 (with 1 element) and S 2 (with 5 elements) of C such that any two elements f and g in E.C/ sharing S 1 and S 2 CM must be identically equal, which completely answered Question 1.1. In [5] and [6], Fang and Xu and independently Yi proved that there exist two finite sets S 1 (with 1 element) and S 2 (with 3 elements) of C such that any two elements f and g in E.C/ sharing S 1 and S 2 CM must be identically equal, which also answered Question 1.1.
For the case G D M.C/, choosing S i D fa i g .i D 1; 2; ; q/ for distinct elements a i of C [ f1g, when q 4, Question 1.2 was completely settled by famous four-value theorem due to Nevanlinna (see e.g. [7] or [1,2]). However, Question 1.2 is still interesting for the cases q Ä 3. In [8], Li and Yang proved that there exist two finite sets S 1 (with 15 elements) of C and S 2 D f1g such that any two elements f and g in M.C/ sharing S 1 and S 2 CM must be identically equal. In [9] and [10], Yi and independently Li and Yang proved that there exist two finite sets S 1 (with 11 elements) of C and S 2 D f1g such that any two elements f and g in M.C/ sharing S 1 and S 2 CM must be identically equal. In [11], Fang and Guo proved that there exist two finite sets S 1 (with 9 elements) of C and S 2 D f1g such that any two elements f and g in M.C/ sharing S 1 and S 2 CM must be identically equal. In [12], Yi proved that there exist two finite sets S 1 (with 8 elements) of C and S 2 D f1g such that any two elements f and g in M.C/ sharing S 1 and S 2 CM must be identically equal. In [4], Yi proved that there exist two finite sets S 1 (with 2 element) and S 2 (with 9 elements) of C such that any two elements f and g in M.C/ sharing S 1 and S 2 CM must be identically equal. In [13], Yi and Li recently proved that there exist two finite sets S 1 (with 2 element) and S 2 (with 5 elements) of C such that any two elements f and g in M.C/ sharing S 1 and S 2 CM must be identically equal.
For the family G D M 1 .C/, we solve Question 1.2 by proving the following theorems. ;˛kg, S 2 D fˇ1;ˇ2g, where˛1,˛2, ,˛k, If two nonconstant meromorphic functions f .z/ and g.z/ in M 1 .C/ share S 1 CM, S 2 IM, and if the order of f .z/ is neither an integer nor infinite, then f .z/ Á g.z/.
In order to state the next result, we need the following definition related to unique range set.  It is easy to verify that f .z/; g.z/ 2 M 1 .C/, f .z/ and g.z/ share S 1 , S 2 CM. But f .z/ 6 Á g.z/.

Some lemmas
In this section we present some important lemmas which will be needed in the sequel.   which imply that the order and the lower order of T 1 .r/ are not greater than the order and the lower order of T 2 .r/ respectively. Lemma 2.6 (see [2], Theorem 1.14). Let f .z/; g.z/ 2 M.C/. Then .f g/ Ä maxf .f /; .g/g; .f C g/ Ä maxf .f /; .g/g: Lemma 2.7 (see [2], Theorem 2.20). Let a 1 , a 2 , and a 3 be three distinct complex numbers in C [ f1g. If two nonconstant meromorphic functions f .z/ and g.z/ in M.C/ share a 1 , a 2 , and a 3 CM, and if the order of f .z/ and g.z/ is neither an integer nor infinite, then f .z/ Á g.z/. where H.z/ is a rational function such that V .z/ has neither a pole nor a zero in C. It is easy to see that such an H.z/ does exist since f .z/; g.z/ 2 M 1 .C/, and a possible pole or zero of .f .z/ ˛1/.f .z/ ˛2/ .f .z/ ˛k / .g.z/ ˛1/.g.z/ ˛2/ .g.z/ ˛k / may only come from a pole of f .z/ or g.z/, in view of the condition that f .z/ and g.z/ share S 1 D f˛1;˛2; ;˛kg CM. Then by Lemma 2.5 there exists an entire function .z/ 2 E.C/ such that Noting that f .z/ and g.z/ have only finitely many poles, we have N.r; f / D O.log r/; N.r; g/ D O.log r/: Since f .z/ and g.z/ share S 2 D fˇ1;ˇ2g IM, it follows from (2), the first and second fundamental theorems that r ! 1; r 6 2 E. Then by (3) and Lemma 2.4 we obtain Similarly, .g/ Ä .f /: Combining (4) with (5) yields .g/ D .f /: From the first fundamental theorem we have In view of the assumption that f .z/ and g.z/ share S 2 D fˇ1;ˇ2g IM, we deduce from (1) Thus from (9) and (10)  which contradicts the assumption.
Case 3. Suppose that (iii) occurs. Then using the same manner as in Case 2, we also get a contradiction. This completes the proof of Theorem 1.3.

Proof of Theorem 1.5
Note that if f and g share the set S CM (counting multiplicity) then f and g certainly share the set S IM (ignoring multiplicity). Then f and g satisfy the conditions in Theorem 1.3. Therefore the conclusion of Theorem 1.5 follows from Theorem 1.3. This completes the proof of Theorem 1.5.