This paper has twofolds. The first is to prove that there are at most two meromorphic functions sharing a small function with multiplicities truncated by 2 and other three small functions regardless of multiplicities, where all zeros with multiplicities more than a certain number are not counted. This result is an improvement of the four values theorems of Nevanlinna, Gundersen, Fujimito, Thai Tan and others. The second purpose of this paper is to prove that there are at most three meromorphic functions sharing four small functions ignoring multiplicity, where all zeros with multiplicities more than a certain number are omitted
FINITENESS PROBLEM OF MEROMORPHIC FUNCTIONS SHARING FOUR SMALL FUNCTIONS REGARDLESS OF MULTIPLICITIES SI DUC QUANG Abstract. This paper has twofolds. The first is to prove that there are at most two meromorphic functions sharing a small function with multiplicities truncated by 2 and other three small functions regardless of multiplicities, where all zeros with multiplicities more than a certain number are not counted. This result is an improvement of the four values theorems of Nevanlinna, Gundersen, Fujimito, Thai - Tan and others. The second purpose of this paper is to prove that there are at most three meromorphic functions sharing four small functions ignoring multiplicity, where all zeros with multiplicities more than a certain number are omitted. 1. Introduction Let f be a nonzero holomorphic function on C. For each z0 ∈ C, expanding f as i 0 f (z) = ∞ i=0 bi (z − z0 ) around z0 , then we define νf (z0 ) := min{i : bi = 0}. Let k be a positive integer or +∞. We set 0 νf,≤k (z) νf0 (z) 0 := if νf0 (z) ≤ k, if νf0 (z) > k. 0 0 Similarly, we define νf,>k and νf,=k . Let ϕ be a nonconstant meromorphic function on C with a reduced representation ϕ = (ϕ1 : ϕ2 ), where ϕ1 , ϕ2 are holomorphic functions on C having no common zeros ϕ1 0 ∞ . We define νϕ0 := νϕ0 1 , νϕ,≤k = νϕ0 1 ,≤k , νϕ∞ := νϕ0 2 and νϕ,≤k = νϕ0 2 ,≤k . The and ϕ = ϕ2 proximity function of ϕ is defined by: 2π 1 m(r, ϕ) := 2π log+ |ϕ(reiθ )|dθ (r > 1), 0 + here log x = max{0, log x} for x ∈ (0, ∞). Then the Nevanlinna characteristic function of ϕ is defined by T (r, ϕ) := m(r, ϕ) + N (r, νϕ∞ ), where by N (r, ν) we denote the counting function of the divisor ν. Let f, a be two meromorphic functions on C. The function a is said to be small (with respect to f ) if || T (r, a) = o(T (r, f )). Here, the notion || P means that the assertion P holds for all r ∈ [1, ∞) outside a finite Borel measure set. We denote by Rf the field of all small (with respect to f ) functions on C. 2010 Mathematics Subject Classification: Primary 32H30, 32A22; Secondary 30D35. Key words and phrases: meromorphic function, small function, truncated multiplicity. 1 2 SI DUC QUANG In 1926, Nevanlinna [2] showed that for two nonconstant meromorphic functions f and g on C, if they share four distinct values with multiplicity, then that g is a m¨obius transformation of f . In [1] Fujimoto improved the result of Nevanlinna by proving that there are at most two meromorphic functions on C which share four distinct values with multiplicities truncated by 2. Later, Thai and Tan [7] generalized the result of Fujimoto to the case of small functions. In [5] Quang improved the above mentioned results by considering the case where the meromorphic functions share a small functions with multiplicities truncated to level 2 and share other three small functions regardless of multiplicity. His result is stated as follows. Theorem A. (see [5, Theorem 1.2]) Let f1 , f2 , f3 be three nonconstant meromorphic functions on C. Let a1 , . . . , a4 be distinct small (with respect to fi , ∀1 ≤ i ≤ 3) functions on C. Assume that (i) min{νf01 −ai , 1} = min{νf02 −ai , 1} = min{νf03 −ai , 1} ∀1 ≤ i ≤ 3, (ii) min{νf01 −a4 , 2} = min{νf02 −a4 , 2} = min{νf03 −a4 , 2}. Then f1 = f2 or f2 = f3 or f3 = f1 . We see that, in the above result, all zeros of the functions (fs − ai ) are considered. The first purpose of this paper is to improve this result by omitting all such zeros with multiplicity more than a certain number. Namely, we will prove the following. Theorem 1.1. Let f1 , f2 , f3 be three nonconstant meromorphic functions on C. Let a1 , . . . , a4 be distinct small (with respect to fi , ∀1 ≤ i ≤ 3) functions on C. Let k1 , . . . , k4 be four positive integers or +∞ with 14 3 4 i=1 1 +3 ki 4 i=1 1 28 1 < + , ki + 1 3 3k0 where k0 = max1≤i≤4 ki . Assume that (i) min{νf01 −a1 ,≤k1 , 2} = min{νf02 −a1 ,≤k1 , 2} = min{νf03 −a, ≤k1 , 2}, (ii) min{νf01 −ai ,≤ki , 1} = min{νf02 −ai ,≤ki , 1} = min{νf03 −ai ,≤ki , 1} ∀2 ≤ i ≤ 4. Then f1 = f2 or f2 = f3 or f3 = f1 . Let k1 = · · · = k4 = k, we have the following corollary. Corollary 1.2. Let f1 , f2 , f3 be three nonconstant meromorphic functions on C. Let a1 , . . . , a4 be distinct small (with respect to fi , ∀1 ≤ i ≤ 3) functions on C. Let k ≥ 24 be a positive integer or +∞. Assume that (i) min{νf01 −a1 ,≤k , 2} = min{νf02 −a1 ,≤k , 2} = min{νf03 −a, ≤k , 2}, (ii) min{νf01 −ai ,≤k , 1} = min{νf02 −ai ,≤k , 1} = min{νf03 −ai ,≤k , 1} ∀2 ≤ i ≤ 4. Then f1 = f2 or f2 = f3 or f3 = f1 . However, as far as we know, there has no finiteness theorem for the case meromorphic functions sharing four small functions regardless of multiplicities. The second purpose of this paper is to prove a such finiteness theorem. For detail, we will show that there are at FINITENESS PROBLEM OF MEROMORPHIC FUNCTIONS SHARING FOUR SMALL FUNCTIONS 3 most three meromorphic functions sharing four small functions regardless of multiplicity. To state our result, we need the following. Let f be a nonconstant meromorphic function on C. Let a1 , . . . , a4 be four distinct small (with respect to f ) functions and let k1 , . . . , k4 be four positive integers or maybe +∞. We consider the set F(f, {ai , ki }4i=1 , 1) of all meromorphic functions g defined on C satisfying 0 min{1, νf0−ai ,≤ki (z)} = min{1, νg−a (z)} (1 ≤ i ≤ 4) i ,≤ki for every z ∈ C. Theorem 1.3. Let f be a nonconstant meromorphic function on C. Let a1 , . . . , a4 be four meromorphic functions which are small with respect to f . Let k1 , . . . , k4 be positive integers or maybe +∞ with 25 64 4 i=1 17 1 + ki 16 4 i=1 1 25 1 < + . ki + 1 32 64k0 Then F (f, {ai , ki }4i=1 , 1) ≤ 3. Letting k1 = · · · = k4 = k, we have the following corollary. Corollary 1.4. Let f be a nonconstant meromorphic function on C. Let a1 , . . . , a4 be four meromorphic functions which are small with respect to f . Let k > 314 be positive integers or maybe +∞. Then F (f, {ai , k}4i=1 , 1) ≤ 3. 2. Some second main theorems and lemmas For a divisor ν on C, which we may regard as a function on C with values in Z whose support is discrete subset of C, and for a positive integer M (maybe M = ∞) , we define the counting function of ν as follows n[M ] (t) = min{M, ν(z)} |z|≤t r N [M ] n(t) dt (1 < r < ∞). t (r, ν) = 1 For brevity we will omit the character (M ) if M = ∞. For a divisor ν and a positive integer k (maybe k = +∞), we define ν≤k (z) = ν(z) 0 if ν(z) ≤ k and ν>k (z) = otherwise ν(z) 0 if ν(z) > k otherwise. Let ϕ be a meromorphic function on C. Define: • νϕ0 (resp. νϕ∞ ) the divisor of zeros (resp. divisor of poles) of ϕ. • νϕ = νϕ0 − νϕ∞ . 0 0 • νϕ,≤k = (νϕ0 )≤k , νϕ,>k = (νϕ0 )>k . 4 SI DUC QUANG ∞ ∞ , νϕ,≤k , νϕ,≤k and their counting functions. , νϕ,≤k Similarly, we define νϕ,≤k For a discrete subset S ⊂ C, we consider it as a reduced divisor (denoted again by S) whose support is S, and denote by N (r, S) its counting function. Let f be a meromorphic function on C. A function h is said to be equal Sf (r) if for every positive number > 0, there exists a Lebesgue subset I ⊂ [1, +∞) of finite measure such that h(r) ≤ T (r, f ) for every r ∈ I. A function a is said to be small with respect to f if || T (r, a) = o(T (r)). Hence if a is small with respect to f then T (r, a) = Sf (r) The following second main theorem is due to Yamanoi [8]. Theorem 2.1 (see [8, Corollary 1]). Let f be a nonconstant meromorphic function on C. Let a1 , . . . , aq (q ≥ 3) be q distinct meromorphic functions on C. Then the following holds q N [1] (r, νf0−ai ) + Sf (r), || (q − 2)T (r, f ) ≤ i=1 Lemma 2.2. Let f be a nonconstant meromorphic function on C and let a be a small function with respect to f then for every positive integer k (maybe k = +∞). We have 1 || N [1] (r, νf0−a,>k ) ≤ T (r, f ) + o(T (r, f )) k+1 1 k + 1 [1] N (r, νf0−a ) − T (r, f ) + o(T (r, f )). and || N [1] (r, νf −ai ,≤k ) ≥ k k Proof. First, we have || N [1] (r, νf0−a,>k ) ≤ 1 1 N (r, νf0−a,>k ) ≤ T (r, f ) + o(T (r, f )). k+1 k+1 Second, we have || N [1] (r, νf −ai ,≤k ) = N [1] (r, νf −ai ) − N [1] (r, νf −ai ,>k ) 1 N (r, νf0−a,>k ) ≥ N [1] (r, νf −ai ) − k+1 1 = N [1] (r, νf −ai ) − (N (r, νf0−a ) − N (r, νf0−a,≤k )) k+1 1 (T (r, f ) − N [1] (r, νf −ai ,≤k )) + o(T (r, f )). ≥ N [1] (r, νf −ai ) − k+1 Thus k + 1 [1] 1 N (r, νf −ai ) − T (r, f ) + o(T (r, f )). || N [1] (r, νf −ai ,≤k ) ≥ k k The lemma is proved. Lemma 2.3. Let f be a meromorphic function on C and let a1 , . . . , a4 be four distinct small (with respect to f ) meromorphic functions. Let k1 , . . . , k4 be four positive integers or maybe +∞ such that 4 1 < 2. k +1 i=1 i FINITENESS PROBLEM OF MEROMORPHIC FUNCTIONS SHARING FOUR SMALL FUNCTIONS 5 Then for every g ∈ F(f, {ai , ki }4i=1 , 1), we have || T (r, f ) = O(T (r, g)) and || T (r, g) = O(T (r, f )). Proof. By Lemma 2.1 we have 4 N [1] (r, νf0−ai ) + Sf (r) || 2T (r, f ) ≤ i=1 4 ≤ i=1 4 ≤ i=1 4 ≤ i=1 ki N [1] (r, νf0−ai ,≤ki ) + ki + 1 ki 0 N [1] (r, νg−a )+ i ,≤ki ki + 1 ki T (r, g) + ki + 1 4 i=1 4 i=1 4 i=1 1 T (r, f ) + Sf (r) ki + 1 1 T (r, f ) + Sf (r) ki + 1 1 T (r, f ) + Sf (r). ki + 1 Thus 4 || 2 − i=1 1 T (r, f ) ≤ ki + 1 4 i=1 ki T (r, g) + Sf (r). ki + 1 This implies that || T (r, f ) = O(T (r, g)). Similarly, we have || T (r, g) = O(T (r, g)). The lemma is proved. 3. Proof of Theorem 1.1 Let f1 , f2 , f3 and {ai }4i=1 be as in Theorem 1.1. By Lemma 2.3, we see that || T (r, fk ) = O(T (r, fl )) (1 ≤ k, l ≤ 3). We set T (r) = T (r, f1 ) + T (r, f2 ) + T (r, f3 ). For i, j ∈ {1, . . . , 4}, we put Fijk = f k − ai . Then f k − aj T (r, Fijk ) = T (r, fk ) + o(T (r)). We define • S = i=j {z; ai (z) = aj (z)} (then S has counting function small with respect to fs (s = 1, 2, 3)), • νi = {z; νf01 −ai ,≤ki (z) > 0}, Si = ∪3s=1 {z; νf0s −ai ,>ki (z) > 0}, • Ti : the set of all z ∈ νi such that νf0s −ai ,≤ki (z) ≥ νf0t −ai ,≤ki (z) = νf0l −ai ,≤ki (z) = 1 for a permutation (s, t, l) of (1, 2, 3), • Ti = νi \ Ti • µi : the set of all z ∈ νi such that νf0s −ai ,≤ki (z) = νf0t −ai ,≤ki (z) = νf0l −ai ,≤ki (z) 6 SI DUC QUANG Now we recall the Cartan’s auxiliary function (see [1, Definition 3.1]). Let F, G, H be three nonzero meromorphic functions, we define Cartan’s auxiliary function by Φ(F, G, H) := F · G · H · (3.1) =F ( H1 ) 1 H − ( G1 ) 1 G 1 1 1 1 F ( F1 ) 1 G ( G1 ) 1 H ( H1 ) ( F1 ) +G 1 F − ( H1 ) 1 H +H ( G1 ) 1 G − ( F1 ) 1 F . It is easy to see that, for every meromorphic function h we have the following property Φ(hF, hG, hH) = h · Φ(F, G, H). Lemma 3.2. With the same assumption as in Theorem 1.1, if f1 , f2 , f3 are distinct then 4 4 N (r, Ti ) ≤ ( i=1 i=1 1 2 − )T (r) + ki k0 3 Sfs (r). s=1 Proof. For k, l ∈ {1, 2, 3}, k = l, we have 4 N (r, min{νf0k −ai ,≤ki , νf0l −ai ,≤ki }) ≤ N (r, νf0k −fl ) ≤ T (r, fk ) + T (r, fl ). i=1 Therefore 4 N (r, min{νf0k −ai ,≤ki , νf0l −ai ,≤ki }) ≤ 2T (r). (3.3) i=1 1≤k 0}, Si = ∪4s=1 {z; νf0s −ai ,>ki (z) > 0}, • Ti : the set of all z ∈ νi such that νf0s1 −ai ,≤ki (z) ≥ νf0s2 −ai ,≤ki (z) = · · · = νf0s3 −ai ,≤ki (z) = 1 for a permutation (s1 , s2 , s3 , s4 ) of (1, 2, 3, 4), • Ti = νi \ Ti 14 SI DUC QUANG Lemma 4.1. With the same assumption as in Theorem 1.1, if f1 , f2 , f3 , f4 are distinct then 4 4 N (r, Ti ) ≤ ( i=1 i=1 1 2 − )T (r) + ki k0 4 Sfs (r). s=1 Proof. For k, l ∈ {1, 2, 3, 4}, k = l, we have 4 N (r, min{νf0k −ai ,≤ki , νf0l −ai ,≤ki }) ≤ N (r, νf0k −fl ) ≤ T (r, fk ) + T (r, fl ). i=1 Therefore 4 N (r, min{νf0k −ai ,≤ki , νf0l −ai ,≤ki }) ≤ 3T (r). (4.2) i=1 1≤k[...]... Meromorphic functions sharing small functions as targets, Internat J Math 16 (2005), 437-451 [8] K Yamanoi, The second main theorem for small functions and related problems, Acta Math 192 (2004), 225-294 ¯k)(β,f ) = E ¯k) (β, g), [9] W Yao, Two meromorphic functions sharing five small functions in the sense E Nagoya Math J 167 (2002), 35-54 [10] H-X Yi, On one problem of uniqueness of meromorphic functions. . .FINITENESS PROBLEM OF MEROMORPHIC FUNCTIONS SHARING FOUR SMALL FUNCTIONS1 1 In order to prove the above inequality, it is enough to show that the inequality 3 νΦ∞ + 2χνj ≤ χSi ∪Sj + (3.6) νF∞k ij k=1 holds outside a discrete set of counting function small with respect to fs (s = 1, 2, 3) For each z ∈ S, we see that if νΦ∞ (z) + 2χTj (z) > 0 then z must be zero of some (fs − ai ) or zero of some... {s1 , s2 , s3 } = {1, 2, 3, 4} \ {s} Then 4 3 N (r, νv ) ≤ N (r, µsi ) + 2 v=1 v∈{i} T (r, fst ) + N (r, Si ∪ Sj ) + o(T (r)) t=1 FINITENESS PROBLEM OF MEROMORPHIC FUNCTIONS SHARING FOUR SMALL FUNCTIONS1 7 Proof of Theorem 1.3 Suppose that f1 , f2 , f3 , f4 are distinct functions By Lemma 4.3, for each 1 ≤ s ≤ 4 and i = j, Φ(Fijs1 , Fijs2 , Fijs3 ) ≡ 0, where {s1 , s2 , s3 } = {1, 2, 3, 4} \ {s} Then... (3.9) ≤ 5T (r) + 2 4 ≤ 5T (r) + 2( i=1 N (r, Sj ) + o(T (r)) j=1 4 N (r, Ti ) + 3 i=1 4 N (r, Sj ) + o(T (r)) i=1 2 1 − )T (r) + 3 ki k0 4 i=1 1 T (r) + ki + 1 3 Sfs (r) s=1 FINITENESS PROBLEM OF MEROMORPHIC FUNCTIONS SHARING FOUR SMALL FUNCTIONS1 3 On the other hand we have 4 || 8 N (r, νs ) = s=1 3 8 3 (N [1] (r, νf0k −as ,≤ks ) k=1 s=1 3 8 ≥ 3 (3.10) 4 4 ( k=1 s=1 ks + 1 [1] 1 N (r, νf0k −as ) − T (r,... min{1, νf0s −ai ,≤ki (z)} + χTi , s=1 for every z This yields that N (r, min{νf0k −ai ,≤ki , νf0l −ai ,≤ki }) ≥ 1≤k ... show that there are at FINITENESS PROBLEM OF MEROMORPHIC FUNCTIONS SHARING FOUR SMALL FUNCTIONS most three meromorphic functions sharing four small functions regardless of multiplicity To state... there has no finiteness theorem for the case meromorphic functions sharing four small functions regardless of multiplicities The second purpose of this paper is to prove a such finiteness theorem... Si ∪ Sj ) + o(T (r)) t=1 FINITENESS PROBLEM OF MEROMORPHIC FUNCTIONS SHARING FOUR SMALL FUNCTIONS1 7 Proof of Theorem 1.3 Suppose that f1 , f2 , f3 , f4 are distinct functions By Lemma 4.3, for