Bull. Korean Math. Soc. 52 (2015), No. 2, pp. 345–350 http://dx.doi.org/10.4134/BKMS.2015.52.2.345
A NOTE ON NEVANLINNA’S FIVE VALUE THEOREM Indrajit Lahiri and Rupa Pal Abstract. In the paper we prove a uniqueness theorem which improves and generalizes a number of uniqueness theorems for meromorphic functions related to Nevanlinna’s five value theorem.
1. Introduction, definitions and results In the paper, by meromorphic functions we always mean meromorphic functions in the open complex plane C. Let f be a non-constant meromorphic function. A meromorphic function a = a(z) is said to be a small function of f if either a ≡ ∞ or T (r, a) = S(r, f ). We denote by S(f ) the collection of all small functions of f . Clearly C ∪ {∞} ⊂ S(f ) and S(f ) is a field over the set of complex numbers. For a positive integer p and a ∈ S(f ) we denote by E p) (a; f ) the set of those distinct zeros of f − a whose multiplicities do not exceed p, where we mean by a zero of f − ∞ a pole of f . Also by E ∞) (a; f ) we denote the set of all distinct zeros of f − a. For A ⊂ C we denote by N A (r, a; f ) the reduced counting function of those zeros of f − a which belong to the set A, where a ∈ S(f ). For a positive integer p and a ∈ S(f ) we denote by Np) (r, a; f ) (N p) (r, a; f )) the counting function (reduced counting function) of those zeros of f − a whose multiplicities do not exceed p. Similarly we define N(p (r, a; f ) and N (p (r, a; f )). For standard definitions and notations of Nevanlinna theory we refer the reader to [4]. The modern theory of uniqueness of entire and meromorphic functions was initiated by R. Nevanlinna with his two famous theorems: The Five Value Theorem and The Four Value Theorem. The five value theorem of Nevanlinna may be stated as follows: Theorem A ([4, p. 48]). Let f and g be two non-constant meromorphic functions and aj ∈ C ∪ {∞} be distinct for j = 1, 2, . . . , 5. If E ∞) (aj ; f ) = E ∞) (aj ; g) for j = 1, 2, . . . , 5, then f ≡ g. Received June 16, 2012. 2010 Mathematics Subject Classification. 30D35. Key words and phrases. uniqueness, meromorphic function, small function. c
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In 1976 H. S. Gopalakrishna and S. S. Bhoosnurmath [3] improved Theorem A in the following manner. Theorem B ([3]). Let f , g be distinct non-constant meromorphic functions. If there exist distinct elements a1 , a2 . . . . , ak of C∪{∞} such that E pj ) (aj ; f ) = E pj ) (aj ; g) for j = 1, 2, . . . , k, where p1 , p2 , . . . , pk are positive integers or ∞ Pk pj p1 with p1 ≥ p2 ≥ · · · ≥ pk , then j=2 1+p ≤ 2 + 1+p . j 1
As a consequence of Theorem B we obtain the following result, which is an improvement over Theorem A. Theorem C. Let f and g be two non-constant meromorphic functions. Suppose that there exist distinct elements a1 , a2 , . . . , a5 in C ∪ {∞} such that E pj ) (aj ; f ) = E pj ) (aj ; g) for j = 1, 2, . . . , 5, where p1 , p2 , . . . , p5 are positive integers or ∞ with p1 ≥ p2 ≥ · · · ≥ p5 . If p3 ≥ 3 and p5 ≥ 2, then f ≡ g. C. C. Yang [8, p. 157] improved Theorem A by considering partial sharing of values and proved the following theorem. Theorem D ([8, p. 157]). Let f and g be two non-constant meromorphic functions such that E ∞) (aj ; f ) ⊂ E ∞) (aj ; g) for five distinct elements a1 , a2 , . . . , a5 of C ∪ {∞}. If lim inf r→∞
P5
Pj=1 5
j=1
N (r,aj ;f ) N (r,aj ;g)
> 21 , then f ≡ g.
In 2000 Y. Li and J. Qiao [5] improved Theorem A by considering shared small functions instead of shared values. Their result may be stated as follows: Theorem E. Let f , g be non-constant meromorphic functions and aj ∈ S(f )∩ S(g) be distinct for j = 1, 2, . . . , 5. If E ∞) (aj ; f ) = E ∞) (aj ; g) for j = 1, 2, . . . , 5, then f ≡ g. In 2007 T. B. Cao and H. X. Yi [1] further improved Theorem E and also improved a result of D. D. Thai and T. V. Tan [6]. Following is the result of Cao and Yi. Theorem F ([1]). Let f and g be two non-constant meromorphic functions and aj ∈ S(f ) ∩ S(g) be distinct for j = 1, 2, . . . , k. Suppose further that p1 , p2 , . . . , pk be positive integers or ∞ such that p1 ≥ p2 ≥ · · · ≥ pk and E pj ) (aj ; f ) = E pj ) (aj ; g) for j = 1, 2, . . . , k. Then f ≡ g, if one of the following holds: (i) k = 7, (ii) k = 6 and p3 ≥ 2, (iii) k = 5, p3 ≥ 3 and p5 ≥ 2, (iv) k = 5 and p4 ≥ 4, (v) k = 5, p3 ≥ 5 and p4 ≥ 3, (vi) k = 5, p3 ≥ 6 and p4 ≥ 2. In the same year T. G. Chen, K. Y. Chen and Y. L. Tsai [2] improved Theorem D in the following manner. Theorem G ([2]). Let f and g be two non-constant meromorphic functions such that E ∞) (aj ; f ) ⊂ E ∞) (aj ; g) for distinct elements a1 , a2 , . . . , ak (k ≥ 5) of S(f ) ∩ S(g). If lim inf r→∞
Pk
Pj=1 k
j=1
N (r,aj ;f ) N(r,aj ;g)
>
1 k−3 ,
then f ≡ g.
A NOTE ON NEVANLINNA’S FIVE VALUE THEOREM
347
In the paper we prove the following theorem which includes all the above mentioned results. Theorem 1.1. Let f , g be two non-constant meromorphic functions and aj = aj (z) ∈ S(f ) ∩ S(g) be distinct for j = 1, 2, . . . , k (k ≥ 5). Suppose that p1 ≥ p2 ≥ · · · ≥ pk are positive integers or infinity and δ(≥ 0) is such that k 1 1 X 1 1 . + 1+ + 1 + δ < (k − 2) 1 + p1 p1 j=2 1 + pj p1 P Let Aj = E pj ) (aj ; f )\E pj ) (aj ; g) for j = 1, 2, . . . , k. If kj=1 N Aj (r, aj ; f ) ≤ δT (r, f ) and Pk j=1 N pj ) (r, aj ; f ) lim inf Pk r→∞ j=1 N pj ) (r, aj ; g) p1 , > P 1 (1 + p1 )(k − 2) − p1 (1 + δ) − 1 − (1 + p1 ) kj=2 1+p j then f ≡ g.
After the discovery of the second fundamental theorem for moving targets by K. Yamanoi [7], it becomes indispensable for proving “Five Value” type uniqueness theorems for shared small functions. So we mention below the result of Yamanoi. Lemma 1.1. Let f be a non-constant meromorphic function and aj ∈ S(f ) be distinct for j = 1, 2, . . . , k. Then for any ε(> 0) (k − 2 − ε)T (r, f ) ≤
k X
N (r, aj ; f ) + S(r, f ).
j=1
2. Proof of Theorem 1.1 Proof. Let f 6≡ g. Then by Lemma 1.1 we get for ε(> 0) (k − 2 − ε)T (r, f ) ≤
k X
N (r, aJ ; f ) + S(r, f )
j=1
=
k X j=1
≤
k X N pj ) (r, aj ; f ) + j=1
≤
N pj ) (r, aj ; f ) + N (pj +1 (r, aj ; f ) + S(r, f ) 1 N(pj +1 (r, aj ; f ) + S(r, f ) 1 + pj
k X pj 1 N pj ) (r, aj ; f ) + N (r, aj ; f ) + S(r, f ) 1 + pj 1 + pj j=1
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I. LAHIRI AND R. PAL
≤
k X j=1
i.e., (2.1)
k−2−
k X j=1
k X pj 1 T (r, f ) + S(r, f ) N pj ) (r, aj ; f ) + 1 + pj 1 + p j j=1
k X 1 pj N pj ) (r, aj ; f ). − ε + o(1) T (r, f ) ≤ 1 + pj 1 + pj j=1
Similarly k k X X 1 pj (2.2) k−2− N pj ) (r, aj ; g). − ε + o(1) T (r, g) ≤ 1 + pj 1 + pj j=1
j=1
Let Bj = E pj ) (aj ; f )\Aj for j = 1, 2, . . . , k. Now using (2.1) and (2.2) we get for a sequence of values of r tending to +∞ k X
N pj ) (r, aj ; f ) =
k X
N Aj (r, aj ; f ) +
N Bj (r, aj ; f )
j=1
j=1
j=1
k X
≤ δT (r, f ) + N (r, 0; f − g) ≤ (1 + δ)T (r, f ) + T (r, g) + O(1) i.e., k k X X 1 N pj ) (r, aj ; f ) − ε + o(1) k−2− 1 + pj j=1
j=1
(2.3) ≤ (1 + δ)
k X j=1
k X pj pj Npj ) (r, aj ; f ) + {1 + o(1)} Np ) (r, aj ; g). 1 + pj 1 + pj j j=1
p1 1+p1
p2 pk Since 1 ≥ ≥ 21 , we get from (2.3) for a sequence ≥ 1+p ≥ · · · ≥ 1+p 2 k of values of r tending to +∞ k k X X 1 (1 + δ)p1 k−2− N pj ) (r, aj ; f ) −ε− + o(1) 1 + pj 1 + p1 j=1
≤ {1 + o(1)}
p1 1 + p1
j=1
k X
N pj ) (r, aj ; g).
j=1
Since ε(> 0) is arbitrary, this implies Pk j=1 N pj ) (r, aj ; f ) lim inf Pk r→∞ j=1 N pj ) (r, aj ; g) p1 ≤ Pk (1 + p1 )(k − 2) − p1 (1 + δ) − 1 − (1 + p1 ) j=2
1 1+pj
,
A NOTE ON NEVANLINNA’S FIVE VALUE THEOREM
which is a contradiction. Therefore f ≡ g. This proves the theorem.
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3. Consequences of Theorem 1.1 In this section we discuss some consequences of Theorem 1.1. Pk
N (r,aj ;f )
1 Corollary 3.1. Let pk = ∞ and A = lim inf r→∞ Pj=1 > k−3 . If k j=1 N (r,aj ;g) Pk 1 j=1 N Aj (r, aj ; f ) ≤ δT (r, f ) for some δ, 0 ≤ δ < k − 3 − A , then f ≡ g.
If we suppose E ∞) (aj ; f ) ⊂ E ∞) (aj ; f ) for j = 1, 2, . . . , k, then Aj = ∅ for j = 1, 2, . . . , k and so we can choose δ = 0. Therefore Corollary 3.1 is an improvement over Theorem G. Corollary 3.2. Let f , g be distinct non-constant meromorphic functions. If there exist distinct members a1 , a2 , . . . , ak of S(f )∩S(g) such that E pj ) (aj ; f ) ⊂ E pj ) (aj ; g) for j = 1, 2, . . . , k, where p1 , p2 , . . . , pk are positive integers or Pk pj p1 ∞ with p1 ≥ p2 ≥ · · · ≥ pk , then j=2 1+pj ≤ A(1+p1 ) + 2, where A =
lim inf r→∞
Pk
j=1
N pj ) (r,aj ;f )
j=1
N pj ) (r,aj ;g)
Pk
.
Proof. Since Aj = ∅ for j = 1, 2, . . . , k, putting δ = 0 we get from Theorem 1.1 p1 ≥A Pk 1 (1 + p1 )(k − 2) − p1 − 1 − (1 + p1 ) j=2 1+p j Pk pj p1 and so j=2 1+pj ≤ A(1+p1 ) + 2. This proves the corollary.
If we suppose that E pj ) (aj ; f ) = E pj ) (aj ; g), then A = 1 and so Corollary 3.2 improves B. If, further, we suppose that k = 5, p3 ≥ 3 and p5 ≥ 2, P5 Theorem pj p1 > then j=1 1+pj 1+p1 + 2 and so by Corollary 3.2 we get f ≡ g. Hence Corollary 3.2 improves Theorem C. Also if we put p1 = p2 = · · · = p5 = ∞, then Theorem E follows from Corollary 3.2. Under the hypotheses of Theorem Pk pj p1 F we see that A = 1, j=2 1+p > 1+p + 2 and so by Corollary 3.2 we get j 1 f ≡ g. So Corollary 3.2 includes Theorem F. References [1] T. B. Cao and H. X. Yi, On the multiple values and uniqueness of meromorphic functions sharing small functions as targets, Bull. Korean Math. Soc. 44 (2007), no. 4, 631–640. [2] T. G. Chen, K. Y. Chen, and Y. L. Tsai, Some generalizations of Nevanlinna’s five value theorem, Kodai Math. J. 30 (2007), no. 3, 438–444. [3] H. S. Gopalakrishna and S. S. Bhoosnurmath, Uniqueness theorems for meromorphic functions, Math. Scand. 39 (1976), no. 1, 125–130. [4] W. K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964. [5] Y. Li and J. Qiao, The uniqueness of meromorphic functions concerning small functions, Sci. China Ser. A 43 (2000), no. 6, 581–590. [6] D. D. Thai and T. V. Tan, Meromorphic functions sharing small functions as targets, Internat. J. Math. 16 (2005), no. 4, 437–451.
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[7] K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), no. 2, 225–294. [8] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Science Press (Beijing/New York) and Kluwer Academic Publishers (Dordrecht/Boston/London), 2003. Indrajit Lahiri Department of Mathematics University of Kalyani West Bengal 741235, India E-mail address:
[email protected] Rupa Pal Department of Mathematics Jhargram Raj College Jhargram, Midnapur(W), West Bengal 721507, India Present Address: PG Department of Mathematics Bethune College 181 Bidhan Sarani, Kolkata 700006, India E-mail address:
[email protected]
