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
2015
Korean Mathematical Society
345
346
I. LAHIRI AND R. PAL
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
348
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.
349
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.
350
I. LAHIRI AND R. PAL
[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]