Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 873184, 16 pages
doi:10.1155/2011/873184
Research Article
Normality Criteria of Lahiri’s Type and
Their Applications
Xiao-Bin Zhang,
1
Jun-Feng Xu,
1, 2
and Hong-Xun Yi
1
1
Department of Mathematics, Shandong University, Jinan, Shandong 250100, China
2
Department of Mathematics, Wuyi University, Jiangmen, Guangdong 529020, China
Correspondence should be addressed to Jun-Feng Xu, [email protected]
Received 22 September 2010; Revised 9 January 2011; Accepted 9 February 2011
Academic Editor: Siegfried Carl
Copyright q 2011 Xiao-Bin Zhang et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We prove two normality criteria for families of some functions concerning Lahiri’s type, the results
generalize those given by Charak and Rieppo, Xu and Cao. As applications, we study a problem
related to R. Br
¨
uck’s Conjecture and obtain a result that generalizes those given by Yang and
Zhang, L
¨
u, Xu and Chen.
M
1
f, f
, ,f
k
f
n
f
n
1
···
f
k
n
k
,
M
2
f, f
, ,f
··· m
k
,
γ
∗
M
1
k−1
j1
n
j
, Γ
M
1
k
j1
jn
j
,γ
∗
M
2
k−1
j1
i
is called the degree of M
i
f, f
, ,f
k
i 1, 2.
Let F be a family of meromorphic functions defined in a domain D ⊂
. F is said to
be normal in D, in the sense of Montel, if for any sequence f
n
∈F, there exists a subsequence
f
n
j
such that f
n
j
converges spherically locally uniformly in D, to a meromorphic function or
∞.
According to Bloch’s principle, every condition which reduces a meromorphic
function in
to a constant makes a family of meromorphic functions in a domain D normal.
Although the principle is false in general, many authors proved normality criteria for families
of meromorphic functions starting from Picard type theorems, for instance.
Theorem A see 5. Let n ≥ 5 be an integer, a, b ∈
and a
/
0. If, for a meromorphic function f,
z
a
f
z
b
.
1.2
If there exists a positive constant M such that |fz|≥M for all f ∈Fwhenever z ∈ E
f
,thenF is a
normal family.
In 2009, Charak and Rieppo 9 generalized Theorem C and obtained two normality
criteria of Lahiri’s type.
Journal of Inequalities and Applications 3
Theorem D. Let F be a family of meromorphic functions in a complex domain D.Leta, b ∈
such
that a
/
0.Letm
1
, m
2
, n
1
, n
n
1
f
z
m
1
a
f
z
n
2
f
z
m
2
z ∈ D :
f
z
n
1
f
z
m
1
a
f
z
n
2
1
, n
2
be nonnegative integers
such that m
1
n
2
− m
2
n
1
> 0, m
1
m
2
≥ 1, n
1
n
2
≥ 2,(ifn
1
n
2
1, k ≥ 5), and put
E
f
z ∈ D :
z
m
2
b
. 1.5
If there exists a positive constant M such that |fz|≥M for all f ∈Fwhenever z ∈ E
f
,thenF is a
normal family.
Theorem G. Let F be a family of meromorphic functions in a complex domain D, all of whose zeros
have multiplicity at least k.Leta, b ∈
such that a
/
0.Letm
1
≥ 2, m
2
, n
1
, n
2
be positive integers
such that m
1
n
2
f
z
n
2
f
k
z
m
2
b
. 1.6
If there exists a positive constant M such that |fz|≥M for all f ∈Fwhenever z ∈ E
f
,thenF is a
normal family.
To prove Theorems D–G, the authors used a key lemma Lemma 2.4 in this paper
besides Zalcman-Pang’s Lemma. It’s natural to ask whether such normality criteria of Lahiri’s
4 Journal of Inequalities and Applications
type still hold for the general differential monomial Mf, f
, ,f
k
E
f
z ∈ D : M
1
f, f
, ,f
k
a
M
2
f, f
, ,f
k
b
. 1.8
If there exists a positive constant M such that |fz|≥M for all f ∈Fwhenever z ∈ E
f
,thenF is a
normal family.
Theorem 1.2. Let F be a family of meromorphic functions in a complex domain D, for every f ∈F,
all zeros of f have multiplicity at least k.Leta, b ∈
, 1 ≤ j ≤ k.Put
E
f
z ∈ D : M
1
f, f
, ,f
k
a
M
2
f, f
, ,f
k
b
. 1.9
If there exists a positive constant M such that |fz|≥M for all f ∈Fwhenever z ∈ E
f
,thenF is a
normal family.
, ,f
k
share 0 in D. However, it can be easily verified that F is
not normal in D. Example 1.4 shows that the condition b
/
0inTheorem 1.3 is sharp.
Example 1.5. Let D {z : |z| < 1} and F {f
m
}.Ifa
/
0, let f
m
: me
λz
− e
−λz
,where
λ is the root of z
2
b/a. For each function f ∈F, f
b/af, f
n1
a ⇔ f
n
f
b in
D. However, it can be easily verified that F is not normal in D. Example 1.5 shows that the
multiplicity restriction on zeros of f in Theorem 1.3 is sharp at least for k 2.
k,suchthatg
#
ξ ≤ g
#
0. Here, as usual, g
#
z|g
z|/1 |gz|
2
is the spherical derivative.
Lemma 2.2 see 1, page 158. Let F {f} be a family of meromorphic functions in a domain
D ⊂
.ThenF is normal in D if and only if the spherical derivatives of functions f ∈Fare uniformly
bounded on each compact subset of D.
Lemma 2.3 see 12. Let f be an entire function and M a positive integer. If f
#
z ≤ M for all
z ∈
,thenf has the order at most one.
Lemma 2.4 see 13. Take nonnegative integers n, n
1
, , n
k
with n ≥ 1, n
1
n
2
··· n
k
1
n
2
··· n
k
.Letf be a transcendental meromorphic function whose zeros have multiplicity at l east k.
Then for any nonzero value c, th e function f
n
f
n
1
···f
k
n
k
−c has infinitely many zeros, provided
that n
1
n
2
··· n
k−1
≥ 1 and k
/
2 when n 1. Specially, if f is transcendental entire, the
function f
f
n
f
k
n
k
, this case has been
considered see 5, 12–20.
If n
1
n
2
··· n
k−1
≥ 1andifn ≥ 2, we immediately get the conclusion from
Lemma 2.4. Next we consider the case n 1.
Let Ψf
n
f
n
1
···f
k
n
k
. Using the proof of Lemma 2.4 see 13, page 161–163,
r,
Ψ − c
Ψ
− 3N
r,
Ψ − c
Ψ
S
r, f
.
2.1
6 Journal of Inequalities and Applications
Suppose that z
0
is a zero of f of multiplicity p≥ k,thenz
0
is a zero of Ψ of multiplicity
dp − Σ
k
j1
jn
j
r,
1
f
4
N
r,
1
Ψ − c
S
r, f
≤
3
k
j1
jn
j
5
k
N
r,
1
f
r,
1
Ψ − c
S
r, f
.
2.3
If k 1, then Ψf
n
f
n
1
; this case has been proved as mentioned above see 13–16.
If k ≥ 5, then we have k − 5 3
k−1
j1
k − jn
j
> 0; the conclusion is evident.
If 3 ≤ k ≤ 4, note that n
1
n
2
Ψ − c
Ψ
S
r, f
.
2.4
With similar discussion as above, we obtain
⎛
⎝
n
k−1
j1
k − j
n
j
− 1
k
⎞
⎠
T
r, f
2
··· n
k
.Letf be a nonconstant rational function whose zeros have multiplicity
at least k. Then for any nonzero value c, the function f
n
f
n
1
···f
k
n
k
− c hasatleastonefinite
zero.
Proof. Since the case k 1 has been proved by Charak and Rieppo 9, we only need to
consider k ≥ 2.
Journal of Inequalities and Applications 7
Suppose that f
n
f
n
1
···f
k
at least one zero, which contradicts our assumption.
Case 2. If f is a nonconstant rational function but not a polynomial. Set
f
z
A
z − a
1
m
1
z − a
2
m
2
···
z − a
s
m
s
z − b
1
z − a
1
m
1
−k
z − a
2
m
2
−k
···
z − a
s
m
s
−k
g
k
z
z − b
1
, c
m
, ,c
0
are constants and
m
1
m
2
··· m
s
M ≥ ks, 2.8
l
1
l
2
··· l
t
N ≥ t. 2.9
It is easily obtained that
deg
g
k
≤ k
s t − 1
. 2.10
···
z − a
s
dm
s
−
k
j1
jn
j
g
z
z − b
1
dl
1
k
j1
jn
j
···
j
is a constant for j 1, ,k.The
leading coefficient of g
j
is M − N − j − 1s t.
If g
1
is a constant, then we get
M N. 2.12
If g
k
is a constant, then we get
k − 1
s t
0, 2.13
which implies k 1, a contradiction with the assumption k ≥ 2.
8 Journal of Inequalities and Applications
Then from 2.11,weobtain
f
n
f
n
1
s
−
k
j1
jn
j
−1
h
z
z − b
1
dl
1
k
j1
jn
j
1
···
z − b
t
/
0, we obtain from 2.11 that
f
n
f
n
1
···
f
k
n
k
c
B
z − b
1
dl
1
k
j1
jn
n
k
B · H
z
z − b
1
dl
1
k
j1
jn
j
1
···
z − b
t
dl
t
t − 1, 2.17
in view of degh ≥ s t 1, together with 2.8,wehave
dks ≤
k
j1
jn
j
s,
2.18
namely
nks
k
j1
k − j
n
j
s ≤ 0.
2.19
which is a contradiction since n ≥ 1.
Hence f
n
f
n
1
, n
j
j 1, 2, ,k be nonnegative
integers such that mnm
k
n
k
γ
∗
M
1
γ
∗
M
2
> 0,(k
/
2 when n 1 or m 1), m/n m
j
/n
j
for all positive
integers m
j
and n
j
, 1 ≤ j ≤ k.Letf be a meromorphic function in ;allzerosoff have multiplicity
at least k.Define
Φ
0
∈ . By Lemmas 2.5 and 2.6,thedifferential
monomial M
1
f, f
, ,f
k
cannot avoid it and thus there exists z
0
∈ such tha t
M
1
fz
0
,f
z
0
, ,f
k
z
0
x
0
.
Under the assumptions, for all positive integers m, n, m
j
, n
j
0
, ,f
k
z
0
a
M
m/n
1
f
z
0
,f
z
0
, ,f
k
z
dt/t < ∞ such that
g
m
z
g
z
ν
r, g
z
m
1 o
1
2.25
holds for all m ≥ 0 and all r/∈ F.
10 Journal of Inequalities and Applications
3. Proof of Theorem 1.1
Without loss of generality, we may assume D Δ{z : |z| < 1}. Suppose that F is not
normal at z
ρ
j
ξ by f
j
. By Lemmas 2.4
and 2.6,thereexistsξ
0
∈ such that
g
ξ
0
n
g
ξ
0
n
1
···
g
k
ξ
m
k
0.
3.1
Obviously, gξ
0
/
0, ∞, so in some neighborhood of ξ
0
, g
j
ξ converges uniformly to gξ.
We have
g
j
ξ
n
g
j
ξ
1
···
g
k
j
ξ
m
k
− ρ
αγ
M
2
−Γ
M
2
j
b
ρ
−αγ
M
1
Γ
M
1
j
f
f
j
m
1
···
f
k
j
m
k
− ρ
αγ
M
2
−Γ
M
2
j
b
ρ
αγ
M
2
−Γ
M
f
k
j
n
k
a
f
m
j
f
j
m
1
···
f
k
j
m
k
− b
⎤
⎥
n
g
n
1
···
g
k
n
k
a
g
m
g
m
1
···
g
k
m
f
j
n
1
···
f
k
j
n
k
a
f
m
j
f
j
m
1
···
f
ζ
j
n
f
j
ζ
j
n
1
···
f
k
j
ζ
j
n
k
a
k
b.
3.5
Journal of Inequalities and Applications 11
Hence for all large values of j, ζ
j
z
j
ρ
j
ξ
j
∈ E
f
, it follows from the condition that
g
j
ξ
j
f
j
g
j
ξ
− g
ξ
< 1. 3.7
By 3.7,weget
K ≥
g
ξ
j
≥
g
j
j
→ 0asj →∞.
This completes the proof of Theorem 1.1.
4. Proof of Theorem 1.2
Without loss of generality, we may assume D Δ{z : |z| < 1}. Suppose that F is not
normal in D.ByLemma 2.1,for0≤ α<k,thereexistr<1, z
j
∈ Δ, f
j
∈Fand ρ
j
→ 0
such
that g
j
ξρ
−α
j
f
j
z
j
ρ
j
ξ → gξ locally uniformly with respect to the spherical metric,
where gξ is a nonconstant meromorphic function on
, all of whose zeros have multiplicity
at least k.ByLemma 2.8,weget
g
ξ
0
m
g
ξ
0
m
1
···
g
k
ξ
0
m
k
− b 0,
4.1
for some ξ
0
∈ .
and |fz|≥M for all f ∈Fwhenever z ∈ E
f
; by Lemmas 2.5 and 2.6, using the similar proof
of Theorem 1.1, we obtain the conclusion.
Case 2 a 0.Forf ∈F,iffz
0
0forz
0
∈ D,sincePf0 ⇒ M
1
f, f
, ,f
k
b,we
have b 0, which is a contradiction, hence f
/
0.
If M
1
fz
0
,f
z
0
, ,f
k
z
0
j
∈F,
and ρ
j
→ 0
such that g
j
ξρ
−Γ
M
1
/γ
M
1
j
f
j
z
j
ρ
j
ξ → gξ locally uniformly with respect
to the spherical metric, where gξ is a nonconstant entire function, all of whose zeros have
multiplicity at least k. By Hurwitz’s theorem, we have
i g ≡ 0org
/
0, and
ii g
n
/
0, d are two constants. Thus
g
n
ξ
g
n
1
ξ
···
g
k
n
k
ξ
c
Γ
M
1
g
k
n
k
/
b,
5.3
but by Lemmas 2.5 and 2.6 we get a contradiction again.
This proves Theorem 5.1.
Further more, using Theorem 1.3, we obtain a uniqueness theorem related to R. Br
¨
uck’s
Conjecture. Firstly, we recall this conjecture.
R. Br
¨
uck’s Conjecture
Let f be a nonconstant entire function such that the hyper-order σ
2
f is not a positive integer and
σ
2
f < ∞.If f and f
shareafinitevaluea CM, then
f
− a
f − a
c,
in Br
¨
uck’s Conjecture.
Recently, Yang and Zhang 29 considered this problem and got the following theorem.
Theorem H. Let f be a nonconstant entire function. n ≥ 7 be an integer, and let F f
n
.IfF and F
share 1 CM, then F F
,andf assumes the form
f
z
ce
z/n
,
5.6
where c is a nonzero constant.
L
¨
uetal.30 improves Theorem H and obtained the following theorem.
Theorem I. Let Q
1
/
≡ 0 be a polynomial, and let n ≥ 2 be an intege; let fz be a transcendental
entire function, and let Fzfz
n
f
n
1
···f
k
n
k
b,then
f
n
f
n
1
···
f
k
n
k
− b
f
nn
1
,z∈ D Δ. 5.9
Under the assumptions of Theorem 1.3,wegetthatF is a normal family of holomorphic
functions in D.ByLemma 2.2,thereexistsaconstantM such that
f
#
ω
f
ω
1
f
ω
2
.HencebyLemma 2.3, f has the order at most 1.
14 Journal of Inequalities and Applications
Since f
nn
1
···n
k
a f
n
f
n
1
···f
k
n
k
b, f must be a transcendental entire
function and
f
n
f
n
1
···
n
e
iθ
n
,wededuce
lim
r
n
→∞
1
f
z
n
lim
r
n
→∞
1
M
r
n
,f
0.
5.12
By Lemma 2.10,thereexistsasubsetF
1
⊂ 1, ∞ of finite logarithmic measure, namely
z
n
ν
r
n
,f
z
n
k
1 o
1
,
5.13
as r →∞.
Rewrite 5.11 as
f
/f
|
log e
αz
n
Γ
M
1
log ν
r
n
,f
− log
r
n
o
1
≤ O
log r
n
,
5.15
as r
n
→∞.Sinceαz is a polynomial, from 5.15,wededucethatαz is a constant. Let
e
α
c,thenc is a nonzero constant. Thus
f
n
f
n
1
···
n
1
···
f
k
n
k
.
5.17
Journal of Inequalities and Applications 15
Suppose that z
0
is a zero of f with multiplicity p≥ k,thenz
0
is a zero of f
n
1
···n
k
with
multiplicity n
1
···n
k
p,andazerooff
n
Γ
M
1
− c
e
γ
M
1
tz
b
1 − c
;
5.18
hence c 1andt
Γ
M
1
c 1. f c
1
e
ωz
, ω is the root of t
Γ
M
1
1.
This completes the proof of Theorem 5.1.
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n
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atica Iberoamericana, vol. 11, no. 2, pp. 355–373, 1995.
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