Volume 2011, Article ID 873184, 16 pagesdoi:10.1155/2011/873184 Research Article Normality Criteria of Lahiri’s Type and Their Applications Xiao-Bin Zhang,1 Jun-Feng Xu,1, 2and Hong-Xun
Trang 1Volume 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, 2and Hong-Xun Yi1
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,xujunf@gmail.com
Received 22 September 2010; Revised 9 January 2011; Accepted 9 February 2011
Academic Editor: Siegfried Carl
Copyrightq 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
1 Introduction and Main Results
is assumed that the reader is familiar with the standard notion used in the Nevanlinna value
distribution theory such as the characteristic function Tr, f, the proximity function mr, f, the counting function Nr, f see, e.g., 1 4, and Sr, f denotes any quantity that satisfies the condition Sr, f oTr, f as r → ∞ outside of a possible exceptional set of finite linear measure A meromorphic function az is called a small function with respect to fz, provided that Tr, a Sr, f.
Let fz and gz be two nonconstant meromorphic functions Let az and bz be
g have the same poles counting multiplicity If gz −bz 0 whenever fz −az 0, we write fz az ⇒ gz bz If fz az ⇒ gz bz and gz bz ⇒ fz az,
share a.
Trang 2P
f
f n n1···n k ,
M1
f, f, , f k
f n
fn1
· · ·f kn k
,
M2
f, f, , f k
f m
fm1
· · ·f km k
,
γ M1 n n1 · · · n k , γ M2 m m1 · · · m k ,
γ M∗
1k−1
j1
n j , ΓM1k
j1
jn j , γ M∗
2k−1
j1
m j , ΓM2k
j1
jm j ,
1.1
where n, n1, , n k , m, m1, , m k are nonnegative integers M i f, f, , f k is called the differential monomial of f and γM i is called the degree of M i f, f, , f k i 1, 2.
∞
According to Bloch’s principle, every condition which reduces a meromorphic
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,
f af n / b for all z ∈ , then f must be a constant.
Theorem B see 6,7 Let n ≥ 3 be an integer, a, b ∈ , a / 0, and let F, be a family of meromorphic
functions in a domain D If f af n / b for all f ∈ F, then F is a normal family.
In 2005, Lahiri8 got a normality criterion as follows
that a / 0 Define
E f
f z b
If there exists a positive constant M such that |fz| ≥ M for all f ∈ F whenever z ∈ E f , then F is a
normal family.
criteria of Lahiri’s type
Trang 3Theorem D Let F be a family of meromorphic functions in a complex domain D Let a, b ∈ such
that a / 0 Let m1, m2, n1, n2 be positive integers such that m1n2 − m2n1 > 0, m1 m2 ≥ 1,
n1 n2≥ 2, and put
E f
z ∈ D :f zn1
fzm1 a
f zn2
If there exists a positive constant M such that |fz| ≥ M for all f ∈ F whenever z ∈ E f , then F is a
normal family.
that a / 0 Let m1, m2, n1, n2be nonnegative integers such that m1n2 m2n1, and put
E f
z ∈ D :f zn1
fzm1 a
f zn2
If there exists a positive constant M such that |fz| ≥ M for all f ∈ F whenever z ∈ E f , then F is a
normal family.
with f k; they got
Theorem F Let F be a family of meromorphic functions in a complex domain D, all of whose zeros
have multiplicity at least k Let a, b ∈ such that a / 0 Let m1, m2, n1, n2be nonnegative integers such that m1n2− m2n1> 0, m1 m2≥ 1, n1 n2 ≥ 2, (if n1 n2 1, k ≥ 5), and put
E f
z ∈ D :f zn1
f k zm1
f zn2
If there exists a positive constant M such that |fz| ≥ M for all f ∈ F whenever z ∈ E f , then F 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 Let a, b ∈ such that a / 0 Let m1≥ 2, m2, n1, n2be positive integers such that m1n2 m2n1, and put
E f
z ∈ D :f zn1
f k zm1 a
f zn2
If there exists a positive constant M such that |fz| ≥ M for all f ∈ F whenever z ∈ E f , then F is a
normal family.
besides Zalcman-Pang’s Lemma It’s natural to ask whether such normality criteria of Lahiri’s
Trang 4type still hold for the general differential monomial Mf, f, , f k We study this problem and obtain the following theorem
Theorem 1.1 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 Let a, b ∈ such that a / 0, let m, n, k≥ 1, m j ,
n j j 1, 2, , k be nonnegative integers such that
γ M2ΓM1− γ M1ΓM2> 0, n k m k > 0, m n ≥ 2. 1.7
Put
E f
f, f, , f k
M2
f, f, , f k b 1.8
If there exists a positive constant M such that |fz| ≥ M for all f ∈ F whenever z ∈ E f , then F 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 Let a, b ∈ such that a / 0, let m, n, k≥ 1, m j ,
n j j 1, 2, , k be nonnegative integers such that mnm k n k γ M∗
1γ 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 Put
E f
f, f, , f k
M2
f, f, , f k b 1.9
If there exists a positive constant M such that |fz| ≥ M for all f ∈ F whenever z ∈ E f , then F is a
normal family.
Theorem 1.3 Let F be a family of holomorphic functions in a domain D, for every f ∈ F, all zeros of
f have multiplicity at least k Let a, b / 0 be two finite values and n, k, n1, , n k be nonnegative integers with n ≥ 1, k ≥ 1, n k ≥ 1 For every f ∈ F, all zeros of f have multiplicity at least k, if
P f a ⇔ M1f, f, , f k b, then F is normal in D.
Example 1.4 Let D {z : |z| < 1} and F {f m } If a 0, let f m : emz For each function
f ∈ F, Pf and M1f, f, , f k share 0 in D However, it can be easily verified that F is
Example 1.5 Let D {z : |z| < 1} and F {f m } If a / 0, let f m : meλz − e −λz, where
λ is the root of z2 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 1.5shows that the
Trang 52 Preliminary Lemmas
Lemma 2.1 see 11 Let F be a family of meromorphic functions on the unit disc Δ, all of whose
zeros have the multiplicity at least k, then if F is not normal, there exist, for each 0 ≤ α < k
a a number r, 0 < r < 1,
b points z n , |z n | < r,
c functions f n ∈ F, and
such that ρ n −α f n z n ρ n ξ g n ξ → gξ locally uniformly with respect to the spherical metric,
where g ξ is a nonconstant meromorphic function on , all of whose zeros have multiplicity at least
k, such that g#ξ ≤ g#0 Here, as usual, g#z |gz|/1 |gz|2 is the spherical derivative.
D ⊂ Then F is normal in D if and only if the spherical derivatives of functions f ∈ F are 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 ∈ , then f has the order at most one.
Lemma 2.4 see 13 Take nonnegative integers n, n1, , n k with n ≥ 1, n1 n2 · · · n k≥ 1
and define d nn1n2· · ·n k Let f be a transcendental meromorphic function with the deficiency
δ 0, f > 3/3d1 Then for any nonzero value c, the function f n fn1· · · f kn k −c has infinitely
many zeros Moreover, if n ≥ 2, the deficient condition can be omitted.
Lemma 2.5 Take nonnegative integers n, n1, , n k with n ≥ 1, n k ≥ 1 and define d n n1
n2 · · · n k Let f be a transcendental meromorphic function whose zeros have multiplicity at least k Then for any nonzero value c, the function f n fn1· · · f kn k −c has infinitely many zeros, provided
that n1 n2 · · · n k−1≥ 1 and k / 2 when n 1 Specially, if f is transcendental entire, the
function f n fn1· · · f kn k − c has infinitely many zeros.
Proof If n1 n2 · · · n k−1 0, then f n fn1· · · f kn k f n f kn k, this case has been consideredsee 5,12–20
Lemma 2.4 Next we consider the case n 1
LetΨ f n fn1· · · f kn k Using the proof ofLemma 2.4see 13, page 161–163 ,
we obtain
3d − 2Tr, f
≤ 3dN
r,1 f
N
r, 1 f
4N
r, 1
Ψ − c
N
r, Ψ − c
Ψ
− 3N
r, Ψ − c
Ψ
Sr, f
.
2.1
Trang 6Suppose that z0 is a zero of f of multiplicity p≥ k, then z0 is a zero of Ψ of multiplicity
dp− Σk
j1jn j, and thus is a pole ofΨ − c/Ψof multiplicity dp− Σk
j1jn j− 1 Thereby, from
2.1 we get
3d − 2Tr, f
≤
⎛
j1
jn j 5
⎞
f
4N
r, 1
Ψ − c
Sr, f
j1jn j 5
r, 1 f
4N
r, 1
Ψ − c
Sr, f
.
2.2
k− 5 3k−1
j1
k − jn j
r, f
≤ 4N
r, 1
Ψ − c
Sr, f
If k 1, then Ψ f n fn1; this case has been proved as mentioned abovesee 13–16
If k ≥ 5, then we have k − 5 3k−1
j1k − jn j > 0; the conclusion is evident.
If 3≤ k ≤ 4, note that n1 n2 · · · n k−1≥ 1 and we deduce that k − 5 3k−1
j1k−
j nj > 0, thus the conclusion holds.
thatsee Hu et al 21, page 67
dT
r, f
≤ dN
r,1 f
N
r, 1
Ψ − c
− N
r, Ψ − c
Ψ
Sr, f
With similar discussion as above, we obtain
⎛
⎝n
k−1
j1
k − jn j− 1
k
⎞
⎠Tr, f ≤ N r, Ψ − c1
Sr, f
In view of n ≥ 1 and k ≥ 2, we get n k−1
j1k − jn j − 1/k > 0, thus we immediately obtain
Lemma 2.6 Take nonnegative integers n, n1, , n k , k with n ≥ 1, n k ≥ 1, k ≥ 1 and define
d n n1 n2 · · · n k Let f be a nonconstant rational function whose zeros have multiplicity
at least k Then for any nonzero value c, the function f n fn1· · · f kn k − c has at least one finite
zero.
Proof Since the case k 1 has been proved by Charak and Rieppo 9, we only need to
Trang 7Suppose that f n fn1· · · f kn k − c has no zero.
Case 1 If f is a nonconstant polynomial, since the zeros of f have multiplicity at least k, we
know that f n fn1· · · f kn k is also a nonconstant polynomial, so f n fn1· · · f kn k − c has
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 − a1m1z − a2m2· · · z − a sm s
where A is a nonzero constant and m i ≥ k i 1, 2, , s, l j ≥ 1 j 1, 2, , t.
Then by mathematical induction, we get
f k z A z − a1m1−k z − a2m2−k · · · z − a sm s −k g k z
z − b1l1k z − b2l2k · · · z − b tl t k , 2.7
where g k z M −NM −N −1 · · · M −N −k 1z k st−1 c m z k st−1−1 · · ·c0, c m , , c0
are constants and
It is easily obtained that
g k
Combining2.6 and 2.7 yields
f n
fn1
· · ·f kn k
A d z − a1dm1 −k
j1jn j · · · z − a sdm s−k
j1jn j g z
z − b1dl1 k
j1jn j · · · z − b tdl tk
j1jn j
j1g n j
j1jn j s t − 1.
Moreover, gz is not a constant, or else, we get g j is a constant for j 1, , k The
If g1is a constant, then we get
If g kis a constant, then we get
Trang 8Then from2.11, we obtain
f n
fn1· · ·f k
n k
A d z − a1dm1 −k
j1jn j−1· · · z − a sdm s−k
j1jn j−1h z
z − b1dl1 k
j1jn j1· · · z − b tdl tk
j1jn j 1 s t − 1.
Since f n fn1· · · f kn k − c / 0, we obtain from 2.11 that
f n
fn1
· · ·f kn k
z − b1dl1 k
j1jn j · · · z − b tdl tk
j1jn j
where B is a nonzero constant Then
f n
fn1
· · ·f kn k
z − b1dl1 k
j1jn j1· · · z − b tdl tk
where Hz is a polynomial with degH t − 1.
From2.14 and 2.16, we deduce that
dM−
⎛
j1
jn j 1
⎞
dks≤k
j1
namely
nksk
j1
Hence f n fn1· · · f kn k − c has at least one finite zero.
Remark 2.7. Lemma 2.6is a generalization of Lemma 2.2 in10 The proof ofLemma 2.6is
zero of f n is of multiplicity at least nk, but this does not mean that each zero of f n f kmis
implies that the proof of Lemma 2.2 in10 is not correct
Trang 9Lemma 2.8 Let a, b ∈ such that a / 0 Let m, n, k≥ 1, m j , n j j 1, 2, , k be nonnegative
integers such that mnm k n k γ M∗
1γ 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 Let f be a meromorphic function in ; all zeros of f have multiplicity
at least k Define
f, f, , f k
M2
f, f, , f k − b. 2.20
Then Φz has a finite zero.
Proof The algebraic complex equation
x a
M1fz0, fz0, , f k z0 x0
m n m
n , m j n j m
Thus
Φz0 M1
M1m/n
f z0, fz0, , f k z0 − b 0 2.23
Lemma 2.9 see 2, page 51 If f is an entire function of order σf, then
σ
f
lim sup
r→ ∞
log ν
r, f
where ν r, f denotes the central-index of fz.
Lemma 2.10 see 22, page 187–199 or 2, page 51 If g is a transcendental entire function, let
0 < δ < 1/4 and z be such that |z| r and that |gz| Mr, gνr, g −1/4δ holds Then there exists a set F⊂ of finite logarithmic measure, that is,
F dt/t < ∞ such that
g m z
g z
ν
r, g
z
holds for all m ≥ 0 and all r / ∈ F.
Trang 103 Proof of Theorem 1.1
normal at z0 ∈ D ByLemma 2.1, for 0≤ α < k, there exist r < 1, z j ∈ Δ such that z j → z0,
f j ∈ F and ρ j → 0 such that g j ξ ρ −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 For simplicity, we denote f j z j ρ j ξ by f j By Lemmas2.4
g ξ0n
gξ0n1· · ·g k ξ0n k a
g ξ0m
gξ0m1· · ·g k ξ0m k 0. 3.1
We have
g j ξ n
g jξn1
· · ·g j k ξn k
g j ξ m
gj ξm1· · ·g k j ξm k − ρ
αγ M2−ΓM2
ρ −αγ M1ΓM1
j f j n
f jn1
· · ·f j kn k
ρ −αγ j M2ΓM2 f j m
f j
m1
· · ·f j k
m k − ρ αγ M2−ΓM2
ρ αγ M2−ΓM2
j
⎡
⎢
⎣ρ −αγ M1 γ M2ΓM1ΓM2
f jn1
· · ·f j kn k
f m j
f jm1
· · ·f j km k − b
⎤
⎥
⎦.
3.2
Let α ΓM1 ΓM2/γ M1 γ M2 < k, and under the assumption γ M2ΓM1 − γ M1ΓM2 > 0, we
obtain
g n
gn1
· · ·g kn k
g m
gm1
is the uniform limit of
ρ γ j M2ΓM1 −γ M1ΓM2 /γ M1 γ M2
⎡
⎢
⎣f n j
f jn1
· · ·f j kn k
f m j
f jm1
· · ·f j km k − b
⎤
⎥
values of j and ζ j z j ρ j ξ j,
f j
ζ j
n
f j
ζ j
n1
· · ·f j k
ζ j
n k
f j
ζ j
m
f j
ζ j
m1
· · ·f j k
ζ j
...holds for all m ≥ and all r / ∈ F.
Trang 103 Proof of Theorem 1.1
normal...
Proof Since the case k has been proved by Charak and Rieppo 9, we only need to
Trang 7Suppose... kmis
implies that the proof of Lemma 2.2 in10 is not correct
Trang 9Lemma 2.8 Let a, b