- Research Article
- Open access
- Published:
Normality Criteria of Lahiri's Type and Their Applications
Journal of Inequalities and Applications volume 2011, Article number: 873184 (2011)
Abstract
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. Brück's Conjecture and obtain a result that generalizes those given by Yang and Zhang, Lü, Xu and Chen.
1. Introduction and Main Results
Let denote the complex plane, and let
be a nonconstant meromorphic function in
. It is assumed that the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as the characteristic function
, the proximity function
, the counting function
(see, e.g., [1–4]), and
denotes any quantity that satisfies the condition
as
outside of a possible exceptional set of finite linear measure. A meromorphic function
is called a small function with respect to
, provided that
.
Let and
be two nonconstant meromorphic functions. Let
and
be small functions of
and
.
means
and
have the same zeros (counting multiplicity) and
means that
and
have the same poles (counting multiplicity). If
whenever
, we write
. If
and
, we write
. If
, then we say that
and
share
.
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ1_HTML.gif)
where ,
are nonnegative integers.
is called the differential monomial of
and
is called the degree of
.
Let be a family of meromorphic functions defined in a domain
.
is said to be normal in
, in the sense of Montel, if for any sequence
, there exists a subsequence
such that
converges spherically locally uniformly in
, 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
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 be an integer,
and
. If, for a meromorphic function
,
for all
, then
must be a constant.
Let be an integer,
,
, and let
, be a family of meromorphic functions in a domain
. If
for all
, then
is a normal family.
In 2005, Lahiri [8] got a normality criterion as follows.
Theorem C.
Let be a family of meromorphic functions in a complex domain
. Let
such that
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ2_HTML.gif)
If there exists a positive constant such that
for all
whenever
, then
is a normal family.
In 2009, Charak and Rieppo [9] generalized Theorem C and obtained two normality criteria of Lahiri's type.
Theorem D.
Let be a family of meromorphic functions in a complex domain
. Let
,
such that
. Let
,
,
,
be positive integers such that
,
,
, and put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ3_HTML.gif)
If there exists a positive constant such that
for all
whenever
, then
is a normal family.
Theorem E.
Let be a family of meromorphic functions in a complex domain
. Let
,
such that
. Let
,
,
,
be nonnegative integers such that
, and put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ4_HTML.gif)
If there exists a positive constant such that
for all
whenever
, then
is a normal family.
Very recently, Xu and Cao [10] further extended Theorems D and E by replacing with
; they got
Theorem F.
Let be a family of meromorphic functions in a complex domain
, all of whose zeros have multiplicity at least
. Let
such that
. Let
,
,
,
be nonnegative integers such that
,
,
, (if
,
), and put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ5_HTML.gif)
If there exists a positive constant such that
for all
whenever
, then
is a normal family.
Theorem G.
Let be a family of meromorphic functions in a complex domain
, all of whose zeros have multiplicity at least
. Let
such that
. Let
,
,
,
be positive integers such that
, and put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ6_HTML.gif)
If there exists a positive constant such that
for all
whenever
, then
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 type still hold for the general differential monomial . We study this problem and obtain the following theorem
Theorem 1.1.
Let be a family of meromorphic functions in a complex domain
, for every
, all zeros of
have multiplicity at least
. Let
,
such that
, let
,
,
,
,
be nonnegative integers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ7_HTML.gif)
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ8_HTML.gif)
If there exists a positive constant such that
for all
whenever
, then
is a normal family.
Theorem 1.2.
Let be a family of meromorphic functions in a complex domain
, for every
, all zeros of
have multiplicity at least
. Let
,
such that
, let
,
,
,
,
be nonnegative integers such that
, (
when
or
),
for all positive integers
and
,
. Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ9_HTML.gif)
If there exists a positive constant such that
for all
whenever
, then
is a normal family.
As an application of Theorem 1.1, we obtain the following theorem.
Theorem 1.3.
Let be a family of holomorphic functions in a domain
, for every
, all zeros of
have multiplicity at least
. Let
be two finite values and
be nonnegative integers with
,
,
. For every
, all zeros of
have multiplicity at least
, if
, then
is normal in
.
Example 1.4.
Let and
. If
, let
. For each function
,
and
share 0 in
. However, it can be easily verified that
is not normal in
. Example 1.4 shows that the condition
in Theorem 1.3 is sharp.
Example 1.5.
Let and
. If
, let
, where
is the root of
. For each function
,
,
in
. However, it can be easily verified that
is not normal in
. Example 1.5 shows that the multiplicity restriction on zeros of
in Theorem 1.3 is sharp (at least for
).
2. Preliminary Lemmas
Lemma 2.1 (see [11]).
Let be a family of meromorphic functions on the unit disc
, all of whose zeros have the multiplicity at least
, then if
is not normal, there exist, for each
(a)a number ,
,
(b)points ,
,
(c)functions , and
(d)positive numbers
such that locally uniformly with respect to the spherical metric, where
is a nonconstant meromorphic function on
, all of whose zeros have multiplicity at least
, such that
. Here, as usual,
is the spherical derivative.
Lemma 2.2 (see [1, page 158]).
Let be a family of meromorphic functions in a domain
. Then
is normal in
if and only if the spherical derivatives of functions
are uniformly bounded on each compact subset of
.
Lemma 2.3 (see [12]).
Let be an entire function and
a positive integer. If
for all
, then
has the order at most one.
Lemma 2.4 (see [13]).
Take nonnegative integers with
,
and define
. Let
be a transcendental meromorphic function with the deficiency
. Then for any nonzero value
, the function
has infinitely many zeros. Moreover, if
, the deficient condition can be omitted.
The following two lemmas can be seen as supplements of Lemma 2.4.
Lemma 2.5.
Take nonnegative integers with
,
and define
. Let
be a transcendental meromorphic function whose zeros have multiplicity at least
. Then for any nonzero value
, the function
has infinitely many zeros, provided that
and
when
. Specially, if
is transcendental entire, the function
has infinitely many zeros.
Proof.
If , then
, this case has been considered (see [5, 12–20]).
If and if
, we immediately get the conclusion from Lemma 2.4. Next we consider the case
.
Let . Using the proof of Lemma 2.4 (see [13, page 161–163] ), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ10_HTML.gif)
Suppose that is a zero of
of multiplicity
, then
is a zero of
of multiplicity
, and thus is a pole of
of multiplicity
. Thereby, from (2.1) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ11_HTML.gif)
Note that , we deduce from (2.2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ12_HTML.gif)
If , then
; this case has been proved as mentioned above (see [13–16]).
If , then we have
; the conclusion is evident.
If , note that
and we deduce that
, thus the conclusion holds.
If is a transcendental entire function, we only need to consider the case
. Note that (see Hu et al. [21, page 67])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ13_HTML.gif)
With similar discussion as above, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ14_HTML.gif)
In view of and
, we get
, thus we immediately obtain the conclusion. This completes the proof of Lemma 2.5.
Lemma 2.6.
Take nonnegative integers with
,
,
and define
. Let
be a nonconstant rational function whose zeros have multiplicity at least
. Then for any nonzero value
, the function
has at least one finite zero.
Proof.
Since the case has been proved by Charak and Rieppo [9], we only need to consider
.
Suppose that has no zero.
Case 1.
If is a nonconstant polynomial, since the zeros of
have multiplicity at least
, we know that
is also a nonconstant polynomial, so
has at least one zero, which contradicts our assumption.
Case 2.
If is a nonconstant rational function but not a polynomial. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ15_HTML.gif)
where is a nonzero constant and
,
.
Then by mathematical induction, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ16_HTML.gif)
where ,
are constants and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ18_HTML.gif)
It is easily obtained that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ19_HTML.gif)
Combining (2.6) and (2.7) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ20_HTML.gif)
where with
.
Moreover, is not a constant, or else, we get
is a constant for
. The leading coefficient of
is
.
If is a constant, then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ21_HTML.gif)
If is a constant, then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ22_HTML.gif)
which implies , a contradiction with the assumption
.
Then from (2.11), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ23_HTML.gif)
where is a polynomial with
.
Since , we obtain from (2.11) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ24_HTML.gif)
where is a nonzero constant. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ25_HTML.gif)
where is a polynomial with
.
From (2.14) and (2.16), we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ26_HTML.gif)
in view of , together with (2.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ27_HTML.gif)
namely
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ28_HTML.gif)
which is a contradiction since .
Hence has at least one finite zero.
This proves Lemma 2.6.
Remark 2.7.
Lemma 2.6 is a generalization of Lemma 2.2 in [10]. The proof of Lemma 2.6 is quite different from its proof. Actually, the proof of Lemma 2.2 in [10] is incorrect. The main problem appears at (2.2) in [10]. Since has only zero with multiplicity at least
, then each zero of
is of multiplicity at least
, but this does not mean that each zero of
is of multiplicity at least
because the zeros of
may not be the zeros of
, and thus their multiplicity may less than
. Therefore, the inequality of (2.2) in [10] is not valid, which implies that the proof of Lemma 2.2 in [10] is not correct.
Lemma 2.8.
Let ,
such that
. Let
,
,
,
,
be nonnegative integers such that
, (
when
or
),
for all positive integers
and
,
. Let
be a meromorphic function in
; all zeros of
have multiplicity at least
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ29_HTML.gif)
Then has a finite zero.
Proof.
The algebraic complex equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ30_HTML.gif)
has always a nonzero solution, say . By Lemmas 2.5 and 2.6, the differential monomial
cannot avoid it and thus there exists
such that
.
Under the assumptions, for all positive integers ,
,
,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ31_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ32_HTML.gif)
This proves Lemma 2.8.
Lemma 2.9 (see [2, page 51]).
If is an entire function of order
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ33_HTML.gif)
where denotes the central-index of
.
Lemma 2.10 (see [22, page 187–199] or [2, page 51]).
If is a transcendental entire function, let
and
be such that
and that
holds. Then there exists a set
of finite logarithmic measure, that is,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ34_HTML.gif)
holds for all and all
.
3. Proof of Theorem 1.1
Without loss of generality, we may assume . Suppose that
is not normal at
. By Lemma 2.1, for
, there exist
,
such that
,
and
such that
locally uniformly with respect to the spherical metric, where
is a nonconstant meromorphic function on
, all of whose zeros have multiplicity at least
. For simplicity, we denote
by
. By Lemmas 2.4 and 2.6, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ35_HTML.gif)
Obviously, , so in some neighborhood of
,
converges uniformly to
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ36_HTML.gif)
Let , and under the assumption
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ37_HTML.gif)
is the uniform limit of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ38_HTML.gif)
in some neighborhood of .
By (3.1) and Hurwitz's theorem, there exists a sequence such that for all large values of
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ39_HTML.gif)
Hence for all large values of ,
, it follows from the condition that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ40_HTML.gif)
Since is not a pole of
, there exists a positive number
such that in some neighborhood of
we get
.
Since converges uniformly to
in some neighborhood of
, we get for all large values of
and for all
in that neighborhood of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ41_HTML.gif)
By (3.7), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ42_HTML.gif)
which is a contradiction since as
.
This completes the proof of Theorem 1.1.
4. Proof of Theorem 1.2
Without loss of generality, we may assume . Suppose that
is not normal in
. By Lemma 2.1, for
, there exist
,
,
and
such that
locally uniformly with respect to the spherical metric, where
is a nonconstant meromorphic function on
, all of whose zeros have multiplicity at least
. By Lemma 2.8, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ43_HTML.gif)
for some .
We can arrive at a contradiction by using the same argument as in the latter part of proof of Theorem 1.1.
This completes the proof of Theorem 1.2.
5. Applications
Proof of Theorem 1.3.
We shall divide our argument into two cases.
Case 1 ().
Let be a positive constant with
; under the assumptions, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ44_HTML.gif)
and for all
whenever
; by Lemmas 2.5 and 2.6, using the similar proof of Theorem 1.1, we obtain the conclusion.
Case 2 ().
For , if
for
, since
, we have
, which is a contradiction, hence
.
If for
, since
, we immediately get
and hence
, which is still a contradiction, hence
.
Suppose that is not normal in
, by Lemma 2.1, there exist
,
,
, and
such that
locally uniformly with respect to the spherical metric, where
is a nonconstant entire function, all of whose zeros have multiplicity at least
. By Hurwitz's theorem, we have
(i) or
, and
(ii) or
.
Since is not a constant, we have
. By Lemma 2.3,
has the order at most 1, so
, where
,
are two constants. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ45_HTML.gif)
If , we immediately get a contradiction. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ46_HTML.gif)
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. Brück's Conjecture. Firstly, we recall this conjecture.
R. Brück's Conjecture
Let
be a nonconstant entire function such that the hyper-order
is not a positive integer and
. If
and
share a finite value
CM, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ47_HTML.gif)
whereis a nonzero constant and the hyper-order
is defined as follow:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ48_HTML.gif)
Since then, many results related to this conjecture have been obtained. We refer the reader to Brück [23], Gundersen and Yang [24], Yang [25], Chen and Shon [26], Li and Gao [27], and Wang [28].
It's interesting to ask what happens if is replaced by
in Brück's Conjecture. Recently, Yang and Zhang [29] considered this problem and got the following theorem.
Theorem H.
Let be a nonconstant entire function.
be an integer, and let
. If
and
share 1 CM, then
, and
assumes the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ49_HTML.gif)
where is a nonzero constant.
Lü et al. [30] improves Theorem H and obtained the following theorem.
Theorem I.
Let be a polynomial, and let
be an intege; let
be a transcendental entire function, and let
. If
and
share
CM, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ50_HTML.gif)
where is a nonzero constant.
We obtain a more general result as follows.
Theorem 5.1.
Let be nonnegative integers with
,
,
, and
,
be two finite nonzero values. Let
be a nonconstant entire function whose zeros have multiplicity at least
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ51_HTML.gif)
where is a nonzero constant. Specially, if
, then
, where
is a nonzero constant,
is the root of
.
Proof of Theorem 5.1.
First we assert that . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ52_HTML.gif)
Under the assumptions of Theorem 1.3, we get that is a normal family of holomorphic functions in
. By Lemma 2.2, there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ53_HTML.gif)
for all . Hence by Lemma 2.3,
has the order at most 1.
Since ,
must be a transcendental entire function and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ54_HTML.gif)
From (5.11), we have , hence
and
is a polynomial with
. Note that
is transcendental, we have
, as
. Let
, where
, we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ55_HTML.gif)
By Lemma 2.10, there exists a subset of finite logarithmic measure, namely
such that for some point
satisfying
and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ56_HTML.gif)
as .
Rewrite (5.11) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ57_HTML.gif)
it follows from (5.12)–(5.14) and Lemma 2.8 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ58_HTML.gif)
as . Since
is a polynomial, from (5.15), we deduce that
is a constant. Let
, then
is a nonzero constant. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ59_HTML.gif)
Specially, if , suppose that
has a zero
, by letting
in (5.16), we get
; hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ60_HTML.gif)
Suppose that is a zero of
with multiplicity
, then
is a zero of
with multiplicity
, and a zero of
with multiplicity
, which is a contradiction. So
has no zero, note that
is a transcendental entire function and
, we have
, where
and
are two finite nonzero values. In view of (5.16) and
, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F873184/MediaObjects/13660_2010_Article_2364_Equ61_HTML.gif)
hence and
.
,
is the root of
.
This completes the proof of Theorem 5.1.
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Acknowledgments
The authors thank the referees for reading the manuscript very carefully and making a number of valuable suggestions to improve the readability of the paper. The authors were supported by NSF of China (no. 10771121), NSF of Shandong Province (no. Z2008A01) and NSF of Guangdong Province (no. 9452902001003278).
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Zhang, XB., Xu, JF. & Yi, HX. Normality Criteria of Lahiri's Type and Their Applications. J Inequal Appl 2011, 873184 (2011). https://doi.org/10.1155/2011/873184
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DOI: https://doi.org/10.1155/2011/873184