R E S E A R C H
Open Access
Normality of meromorphic functions and
differential polynomials share values
Wenjun Yuan
1,2, Jinchun Lai
1*, Zifeng Huang
1and Zhi Liu
1*Correspondence:
[email protected] 1School of Mathematics and
Information Science, Guangzhou University, Guangzhou, 510006, China
Full list of author information is available at the end of the article
Abstract
In this paper, we discuss the normality of meromorphic functions which involves differential polynomial sharing values. We obtain two results: Letkbe a positive integer,b(= 0) be a complex number, andh(z) be a polynomial with degree at least 2, andH(f,f,. . .,f(k)) be a differential polynomial withγ|H<k+ 1. LetFbe a family of
meromorphic functions defined inD, all of whose zeros have multiplicity at leastk+ 1. Ifh(z) – 1 has at least two distinct zeros,h(f(k)) +H(f,f,. . .,f(k)) – 1 has at most one distinct zero inDfor eachf∈F, thenFis normal inD. Ifh(z) –bhas at least two distinct zeros and for each pair of functionsfandginF,h(f(k)) +H(f,f,. . .,f(k)) and h(g(k)) +H(g,g,. . .,g(k)) sharebinD, thenFis normal inD, too. Two examples show that a condition in our results is necessary and our results improve Fang and Hong’s, and Zeng’s corresponding results.
MSC: Primary 30D35; secondary 34A05
Keywords: differential polynomial; meromorphic functions; shared values; normal families
1 Introduction and main results
LetDbe a domain inC, andF be a family of meromorphic functions defined in the do-mainD.Fis said to be normal inD, in the sense of Montel, if for every sequence{fn} ⊆F contains a subsequence{fnj}such thatfnjconverges spherically uniformly on compact
sub-sets ofD.
Fis said to be normal at a pointz∈Dif there exists a neighborhood ofzin whichF is normal. It is well known thatFis normal in a domainDif and only if it is normal at each of its points.
Letf andgbe meromorphic functions defined in a domainD, andaandbbe complex numbers. Ifg(z) =bwheneverf(z) =a, we writef(z) =a⇒g(z) =b. Iff(z) =a⇒g(z) =b
andg(z) =b⇒f(z) =a, we writef(z) =a⇐⇒g(z) =b; Iff(z) =a⇐⇒g(z) =a, we say that
f andgshare the valueainD.
Letn,n, . . . ,nkbe non-negative integers and one of them nonzero at least, and set
Mf,f, . . . ,f(k)=fnfn· · ·f(k)nk
,
γM=n+n+n+· · ·+nk,
M=n+ n+ n+· · ·+ (k+ )nk.
M(f,f, . . . ,f(k)) is called the differential monomial off,γ
Mthe degree ofM(f,f, . . . ,f(k)) andMthe weight ofM(f,f, . . . ,f(k)).
Let M(f,f, . . . ,f(k)),M(f,f, . . . ,f(k)), . . . ,Mm(f,f, . . . ,f(k)) be differential monomials off, and leta(z),a(z), . . . ,am(z) be analytic inD. Set
Hf,f, . . . ,f(k)=a(z)M
f,f, . . . ,f(k)+· · ·+am(z)Mm
f,f, . . . ,f(k),
γH=max{γM,γM, . . . ,γMm},
H=max{M,M, . . . ,Mm}.
H(f,f, . . . ,f(k)) is called a differential polynomial off,γ
Hthe degree ofH(f,f, . . . ,f(k)) andH the weight ofH(f,f, . . . ,f(k)). IfγM=γM =· · ·=γMm=t, thenH(f,f, . . . ,f(k)) is
called a homogeneous differential polynomial of degreet. Set
γ
H=
max
M
γM ,M
γM
, . . . ,Mm γMm
.
The following theorem was proved by Fang and Hong [].
Theorem .[] LetFbe a family of meromorphic functions defined in D,k and q(≥)
be two positive integers,and H(f,f, . . . ,f(k))be a differential polynomial with γ|H<k+ .
If the zeros of f(z)are of multiplicity at least k+ and(f(k))q+H(f,f, . . . ,f(k))= for each
f ∈F,thenFis normal in D.
It is natural to ask whether the condition in Theorem . that (f(k))q+H(f,f, . . . ,f(k))= can be relaxed. In this paper we investigate this problem and prove the following result.
Theorem . LetF be a family of meromorphic functions defined in D,k be a positive
integer,let h(z)be a polynomial with degree at least,and H(f,f, . . . ,f(k))be a differential polynomial with γ|H<k+ .If h(z) – has at least two distinct zeros,the zeros of f(z)are
of multiplicity at least k+ and h(f(k)) +H(f,f, . . . ,f(k)) – has at most one distinct zero in D for each f ∈F,thenFis normal in D.
By the idea of shared values, very recently, Zeng [] proved the following theorem.
Theorem .[] Let k and q(≥)be two positive integers,b= be a complex number,
and let H(f,f, . . . ,f(k))be a differential polynomial with
γ|H<k+ .LetFbe a family of meromorphic functions defined in D,all of whose zeros have multiplicity at least k+ .If for each pair of functions f and g inF, (f(k))q+H(f,f, . . . ,f(k))and(g(k))q+H(g,g, . . . ,g(k))
share b in D,thenFis normal in D.
It is natural to ask whether Theorem . can be improved. In this paper, we study this problem and obtain the following theorem.
Theorem . Let k be a positive integer,b(= )be a complex number,h(z)be a polynomial,
and let H(f,f, . . . ,f(k))be a differential polynomial with
γ|H<k+ .LetFbe a family of meromorphic functions defined in D,all of whose zeros have multiplicity at least k+ .If h(z) –b has at least two distinct zeros and for each pair of functions f and g inF,h(f(k)) +
Example . LetD={z:|z|< },h(z) =zk++ andF:={fn(z) =nzk+}. Then
hfn(k)(z)+fn(z) =nnk(k+ )!k++ zk++ .
We can see that γ|H= <k+ ,h(fn(k)(z)) +fn(z) – has only one distinct zero inDfor each functionfninF, andh(fn(k)(z)) +fn(z) andh(fm(k)(z)) +fm(z) share inDfor each pair of functionsfnandfminF. On the other hand,fj() = ,fj(
jk+
) = , for anyj∈N. This
implies that the familyFfails to be equicontinuous at , and thusFis not normal at .
Remark . This example shows thath(z) – to have at least two distinct zeros (h(z) –b
to have at least two distinct zeros) is necessary in Theorem . (Theorem .).
Example . LetD={z:|z|< },h(z) =zk++zk+ andF:={f
n(z) =zk+}. Then
hfn(k)(z)–(k+ )!k+fn(z) +fn(z) =(k+ )!k+k+ zk+ .
We can see that γ|H= <k+ ifk≥, and for each pair of functionsfnandfm inF,
h(fn(k)(z)) – [(k+ )!]k+fn(z) +fn(z) andh(f (k)
m (z)) – [(k+ )!]k+fm(z) +fm(z) share . Therefore,
Fis normal inDby our Theorem ..
Remark . From this example we also know thath(fn(k)(z)) – [(k+ )!]k+fn(z) +fn(z) – has only one solution inDfor eachfninF. The case of sharedbincludes the case of=b, that is to say, Theorem . is a generalization of Theorem . and Theorem . is a generalization of Theorem ..
2 Preliminary lemmas
In order to prove our results, we need the following lemmas. The first one is Zalcman’s Theorem.
Lemma .[] Let k∈N+,letFbe a family of functions meromorphic on the unit disc,
all of whose zeros have multiplicity at least k,and suppose that there exists A≥such that |f(k)(z)| ≤A whenever f(z) = .Then if F is not normal at z,there exist,for each ≤α≤k,
(a) functionsfn∈F;
(b) pointszn∈,zn→z,and
(c) positive numbersρn→+
such that gn(ζ) =ρ–α
n fn(zn+ρnζ)→g(ζ)locally uniformly with respect to the spherical
metric,where g is a nonconstant meromorphic function onC,all of whose zeros have mul-tiplicity at least k,such that g#(ζ)≤g#() =kA+ .In particular,g has order at most.
Hereg#(z) denotes the spherical derivative
g#(z) = |g (z)|
+|g(z)|.
Lemma . [] Let f(z) =anzn+an–zn–+· · ·+a+q(z)/p(z), where a,a, . . . ,anare
identically,withdegq<degp, and let k be a positive integer and b a nonzero complex number.If f(k)=b,and the zeros of f all have multiplicity at least k+ ,then
f(z) =b(z–d) k+
k!(z–c) ,
where c and d are distinct complex numbers.
Lemma . Let g be a nonconstant meromorphic function,and h(z)be a polynomial.If h(z)has at least two distinct zeros and all zeros of g have multiplicity at least k+ ,then h(g(k)(ξ))has at least two distinct zeros.
Proof Case . Ifh(g(k)(ξ)) has only one zeroα, thenh(g(k)(α)) = , andξ=α,h(g(k)(ξ))= . Suppose thatdi(i= , ) are two distinct zeros ofh(z). Without loss of generality, we may assume thatg(k)(α) =d, theng(k)(ξ)=d
forξ=α.
Firstly, we will show thatg(ξ) is not a transcendental meromorphic function. By Nevan-linna Theory, we have
Tr,g(k)≤Nr,g(k)+N r,
g(k)–d
+N r,
g(k)–d
+Sr,g(k)
≤ k+ N
r,g(k)+O(logr) +Sr,g(k)
≤ k+ T
r,g(k)+O(logr) +Sr,g(k).
Hence, we getT(r,g(k)) =O(logr) +S(r,g(k)), it follows thatg(ξ) is not a transcendental meromorphic function.
Ifg(ξ) is a polynomial, then
hg(k)(ξ)=hd+c(ξ–α)n
= (ξ–α)nQ(ξ),
whereQis a polynomial such thatQ(α)= and the degree ofQis not less than . Thus there exists anα=α, such thatQ(α) = . That is to say, there exists anα=α, such that
h(g(k)(α)) = , which is a contradiction.
Thereforeg(ξ) is rational but not a polynomial. Under the conditions of Lemma . on the rational functionsg, we have
g(ξ) =d(ξ–d) k+
k!(ξ–c) ,
wherecanddare distinct complex numbers,d= , and then
g(k)=d+
A
(ξ–c)k+,
whereA= is a complex number. Henceg(k)(ξ) =d
Case .h(g(k)(ξ))= . Sincedegh≥, by Nevanlinna Theory once more, we have follows thatg(k)is a constant. This together with the fact that the zeros ofg have
multi-plicity at leastk+ shows thatgis a constant, a contradiction.
3 Proofs of theorems
Proof of Theorem . We show thatF is normal in D. Otherwise, there exists at least one point z∈Dsuch thatF is not normal atz. Then by Lemma ., we can find a subsequence ofF, which we may denote by{fn},zn∈,zn→zandρn→+such that
gn(ξ) =ρ–k
n fn(zn+ρnξ) converges local uniformly with respect to the spherical metric to a nonconstant meromorphic functiongonC, all of whose zeros have multiplicity at least
k+ .
for sufficiently largen, we deduce from
γ|H<k+ that
Hence, by Hurwitz’s Theorem, the hypothesis of the theorem, and Lemma ., we see thath(g(k)(ξ))≡ orh(g(k)(ξ)) – has at least two distinct zeros onC.
Then byh(z) – having at least two distinct zeros, we find thath(z) –chas at least two distinct zeros except for at most one complex numberc. Therefore Lemma . tells us that
h(g(k)(ξ)) –chas zero for at least two distinctcexcept thatg(z) is a constant function. This
This contradiction shows thatFis normal inDand hence Theorem . is proved.
Thus we know that
Then there exists a positive numberδ> such that
hfn(k)+Hfn,fn, . . . ,fn(k)
(z)=b
for allzinDδ={z:|z–z|<δ}, by sharing condition.
Hence, by Hurwitz’s Theorem, the hypothesis of the theorem, and Lemma ., we see thath(g(k)(ξ))=borh(g(k)(ξ))≡bonC.
We fixmand note thatzn+ρnξn→z,zn+ρnξn→zifn→ ∞. From this we deduce
hfm(k)(z) +H
fm,fm, . . . ,fm(k)
(z) –b= .
Since
hfn(k)+Hfn,fn, . . . ,fn(k)
(zn+ρnξ)=b
ifzn+ρnξ=z, and
hf(k)+Hf,f, . . . ,f(k)(z) =b,
noting that the zeros of
hfm(k)(z) +Hfm,fm, . . . ,fm(k)
(z) –b
have no accumulation point, for sufficiently largen, we have
zn+ρnξn=z, zn+ρnξn=z.
Hence
ξn=
z–zn
ρn
, ξn=z–zn
ρn .
This contradicts the fact thatξn∈D(ξ,δ),ξn∈D(ξ,δ), andD(ξ,δ)∩D(ξ,δ) =∅. So
h(g(k)(ξ)) –bhas just a unique zero ignoring multiplicity. This contradicts the conclusion of Lemma . thath(g(k)(ξ)) –bhas at least two distinct zeros.
HenceFis normal atz, and thenFis normal inD. The proof of Theorem . is
com-plete.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
JL and ZH carried out the design of the study and performed the analysis. WY and ZL participated in its design and coordination. All authors read and approved the final manuscript.
Author details
1School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China.2Key Laboratory of
Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, China.
Acknowledgements
This work was supported by the Visiting Academic Sponsor Project of Department of Mathematics and Statistics at Curtin University of Technology (200001807894), the first author would like to express his hearty thanks to Curtin University of Technology for providing him with very comfortable research environment. This work was completed with the support with the NSF of China (11271090) and NSF of Guangdong Province (S2012010010121). The authors wish also specially to thank the managing editor and referees for their very helpful comments and useful suggestions.
References
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4. Wang, YF, Fang, ML: Picard values and normal families of meromorphic functions with zeros. Acta Math. Sin. New Ser.
14, 17-26 (1998)
10.1186/1687-1847-2014-120
Cite this article as:Yuan et al.:Normality of meromorphic functions and differential polynomials share values.
