Volume 2010, Article ID 417480,9pages doi:10.1155/2010/417480
Research Article
Normality of Composite Analytic Functions and
Sharing an Analytic Function
Qifeng Wu,
1Bing Xiao,
2and Wenjun Yuan
31Shaozhou Normal College, Shaoguan University, Shaoguan 512009, China 2Department of Mathematics, Xinjiang Normal University, Urumqi 830054, China
3School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
Correspondence should be addressed to Wenjun Yuan,wjyuan1957@126.com
Received 17 August 2010; Accepted 15 October 2010
Academic Editor: Manuel De la Sen
Copyrightq2010 Qifeng Wu 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.
A result of Hinchliffe2003is extended to transcendental entire function, and an alternative proof is given in this paper. Our main result is as follows: letαzbe an analytic function,Fa family of analytic functions in a domainD, andHza transcendental entire function. IfH◦fzand
H◦gzshareαzIM for each pairfz, gz∈ F, and one of the following conditions holds:1
Hz−αz0has at least two distinct zeros for anyz0∈D;2αzis nonconstant, and there exists
z0∈Dsuch thatHz−αz0: z−β0pQzhas only one distinct zeroβ0, and suppose that the
multiplicitieslandkof zeros offz−β0andαz−αz0atz0, respectively, satisfyk /lp, for each
fz∈ F, whereQβ0/0;3there exists az0∈Dsuch thatHz−αz0has no zero, andαzis
nonconstant, thenFis normal inD.
1. Introduction and Main Results
Letfzandgzbe two nonconstant meromorphic functions in the whole complex plane
C, and letabe a finite complex value or function. We say thatf andg shareaCMor IM provided that f −aand g −ahave the same zeros counting or ignoringmultiplicity. It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory
Tr, f, mr, f, Nr, fNr, f, . . . 1.1
A meromorphic function αzis called a small function related to fz if Tr, α
Sr, f.
In 1952, Rosenbloom3proved the following theorem.
Theorem A. LetPzbe a polynomial of degree at least2andfza transcendental entire function. Then
lim r→ ∞inf
Nr,1/Pf−z
Tr, f ≥1. 1.2
Influenced from Bloch’s principle 1 or 4, that is, there is a normal criterion corresponding to every Liouville-Picard type theorem, Fang and Yuan 5 proved a corresponding normality criterion for inequality1.2.
Theorem B. LetFbe a family of analytic functions in a domainDandPza polynomial of degree at least2. IfPfz/zfor eachfz∈ F, thenFis normal inD.
In 1995, Zheng and Yang6proved the following result.
Theorem C. LetPzbe a polynomial of degreepat least2,fza transcendental entire function, andαza nonconstant meromorphic function satisfyingTr, α Sr, f. Then,
Tr, f≤μN
r, 1
Pf−αz
Sr, f. 1.3
Hereμ2/p−1ifPzhas only one zero; otherwiseμ2.
In 2000, Fang and Yuan7improved1.3and obtained the best possiblek.
Theorem D. LetPzbe a polynomial of degreepat least2andfza transcendental entire function, andαza nonconstant meromorphic function satisfyingTr, α Sr, f. Ifαzis a constant, we also require that there exists a constantA /αsuch thatPz−Ahas a zero of multiplicity at least 2. Then
Tr, f≤μN
r, 1
Pf−αz
Sr, f. 1.4
Hereμ1/p−1ifPzhas only one zero; otherwiseμ1.
The corresponding normal criterion below to TheoremDwas obtained by Fang and Yuan7.
Theorem E. LetFbe a family of analytic functions in a domainDandPza polynomial of degree at least2. Suppose thatαzis either a nonconstant analytic function or a constant function such that Pz−αhas at least two distinct zeros. IfP◦fz/αzfor eachf∈ F, thenFis normal inD.
In 2003, Hinchliffe8proved the following theorem.
Theorem F. Letαz z,Fa family of analytic functions in a domainD, andhza transcendental meromorphic function. IfC \hC ∅,{∞} or{ξ1, ξ2}, where {ξ1, ξ2} are two distinct values in
In 2004, Bergweiler9deals also with the case thatαzis meromorphic in Theorem
Fand extended TheoremEas follows.
Theorem G. Letαzbe a nonconstant meromorphic function,Fa family of analytic functions in a domainD, andRza rational function of degree at least 2. Suppose thatR◦fz/αzfor each f∈ Fand allz∈D. Then,Fis normal inD.
Recently, Yuan et al.10generalized TheoremG in another manner and proved the following result.
Theorem H. Letαzbe a nonconstant meromorphic function,Fa family of analytic functions in a domainD, andRza rational function of degree at least2. IfR◦fzandR◦gzshareαzIM for each pairfz,gz∈ Fand one of the following conditions holds:
1Rz−αz0has at least two distinct zeros or poles for anyz0∈D;
2there existsz0 ∈Dsuch thatRz−αz0 :Pz/Qzhas only one distinct zero (or pole)β0and suppose that the multiplicitieslandkof zeros offz−β0andαz−αz0at z0, respectively, satisfyk /lp(ork /lq), for eachfz∈ F, wherePzandQzare two of no common zero polynomials with degreepandq, respectively, andαz0∈C∪ {∞}.
Then,Fis normal inD.
In this paper, we improve TheoremsEandFand obtain the main resultTheorem 1.1
which is proved below inSection 3.
Theorem 1.1. Let αz be an analytic function, Fa family of analytic functions in a domain D, andHza transcendental entire function. IfH◦fzandH◦gzshareαzIM for each pair fz, gz∈ F, and one of the following conditions holds:
1Hz−αz0has at least two distinct zeros for anyz0∈D;
2αzis nonconstant, and there existsz0∈Dsuch thatHz−αz0: z−β0pQzhas only one distinct zeroβ0and suppose that the multiplicitieslandk of zeros offz−β0 andαz−αz0atz0, respectively, satisfyk /lp, for eachfz∈ F, whereQβ0/0; 3there exists az0∈Dsuch thatHz−αz0has no zero, andαzis nonconstant.
Then,Fis normal inD.
2. Preliminary Lemmas
In order to prove our result, we need the following lemmas.Lemma 2.1is an extending result of Zalcman11concerning normal families.
Lemma 2.1see12. LetFbe a family of functions on the unit disc. Then,Fis not normal on the unit disc if and only if there exist
aa number0< r <1;
bpointsznwith|zn|< r;
cfunctionsfn ∈ F;
such thatgnζ:fnznρnζconverges locally uniformly to a nonconstant meromorphic function
gζ, which order is at most 2.
Remark 2.2. gζis a nonconstant entire function ifFis a family of analytic functions on the unit disc inLemma 2.1.
The followingLemma 2.3is very useful in the proof of our main theorem. We denote byUz0, rthe open disc of radiusr aroundz0, that is, Uz0, r : {z ∈ C : |z−z0| < r}. U0z
0, r:{z∈C: 0<|z−z0|< r}.
Lemma 2.3see13or14. Let{fnz}be a family of analytic functions inUz0, r. Suppose that{fnz}is not normal atz0but is normal inU0z0, r. Then, there exists a subsequence{fnkz} of{fnz}and a sequence of points{znk}tending toz0such thatfnkznk 0, but{fnkz}tending to infinity locally uniformly onU0z
0, r.
3. Proof of Theorem
Proof ofTheorem 1.1. Without loss of generality, we assume thatD{z∈C,|z|<1}. Then, we consider three cases:
Case 1. Hz−αz0has at least two distinct zeros for anyz0∈D
Suppose thatFis not normal inD. Without loss of generality, we assume thatFis not normal atz0.
SetHz−α0have two distinct zerosβ1andβ2.
ByLemma 2.1, there exists a sequence of pointszn → 0,fn ∈ Fandρn → 0such that
Fnξ:fnznρnξ−→Fξ 3.1
uniformly on any compact subset ofC, whereFξis a nonconstant entire function. Hence,
H◦fnznρnξ−αznρnξ−→H◦Fξ−α0 3.2
uniformly on any compact subset ofC.
We claim thatH◦Fξ−α0had at least two distinct zeros.
IfFξis a nonconstant polynomial, then bothFξ−β1andFξ−β2 have zeros. So H◦Fξ−α0has at least two distinct zeros.
IfFξis a transcendental entire function, then eitherFξ−β1orFξ−β2has infinite
zeros. Indeed, suppose that it is not true, then by Picard’s theorem2, we obtain thatFξis a polynomial, a contradiction.
Thus, the claim gives that there existξ1andξ2such that
We choose a positive numberδsmall enough such thatD1∩D2 ∅andFξ−α0
has no other zeros inD1∪D2except forξ1andξ2, where
D1{ξ∈C;|ξ−ξ1|< δ}, D2{ξ∈C;|ξ−ξ2|< δ}. 3.4
By hypothesis and Hurwitz’s theorem14, for sufficiently largenthere exist points
ξ1n∈D1,ξ2n∈D2such that
H◦fnznρnξ1n−αznρnξ1n0,
H◦fnznρnξ2n−αznρnξ2n0.
3.5
Note thatH◦fmzandH◦fnzshareαzIM; it follows that
H◦fmznρnξ1n
−αznρnξ1n
0,
H◦fmznρnξ2n−αznρnξ2n0.
3.6
Takingn → ∞, we obtain
H◦fm0−α0 0. 3.7
Since the zeros of
H◦fmξ−αξ 3.8
have no accumulation points, we have
znρnξ1n0, znρnξ2n0, 3.9
or equivalently
ξ1n−zρn
n, ξ2n −
zn
ρn. 3.10
This contradicts with the facts thatξ1n∈D1,ξ2n∈D2, andD1∩D2∅.
Case 2. αzis nonconstant, and there existsz0∈Dsuch thatHz−αz0: z−β0pQzhas
only one distinct zeroβ0, and suppose that the multiplicitieslandkof zeros offz−β0and αz−αz0atz0, respectively, satisfyk /lp, possibly outside finitefz∈ F, whereQβ0/0.
We shall prove thatFis normal atz0 ∈D. Without loss of generality, we can assume
thatz00.
Byαznonconstant and analytic, we see that there exists a neighborhoodU0, rsuch that
Hypothesis implies thatHz−α0has only one zeroβ0, that is,Hβ0 α0.
We claim thatFis normal atz0 ∈ U00, rfor small enoughr. In fact,Hz−αz0
has infinite zeros by Picard theorem. Hence, the conclusion of Case1tells us that this claim is true.
Next, we proveFis normal atz0. For any{fnz} ⊂ F, by the former claim, there exists a subsequence of{fnz}, denoted{fnz}for the sake of simplicity, such that
fnz−→Gz, 3.12
uniformly on a punctured discU00, r⊂U.
By hypothesis, we see that{H◦fnz−αz}is an analytic family in the discU0, r. If{fnz}is not normal atz0, thenLemma 2.3gives thatGz ∞,on a punctured discU00, randfnzn 0 for a sequence of pointszn → 0.
We claim that there exists a sequence of pointszn ∈ U0, r zn → 0such thatH◦
fnzn−αzn 0.
In fact we may findρ, >0 such that|Hz−α0|> for|z−β0|ρ.Next, we choose δwith 0< δ < rsuch that|αz−α0|< for|z|< δ.
Sincefnz → ∞onU00, randfnzn 0 for a sequence of pointszn → 0, we know that ifnsufficiently large, then
fnz−β−fnz β ≤ β0 ρ < fnz 3.13
for|z| δ andβ ∈ Uβ0, ρ.For largen, we also have |zn| < δ, and thus we deduce that from Rouch´e’s theorem thatfnztakes the valueβ∈U0, δ, that is, we havefnU0, δ⊃
Uβ, ρfor largen. Since alsofn∂U0, δ∩Uβ, ρ ∅ for largen, we find a component
U of f−1
n Uβ0, ρ contained in U0, δ for such n. Moreover, Uis a Jordan domain, and fn:U → Uβ0, ρis a proper map.
Forz∈∂U, we then havefnz∈∂Uβ0, ρ, and thus|H◦fnz−α0|> . Hence
H◦fnz−αz−H◦fnz−α0 |αz−α0|< < H◦fnz−α0 3.14
forz∈∂U. Nowfn, in particular, takes the valueβ0inU, say,fnzn β0withzn∈U.Hence,
H◦fnzn−α0 0, and thus Rouch´e’s theorem now shows that our claim holds.
By the similar argument as Case 1, we obtain that zn 0 for sufficiently large n. BecauseHz−α0 z−ξ0pHz, we have
H◦fnz−αz fnz−ξ0 pHf
nz−αz−α0,
fn0−ξ0 pHf
n0H◦fn0−α0 0.
Hence,
H◦fnz−αz zk
zlp−kh
nz−βz
, iflp > k;
H◦fnz−αz zlp
hnz−zk−lpβz
, iflp < k,
3.16
wherehnz,βzare analytic functions andhn0/0,β0/0.
SetHnz:zlp−khnz−βz, iflp > k; orHnz:hnz−zk−lpβz,iflp < k. Thus,
Hn0 −β0/0 orHn0 hn0/0. Noting thatlp /k, we see that{Hnz}is an analytic family and normal inU00, r.
By the same argument as above, there exists a sequence of pointsz∗n ∈ Usuch that
z∗
n → 0, andHnz∗n 0. Obviously,z∗n/0 and
H◦fnz∗n−αz∗n z∗nHnz∗n 0. 3.17
Noting thatH◦fnzandH◦fmzshareαzIM, we obtain that
H◦fmz∗n−αz∗n 0 3.18
for each m. That is,z∗nHmz∗n 0. Noting thatz∗n/0, we deduce thatHmz∗n 0. Thus, takingn → ∞,Hm0 0,contradicting the hypothesis forHm0.
Case 3. There exists az0∈Dsuch thatHz−αz0has no zero, andαzis nonconstant.
Suppose thatFis not normal inD. Without loss of generality, we assume thatFis not normal atz0.
By Picard theorem and3.11, we know thatHz−αz0has at least two distinct zeros
at anyz0∈U00, rfor small enoughr. The result of Case1tell us thatFis normal inU00, r.
Thus, for any{fnz} ⊂ F, by the former conclusion and Lemma 2.3, there exists a subsequence of{fnz}, denoted by{fnz}for the sake of simplicity, such that
fnz−→ ∞, 3.19
uniformly on a punctured discU00, r⊂Uandfnz
n 0 for a sequence of pointszn → 0. Obviously,{H◦fnz−αz}is an analytic normal family in the punctured discU00, r for small enoughr. We consider two subcases.
Subcase 1{H◦fnz−αz}is not normal atz0. UsingLemma 2.3for{H◦fnz−αz}, we get that there exists a sequence of pointszn∈U0, rsuch thatzn → 0 andH◦fnzn−αzn 0.
Subcase 2{H◦fnz−αz}is normal atz0. Then,{H◦fnz−α0/αz−α0}is normal inU00, r, which tends to a limit functionhz, which is either identically infinite or
analytic inU00, r. Set
Mn:min fnz :|z|r, 3.20
noting that Mn → ∞ asn → ∞. If n is large enough, we havezn ∈ U0, r, and hence
U0, Mn ⊆ fnU0, r. Denote∂fnU0, rbyΓn, and note that theΓn are closed curves, arbitrarily distant from and surrounding the origin.
Suppose thathz≡ ∞onU00, r. Sincehnz: H◦fnz−α0/αz−α0 → ∞
locally uniformly on∂U0, r, there exists, for arbitrarily large positiveM, ann0Msuch
that, forn≥n0,|hnz| ≥Mon∂U0, r. Thus, we have|H◦fnz−α0| ≥M|αz−α0|
on∂U0, r. Hence, for largen, Hzis bounded away fromα0on the curvesΓn, and this contradicts Iversen’s theorem15.
On the other hand, suppose thathzis analytic onU00, r. Then, there exists some
constantLsuch that|hz| ≤Lon∂U0, r, and so, for largen,|hnz| ≤2Lon∂U0, r. Hence,
|H◦fnz−α0| ≤2L|αz−α0|on∂U0, r. Again,Hzis therefore bounded away from ∞of its omitted value on the curvesΓn, contradicting Iversen’s theorem.
ThereforeFis normal in Case3.
Theorem 1.1is proved completely.
Acknowledgments
The authors would like to express their hearty thanks to Professor Mingliang Fang and Degui Yang for their helpful discussions and suggestions. The authors would like to thank referee for hisor hervery careful comments and helpful suggestions. This paper is supported by the NSF of Chinano. 10771220, Doctorial Point Fund of National Education Ministry of Chinano. 200810780002, and Guangzhou Education Bureauno. 62035.
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