Normality of Composite Analytic Functions and Sharing an Analytic Function

Download (0)

Full text

(1)

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,

1

Bing Xiao,

2

and Wenjun Yuan

3

1Shaozhou 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. IfHfzand

HgzshareαzIM for each pairfz, gz∈ F, and one of the following conditions holds:1

Hzαz0has at least two distinct zeros for anyz0∈D;2αzis nonconstant, and there exists

z0∈Dsuch thatHzαz0: zβ0pQzhas only one distinct zeroβ0, and suppose that the

multiplicitieslandkof zeros offzβ0andαzαz0atz0, respectively, satisfyk /lp, for each

fz∈ F, where0/0;3there exists az0∈Dsuch thatHzαz0has 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 faand gahave 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

(2)

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/Pfz

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, fμN

r, 1

Pfαz

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, fμN

r, 1

Pfαz

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 Pzαhas at least two distinct zeros. IfPfz/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

(3)

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 thatRfz/αzfor each f∈ Fand allzD. 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. IfRfzandRgzshareαzIM for each pairfz,gz∈ Fand one of the following conditions holds:

1Rzαz0has at least two distinct zeros or poles for anyz0∈D;

2there existsz0 ∈Dsuch thatRzαz0 :Pz/Qzhas only one distinct zero (or pole)β0and suppose that the multiplicitieslandkof zeros offzβ0andαzαz0at 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. IfHfzandHgzshareαzIM for each pair fz, gz∈ F, and one of the following conditions holds:

1Hzαz0has at least two distinct zeros for anyz0∈D;

2αzis nonconstant, and there existsz0∈Dsuch thatHzαz0: z−β0pQzhas only one distinct zeroβ0and suppose that the multiplicitieslandk of zeros offz−β0 andαzαz0atz0, respectively, satisfyk /lp, for eachfz∈ F, whereQβ0/0; 3there exists az0∈Dsuch thatHzαz0has 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;

(4)

such thatgnζ:fnznρnζconverges locally uniformly to a nonconstant meromorphic function

gζ, which order is at most 2.

Remark 2.2. is 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 : {zC : |zz0| < r}. U0z

0, r:{zC: 0<|zz0|< 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{zC,|z|<1}. Then, we consider three cases:

Case 1. Hzαz0has at least two distinct zeros for anyz0∈D

Suppose thatFis not normal inD. Without loss of generality, we assume thatFis not normal atz0.

SetHzα0have two distinct zerosβ1andβ2.

ByLemma 2.1, there exists a sequence of pointszn → 0,fn ∈ Fandρn → 0such that

Fnξ:fnznρnξ−→ 3.1

uniformly on any compact subset ofC, whereis a nonconstant entire function. Hence,

Hfnznρnξαznρnξ−→Hα0 3.2

uniformly on any compact subset ofC.

We claim thatHα0had at least two distinct zeros.

Ifis a nonconstant polynomial, then bothβ1andβ2 have zeros. So Hα0has at least two distinct zeros.

IfFξis a transcendental entire function, then eitherβ1orβ2has infinite

zeros. Indeed, suppose that it is not true, then by Picard’s theorem2, we obtain thatis a polynomial, a contradiction.

Thus, the claim gives that there existξ1andξ2such that

(5)

We choose a positive numberδsmall enough such thatD1∩D2 ∅andα0

has no other zeros inD1∪D2except forξ1andξ2, where

D1{ξC;|ξξ1|< δ}, D2{ξC;|ξξ2|< δ}. 3.4

By hypothesis and Hurwitz’s theorem14, for sufficiently largenthere exist points

ξ1nD1,ξ2nD2such that

Hfnznρnξ1nαznρnξ1n0,

Hfnznρnξ2nαznρnξ2n0.

3.5

Note thatHfmzandHfnzshareαzIM; it follows that

Hfmznρnξ1n

αznρnξ1n

0,

Hfmznρnξ2nαznρnξ2n0.

3.6

Takingn → ∞, we obtain

Hfm0−α0 0. 3.7

Since the zeros of

Hfmξ−αξ 3.8

have no accumulation points, we have

znρnξ1n0, znρnξ2n0, 3.9

or equivalently

ξ1nzρn

n, ξ2n

zn

ρn. 3.10

This contradicts with the facts thatξ1nD1,ξ2nD2, andD1∩D2∅.

Case 2. αzis nonconstant, and there existsz0∈Dsuch thatHzαz0: z−β0pQzhas

only one distinct zeroβ0, and suppose that the multiplicitieslandkof zeros offzβ0and αzαz0atz0, respectively, satisfyk /lp, possibly outside finitefz∈ F, where0/0.

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

(6)

Hypothesis implies thatHzα0has only one zeroβ0, that is,0 α0.

We claim thatFis normal atz0 ∈ U00, rfor small enoughr. In fact,Hzαz0

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, rU.

By hypothesis, we see that{Hfnz−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 pointsznU0, r zn → 0such thatH

fnznαzn 0.

In fact we may findρ, >0 such that|Hzα0|> for|zβ0|ρ.Next, we choose δwith 0< δ < rsuch that|αzα0|< for|z|< δ.

Sincefnz → ∞onU00, randfnzn 0 for a sequence of pointszn → 0, we know that ifnsufficiently large, then

fnz−βfnz ββ0 ρ < fnz 3.13

for|z| δ andβ0, ρ.For largen, we also have |zn| < δ, and thus we deduce that from Rouch´e’s theorem thatfnztakes the valueβU0, δ, that is, we havefnU0, δ

Uβ, ρfor largen. Since alsofn∂U0, δUβ, 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:U0, ρis a proper map.

Forz∂U, we then havefnz∈∂Uβ0, ρ, and thus|Hfnz−α0|> . Hence

Hfnz−αzHfnz−α0 |αzα0|< < Hfnz−α0 3.14

forz∂U. Nowfn, in particular, takes the valueβ0inU, say,fnzn β0withznU.Hence,

Hfnznα0 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. BecauseHzα0 z−ξ0pHz, we have

Hfnz−αz fnz−ξ0 pHf

nz−αz−α0,

fn0−ξ0 pHf

n0Hfn0−α0 0.

(7)

Hence,

Hfnz−αz zk

zlpkh

nz−βz

, iflp > k;

Hfnz−αz zlp

hnz−zklpβz

, iflp < k,

3.16

wherehnz,βzare analytic functions andhn0/0,β0/0.

SetHnz:zlpkhnz−βz, iflp > k; orHnz:hnz−zklpβz,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 pointsznUsuch that

z

n → 0, andHnz∗n 0. Obviously,zn/0 and

Hfnz∗nαzn znHnz∗n 0. 3.17

Noting thatHfnzandHfmzshareαzIM, we obtain that

Hfmz∗nαzn 0 3.18

for each m. That is,znHmz∗n 0. Noting thatzn/0, we deduce thatHmz∗n 0. Thus, takingn → ∞,Hm0 0,contradicting the hypothesis forHm0.

Case 3. There exists az0∈Dsuch thatHzαz0has 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 thatHzαz0has 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, rUandfnz

n 0 for a sequence of pointszn → 0. Obviously,{Hfnz−αz}is an analytic normal family in the punctured discU00, r for small enoughr. We consider two subcases.

Subcase 1{Hfnz−αz}is not normal atz0. UsingLemma 2.3for{Hfnz−αz}, we get that there exists a sequence of pointsznU0, rsuch thatzn → 0 andHfnznαzn 0.

(8)

Subcase 2{Hfnz−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 haveznU0, r, and hence

U0, MnfnU0, r. Denote∂fnU0, rbyΓn, and note that theΓn are closed curves, arbitrarily distant from and surrounding the origin.

Suppose thathz≡ ∞onU00, r. Sincehnz: Hfnzα0/αzα0 → ∞

locally uniformly on∂U0, r, there exists, for arbitrarily large positiveM, ann0Msuch

that, fornn0,|hnz| ≥Mon∂U0, r. Thus, we have|Hfnz−α0| ≥M|αzα0|

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,

|Hfnz−α0| ≤2L|αzα0|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.

References

1 W. K. Hayman,Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.

2 L. Yang,Value Distribution Theory, Springer, Berlin, Germany, 1993.

3 P. C. Rosenbloom, “The fix-points of entire functions,” Comm. Sem. Math. Univ. Lund, Tom Supplementaire, vol. 1952, pp. 186–192, 1952.

4 W. Bergweiler, “Bloch’s principle,”Computational Methods and Function Theory, vol. 6, no. 1, pp. 77–108, 2006.

5 M. L. Fang and W. Yuan, “On the normality for families of meromorphic functions,”Indian Journal of Mathematics, vol. 43, no. 3, pp. 341–351, 2001.

6 J. H. Zheng and C.-C. Yang, “Further results on fixpoints and zeros of entire functions,”Transactions of the American Mathematical Society, vol. 347, no. 1, pp. 37–50, 1995.

7 M. L. Fang and W. Yuan, “On Rosenbloom’s fixed-point theorem and related results,”Australian Mathematical Society Journal Series A, vol. 68, no. 3, pp. 321–333, 2000.

8 J. D. Hinchliffe, “Normality and fixpoints of analytic functions,”Proceedings of the Royal Society of Edinburgh Section A, vol. 133, no. 6, pp. 1335–1339, 2003.

(9)

10 W. J. Yuan, Z. R. Li, and B. Xiao, “Normality of composite analytic functions and sharing a meromorphic function,”Science in China Series A, vol. 40, no. 5, pp. 429–436, 2010Chinese.

11 L. Zalcman, “A heuristic principle in complex function theory,”The American Mathematical Monthly, vol. 82, no. 8, pp. 813–817, 1975.

12 L. Zalcman, “Normal families: new perspectives,”American Mathematical Society Bulletin. New Series, vol. 35, no. 3, pp. 215–230, 1998.

13 E. F. Clifford, “Normal families and value distribution in connection with composite functions,”

Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 195–204, 2005.

14 Y. X. Gu, X. C. Pang, and M. L. Fang,Theory of Normal Family and Its Applications, Science Press, Beijing, China, 2007.

Figure

Updating...

References