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,}

_{Bing Xiao,}

**2**

_{and Wenjun Yuan}

_{and Wenjun Yuan}

**3**

*1 _{Shaozhou Normal College, Shaoguan University, Shaoguan 512009, China}*

*2*

_{Department of Mathematics, Xinjiang Normal University, Urumqi 830054, China}*3 _{School 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: let*αz*be an analytic function,Fa family
of analytic functions in a domain*D*, and*Hz*a transcendental entire function. If*H*◦*fz*and

*H*◦*gz*share*αz*IM for each pair*fz, gz*∈ F, and one of the following conditions holds:1

*Hz*−*αz*0has at least two distinct zeros for any*z*0∈*D*;2*αz*is nonconstant, and there exists

*z*0∈*D*such that*Hz*−*αz*0: *z*−*β*0*pQz*has only one distinct zero*β*0, and suppose that the

multiplicities*l*and*k*of zeros of*fz*−*β*0and*αz*−*αz*0at*z*0, respectively, satisfy*k /lp*, for each

*fz*∈ F, where*Qβ*0*/*0;3there exists a*z*0∈*D*such that*Hz*−*αz*0has no zero, and*αz*is

nonconstant, thenFis normal in*D*.

**1. Introduction and Main Results**

Let*fz*and*gz*be two nonconstant meromorphic functions in the whole complex plane

**C**, and let*a*be a finite complex value or function. We say that*f* and*g* share*a*CMor IM
provided that *f* −*a*and *g* −*a*have 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 *αz*is called a small function related to *fz* if *T*r, α

*Sr, f*.

In 1952, Rosenbloom3proved the following theorem.

**Theorem A.** *LetP*z*be a polynomial of degree at least*2*andfza 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.** *Let*F*be a family of analytic functions in a domainDandPza polynomial of degree*
*at least*2*. IfP*fz*/zfor eachfz*∈ F*, then*F*is normal inD.*

In 1995, Zheng and Yang6proved the following result.

**Theorem C.** *LetP*z*be a polynomial of degreepat least*2*,f*z*a transcendental entire function,*
*andαza nonconstant meromorphic function satisfyingTr, α Sr, f*. Then,

*Tr, f*≤*μN*

*r,* 1

*Pf*−*αz*

*Sr, f.* 1.3

*Hereμ*2*/p*−1*ifP*z*has only one zero; otherwiseμ*2*.*

In 2000, Fang and Yuan7improved1.3and obtained the best possible*k*.

**Theorem D.** *LetPzbe a polynomial of degreepat least*2*andfza transcendental entire function,*
*andαza nonconstant meromorphic function satisfyingT*r, α *Sr, f*. If*αzis a constant, we*
*also require that there exists a constantA /αsuch thatP*z−*Ahas a zero of multiplicity at least 2.*
*Then*

*Tr, f*≤*μN*

*r,* 1

*Pf*−*αz*

*Sr, f.* 1.4

Here*μ*1*/p*−1if*P*zhas only one zero; otherwise*μ*1.

The corresponding normal criterion below to TheoremDwas obtained by Fang and Yuan7.

**Theorem E.** *Let*F*be a family of analytic functions in a domainDandPza polynomial of degree*
*at least*2*. 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*, then*F*is normal inD.*

In 2003, Hinchliﬀe8proved the following theorem.

**Theorem F.** *Letαz z,*F*a family of analytic functions in a domainD, andhza transcendental*
*meromorphic function. If***C** \*h***C** ∅*,*{∞} *or*{*ξ*1*, ξ*2}*, where* {*ξ*1*, ξ*2} *are two distinct values in*

In 2004, Bergweiler9deals also with the case that*αz*is meromorphic in Theorem

Fand extended TheoremEas follows.

**Theorem G.** *Letαzbe a nonconstant meromorphic function,*F*a family of analytic functions in*
*a domainD, andRza rational function of degree at least 2. Suppose thatR*◦*fz/αzfor each*
*f*∈ F*and allz*∈*D. Then,*F*is normal inD.*

Recently, Yuan et al.10generalized TheoremG in another manner and proved the following result.

**Theorem H.** *Letαzbe a nonconstant meromorphic function,*F*a family of analytic functions in a*
*domainD, andRza rational function of degree at least*2*. IfR*◦*f*z*andR*◦*gzshareαzIM*
*for each pairfz,gz*∈ F*and one of the following conditions holds:*

1*Rz*−*αz*0*has at least two distinct zeros or poles for anyz*0∈*D;*

2*there existsz*0 ∈*Dsuch thatRz*−*αz*0 :*P*z/Qz*has only one distinct zero (or*
*pole)β*0*and suppose that the multiplicitieslandkof zeros offz*−*β*0*andαz*−*αz*0*at*
*z*0*, respectively, satisfyk /lp(ork /lq), for eachf*z∈ F*, wherePzandQzare two*
*of no common zero polynomials with degreepandq, respectively, andαz*0∈**C**∪ {∞}*.*

*Then,*F*is 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,* F*a family of analytic functions in a domain* *D,*
*andHza transcendental entire function. IfH*◦*fzandH*◦*gzshareαzIM for each pair*
*fz, g*z∈ F*, and one of the following conditions holds:*

1*Hz*−*αz*0*has at least two distinct zeros for anyz*0∈*D;*

2*αzis nonconstant, and there existsz*0∈*Dsuch thatHz*−*αz*0: z−*β*0*pQzhas*
*only one distinct zeroβ*0*and suppose that the multiplicitieslandk* *of zeros off*z−*β*0
*andαz*−*αz*0*atz*0*, respectively, satisfyk /lp, for eachfz*∈ F*, whereQβ*0*/*0*;*
3*there exists az*0∈*Dsuch thatHz*−*αz*0*has no zero, andαzis nonconstant.*

*Then,*F*is 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 2.1**see12. *Let*F*be a family of functions on the unit disc. Then,*F*is not normal on the*
*unit disc if and only if there exist*

a*a number*0*< r <*1;

b*pointsznwith*|*zn|< r*;

c*functionsfn* ∈ F*;*

*such thatgn*ζ:*fn*z*nρnζconverges locally uniformly to a nonconstant meromorphic function*

*gζ, which order is at most 2.*

*Remark 2.2.* *gζ*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
by*Uz*0*, r*the open disc of radius*r* around*z*0, that is, *Uz*0*, r* : {*z* ∈ **C** : |*z*−*z*0| *< r*}.
*U*0_{z}

0*, r*:{*z*∈**C**: 0*<*|*z*−*z*0|*< r*}*.*

**Lemma 2.3**see13or14. *Let*{*fn*z}*be a family of analytic functions inUz*0*, r*. Suppose
*that*{*fn*z}*is not normal atz*0*but is normal inU*0z0*, r. Then, there exists a subsequence*{*fnk*z}
*of*{*fn*z}*and a sequence of points*{*znk*}*tending toz*0*such thatfnk*z*nk* 0*, but*{*fnk*z}*tending*
*to infinity locally uniformly onU*0_{z}

0*, r*.

**3. Proof of Theorem**

*Proof ofTheorem 1.1.* Without loss of generality, we assume that*D*{*z*∈**C***,*|*z*|*<*1}. Then, we
consider three cases:

*Case 1.* *Hz*−*αz*0has at least two distinct zeros for any*z*0∈*D*

Suppose thatFis not normal in*D*. Without loss of generality, we assume thatFis not
normal at*z*0.

Set*Hz*−*α*0have two distinct zeros*β*1and*β*2.

ByLemma 2.1, there exists a sequence of points*zn* → 0,*fn* ∈ Fand*ρn* → 0such that

*Fn*ξ:*fnznρnξ*−→*Fξ* 3.1

uniformly on any compact subset of**C**, where*Fξ*is a nonconstant entire function.
Hence,

*H*◦*fnznρnξ*−*αznρnξ*−→*H*◦*Fξ*−*α*0 3.2

uniformly on any compact subset of**C**.

We claim that*H*◦*Fξ*−*α*0had at least two distinct zeros.

If*Fξ*is a nonconstant polynomial, then both*Fξ*−*β*1and*Fξ*−*β*2 have zeros. So
*H*◦*Fξ*−*α*0has at least two distinct zeros.

If*F*ξis a transcendental entire function, then either*Fξ*−*β*1or*Fξ*−*β*2has infinite

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

Thus, the claim gives that there exist*ξ*1and*ξ*2such that

We choose a positive number*δ*small enough such that*D*1∩*D*2 ∅and*Fξ*−*α*0

has no other zeros in*D*1∪*D*2except for*ξ*1and*ξ*2, where

*D*1{*ξ*∈**C**;|*ξ*−*ξ*1|*< δ*}*,* *D*2{*ξ*∈**C**;|*ξ*−*ξ*2|*< δ*}*.* 3.4

By hypothesis and Hurwitz’s theorem14, for suﬃciently large*n*there exist points

*ξ*1*n*∈*D*1,*ξ*2*n*∈*D*2such that

*H*◦*fnznρnξ*1*n*−*αznρnξ*1*n*0*,*

*H*◦*fnznρnξ*2*n*−*αznρnξ*2*n*0*.*

3.5

Note that*H*◦*fm*zand*H*◦*fn*zshare*αz*IM; it follows that

*H*◦*fmznρnξ*1*n*

−*αznρnξ*1*n*

0*,*

*H*◦*fmznρnξ*2*n*−*αznρnξ*2*n*0*.*

3.6

Taking*n* → ∞, we obtain

*H*◦*fm*0−*α*0 0*.* 3.7

Since the zeros of

*H*◦*fm*ξ−*αξ* 3.8

have no accumulation points, we have

*znρnξ*1*n*0*,* *znρnξ*2*n*0*,* 3.9

or equivalently

*ξ*1*n*−*z _{ρ}n*

*n,* *ξ*2*n* −

*zn*

*ρn.* 3.10

This contradicts with the facts that*ξ*1*n*∈*D*1,*ξ*2*n*∈*D*2, and*D*1∩*D*2∅.

*Case 2.* *αz*is nonconstant, and there exists*z*0∈*D*such that*Hz*−*αz*0: z−*β*0*pQz*has

only one distinct zero*β*0, and suppose that the multiplicities*l*and*k*of zeros of*fz*−*β*0and
*αz*−*αz*0at*z*0, respectively, satisfy*k /lp*, possibly outside finite*fz*∈ F, where*Qβ*0*/*0.

We shall prove thatFis normal at*z*0 ∈*D*. Without loss of generality, we can assume

that*z*00.

By*αz*nonconstant and analytic, we see that there exists a neighborhood*U*0*, r*such
that

Hypothesis implies that*Hz*−*α*0has only one zero*β*0, that is,*Hβ*0 *α*0.

We claim thatFis normal at*z*0 ∈ *U*00*, r*for small enough*r*. In fact,*Hz*−*αz*0

has infinite zeros by Picard theorem. Hence, the conclusion of Case1tells us that this claim is true.

Next, we proveFis normal at*z*0. For any{*fn*z} ⊂ F, by the former claim, there
exists a subsequence of{*fn*z}, denoted{*fn*z}for the sake of simplicity, such that

*fn*z−→*Gz,* 3.12

uniformly on a punctured disc*U*0_{}_{0}_{, r}_{⊂}_{U}_{.}

By hypothesis, we see that{*H*◦*fn*z−*αz*}is an analytic family in the disc*U*0*, r*.
If{*fn*z}is not normal at*z*0, thenLemma 2.3gives that*Gz *∞*,*on a punctured
disc*U*00*, r*and*fn*z*n* 0 for a sequence of points*zn* → 0.

We claim that there exists a sequence of points*zn* ∈ *U*0*, r zn* → 0such that*H*◦

*fn*z*n*−*αzn* 0.

In fact we may find*ρ, >*0 such that|*Hz*−*α*0|*> *for|*z*−*β*0|*ρ.*Next, we choose
*δ*with 0*< δ < r*such that|*αz*−*α*0|*< *for|*z*|*< δ.*

Since*fn*z → ∞on*U*00*, r*and*fn*z*n* 0 for a sequence of points*zn* → 0, we know
that if*n*suﬃciently large, then

*fn*z−*β*−*fn*z *β* ≤ *β*0 *ρ <* *fn*z 3.13

for|*z*| *δ* and*β* ∈ *Uβ*0*, ρ.*For large*n*, we also have |*zn*| *< δ*, and thus we deduce that
from Rouch´e’s theorem that*fn*ztakes the value*β*∈*U*0*, δ*, that is, we have*fn*U0*, δ*⊃

*Uβ, ρ*for large*n*. Since also*fn*∂U0*, δ*∩*Uβ, ρ * ∅ for large*n*, we find a component

*U* of *f*−1

*n* Uβ0*, ρ* contained in *U*0*, δ* for such *n*. Moreover, *U*is a Jordan domain, and
*fn*:*U* → *Uβ*0*, ρ*is a proper map.

For*z*∈*∂U*, we then have*fn*z∈*∂Uβ*0*, ρ*, and thus|*H*◦*fn*z−*α*0|*> *. Hence

*H*◦*fn*z−*αz*−*H*◦*fn*z−*α*0 |*αz*−*α*0|*< <* *H*◦*fn*z−*α*0 3.14

for*z*∈*∂U*. Now*fn*, in particular, takes the value*β*0in*U*, say,*fn*z*n* *β*0with*zn*∈*U.*Hence,

*H*◦*fn*z*n*−*α*0 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*.
Because*Hz*−*α*0 z−*ξ*0*pHz*, we have

*H*◦*fn*z−*αz fn*z−*ξ*0
*p _{H}_{f}*

*n*z−αz−*α*0,

*fn*0−*ξ*0
*p _{H}_{f}*

*n*0*H*◦*fn*0−*α*0 0*.*

Hence,

*H*◦*fn*z−*αz zk*

*zlp*−*k _{h}*

*n*z−*βz*

*,* if*lp > k*;

*H*◦*fn*z−*αz zlp*

*hn*z−*zk*−*lpβz*

*,* if*lp < k,*

3.16

where*hn*z,*βz*are analytic functions and*hn*0*/*0,*β*0*/*0*.*

Set*Hn*z:*zlp*−*khn*z−*βz*, if*lp > k*; or*Hn*z:*hn*z−*zk*−*lpβz,*if*lp < k*. Thus,

*Hn*0 −*β*0*/*0 or*Hn*0 *hn*0*/*0. Noting that*lp /k*, we see that{*Hn*z}is an analytic
family and normal in*U*0_{}_{0}_{, r}_{.}

By the same argument as above, there exists a sequence of points*z*∗* _{n}* ∈

*U*such that

*z*∗

*n* → 0, and*Hn*z∗*n* 0. Obviously,*z*∗*n/*0 and

*H*◦*fn*z∗*n*−*αz*∗*n* *z*∗*nHn*z∗*n* 0*.* 3.17

Noting that*H*◦*fn*zand*H*◦*fm*zshare*αz*IM, we obtain that

*H*◦*fm*z∗*n*−*αz*∗*n* 0 3.18

for each *m*. That is,*z*∗* _{n}Hm*z∗

*n*0. Noting that

*z*∗

*n/*0, we deduce that

*Hm*z∗

*n*0. Thus, taking

*n*→ ∞,

*Hm*0 0

*,*contradicting the hypothesis for

*Hm*0.

*Case 3.* There exists a*z*0∈*D*such that*Hz*−*αz*0has no zero, and*αz*is nonconstant.

Suppose thatFis not normal in*D*. Without loss of generality, we assume thatFis not
normal at*z*0.

By Picard theorem and3.11, we know that*Hz*−*αz*0has at least two distinct zeros

at any*z*0∈*U*00*, r*for small enough*r*. The result of Case1tell us thatFis normal in*U*00*, r*.

Thus, for any{*fn*z} ⊂ F, by the former conclusion and Lemma 2.3, there exists a
subsequence of{*fn*z}, denoted by{*fn*z}for the sake of simplicity, such that

*fn*z−→ ∞*,* 3.19

uniformly on a punctured disc*U*0_{}_{0}_{, r}_{⊂}_{U}_{and}_{f}_{n}_{z}

*n* 0 for a sequence of points*zn* → 0.
Obviously,{*H*◦*fn*z−*αz*}is an analytic normal family in the punctured disc*U*00*, r*
for small enough*r*. We consider two subcases.

*Subcase 1*{*H*◦*fn*z−*αz*}is not normal at*z*0. UsingLemma 2.3for{*H*◦*fn*z−*αz*}, we
get that there exists a sequence of points*zn*∈*U*0*, r*such that*zn* → 0 and*H*◦*fn*z*n*−*αzn*
0.

*Subcase 2*{*H*◦*fn*z−*αz*}is normal at*z*0. Then,{H◦*fn*z−*α*0/αz−*α*0}is
normal in*U*0_{}_{0}_{, r}_{, which tends to a limit function}_{hz}_{, which is either identically infinite or}

analytic in*U*0_{}_{0}_{, r}_{. Set}

*Mn*:min *fn*z :|*z*|*r,* 3.20

noting that *Mn* → ∞ as*n* → ∞. If n is large enough, we have*zn* ∈ *U*0*, r*, and hence

*U*0*, Mn* ⊆ *fn*U0*, r*. Denote*∂fn*U0*, r*byΓ*n*, and note that theΓ*n* are closed curves,
arbitrarily distant from and surrounding the origin.

Suppose that*hz*≡ ∞on*U*0_{}_{0}_{, r}_{. Since}_{h}_{n}_{z}_{:}_{ H}_{◦}_{f}_{n}_{z}_{−}_{α}_{0}_{/αz}_{−}_{α}_{0}_{} _{→ ∞}

locally uniformly on*∂U*0*, r*, there exists, for arbitrarily large positive*M*, an*n*0Msuch

that, for*n*≥*n*0,|*hn*z| ≥*M*on*∂U*0*, r*. Thus, we have|*H*◦*fn*z−*α*0| ≥*M*|*αz*−*α*0|

on*∂U*0*, r*. Hence, for large*n, Hz*is bounded away from*α*0on the curvesΓ*n*, and this
contradicts Iversen’s theorem15.

On the other hand, suppose that*hz*is analytic on*U*0_{}_{0}_{, r}_{. Then, there exists some}

constant*L*such that|*hz*| ≤*L*on*∂U*0*, r*, and so, for large*n,*|*h*nz| ≤2*L*on*∂U*0*, r*. Hence,

|*H*◦*fn*z−*α*0| ≤2*L*|*αz*−*α*0|on*∂U*0*, r*. Again,*Hz*is therefore bounded away from
∞of its omitted value on the curvesΓ*n*, 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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