# Composite Holomorphic Functions and Normal Families

Authors: Xiao Bing; Wu Qifeng; Yuan Wenjun
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study the normality of families of holomorphic functions. We prove the following result. Letα(z),ai(z),i=1,2,…,p, be holomorphic functions and ℱ a family of holomorphic functions in a domain D, P(z,w):=(w-a1(z))(w-a2(z))⋯(w-ap(z)),p≥2. If Pw∘f(z) and Pw∘g(z) share α(z) IM for each pair f(z),g(z)∈ℱ and one of the following conditions holds: (1) P(z0,z)-α(z0) has at least two distinct zeros for any z0∈D; (2) there exists z0∈D such that P(z0,z)-α(z0) has only one distinct zero and α(z) is nonconstant. Assume that β0 is the zero of P(z0,z)-α(z0) and that the multiplicities l and k of zeros of f(z)-β0 and α(z)-α(z0) at z0, respectively, satisfy k≠lp, for all f(z)∈ℱ, then ℱ is normal in D. In particular, the result is a kind of generalization of the famous Montel's criterion. At the same time we fill a gap in the proof of Theorem 1.1 in our original paper (Wu et al., 2010).


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## Body

## 1. Introduction and Main Result

Letf(z) and g(z) be two nonconstant meromorphic functions in a domain D⊆C, and let a be a finite complex value or function. We say that f and g share a CM (or IM) in D provided that f-a and g-a have the same zeros counting (or ignoring) multiplicity in D. It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory ([1, 2] or [3]).The following theorem was proved by Chang et al. [4] in 2005 and [5] in 2009. It is an extension of a result obtained by Fang and Yuan [6] in 2000.Theorem A.
Letα(z) be a nonconstant meromorphic function, ℱ a family of holomorphic functions in a domain D,  and R(z) a rational function of degree at least 3. Suppose that R∘f(z)≠α(z) for each f∈ℱ and all z∈D. Then ℱ is normal in D.In the case whereℱ is a family of holomorphic functions and R(z) is a rational function of degree at least 2, the result was proved by Bergweiler [7] in 2004, by Hinchliffe [8] in 2003, and by Clifford [9] in 2005. It extends a result obtained by Fang and Yuan [6] in 2000, in which R(z) is a polynomial of degree at least 2.Recently, we [10] improved Theorem A in the case of R(z) being polynomial.Theorem B.
Letα(z) be a holomorphic function, ℱ a family of meromorphic functions in a domain D,  and P(z) a polynomial of degree at least 3. If P∘f(z) and P∘g(z) share α(z) IM for each pair f(z),g(z)∈ℱ and one of the following conditions holds:(1)
P(z)-α(z0) has at least three distinct zeros for any z0∈D;(2)
there existsz0∈D such that P(z)-α(z0) has at most two distinct zeros and α(z) is nonconstant. Assume that β0 is the zero of P(z)-α(z0) with multiplicity p and that the multiplicities l and k of zeros of f(z)-β0 and α(z)-α(z0) at z0, respectively, satisfy k≠lp, for all f(z)∈ℱ, then ℱ is normal in D.Remark 1.1.
α(z) assuming the value α(z0) with multiplicity k at z0∈D means that α(z)-α(z0)=(z-z0)kβ(z) or α(z)=(z-z0)-kβ(z) and β(z0)≠0.In this paper, we extend Theorem B in the case ofℱ being holomorphic and prove Theorem 1.2. In order to state it, we need some notations below. Set (1.1)P(z,w)∶=(w-a1(z))(w-a2(z))⋯(w-ap(z)),
where ai(z),i=1,2,…,p, are holomorphic in D;  Pw∘f(z)∶=P(z,f(z)).Theorem 1.2.
Letα(z) be a holomorphic function, ℱ a family of holomorphic functions in a domain D,  and P(z,w) a polynomial in variable w as in (1.1) with p≥2.  If Pw∘f(z) and Pw∘g(z) share α(z) IM for each pair f(z),g(z)∈ℱ and one of the following conditions holds:(1)
P(z0,z)-α(z0) has at least two distinct zeros for any z0∈D;(2)
there existsz0∈D such that P(z0,z)-α(z0) has only one distinct zero and α(z) is nonconstant. Assume that β0 is the zero of P(z0,z)-α(z0) and that the multiplicities l and k of zeros of f(z)-β0 and α(z)-α(z0) at z0, respectively, satisfy k≠lp, for all f(z)∈ℱ, then ℱ is normal in D.Remark 1.3.
Example1.4 shows that p=degwP(z,w)≥2 is best possible in Theorem 1.2.Example 1.4.
LetP(z,w)=w+z2,D={|z|<1}, and let ℱ∶={fn}, where
(1.2)fn(z)∶=nz,n=1,2,….
If Pw(fn(z))=nz+z2=z2, then z=0. Hence Pw∘fn(z) and Pw∘fm(z) share α(z):=z2 IM for each pair fn(z),fm(z)∈ℱ. Obviously, for z0=0∈D, we have that P(0,z)-α(0) has only one distinct zero β0=0 and z2 is nonconstant, noting that the multiplicities l=1 and k=2 of zeros of fn(z) and α(z) at 0, respectively, satisfy k≠lp, for all fn(z)∈ℱ. However, clearly, ℱ is not normal at 0.Remark 1.5.
In Theorem1.2 setting α(z) constant zero and P(z,w) a polynomial in variable w that vanishes exactly on a finite set of holomorphic functions S, we obtain Corollary 1.6 which generalizes the famous Montel’s criterion that a holomorphic family omitting 2 (or more) values is normal.Corollary 1.6.
Letℱ be a family of holomorphic functions on a domain D. Let S be a finite set of holomorphic functions with at least 2 elements. If all functions in ℱ share the set Signoring multiplicities, that is, if for all f(z),g(z)∈ℱ and for all z∈D(1.3)f(z)∈S⇔g(z)∈S,
then ℱ is normal in D.In 2010, we [11] obtained a normal criterion as follows.Theorem C.
Letα(z) be an analytic function, ℱ a family of analytic functions in a domain D, and H(z) a transcendental entire function. If H∘f(z) and H∘g(z) share α(z) IM for each pair f(z),g(z)∈ℱ and one of the following conditions holds:(1)
H(z)-α(z0) has at least two distinct zeros for any z0∈D;(2)
α(z) is nonconstant and there exists z0∈D such that H(z)-α(z0):=(z-β0)pQ(z) has only one distinct zero β0 and suppose that the multiplicities l and k of zeros of f(z)-β0 and α(z)-α(z0) at z0, respectively, satisfy k≠lp, for each f(z)∈ℱ, where Q(β0)≠0;(3)
there exists az0∈D such that H(z)-α(z0) has no zero and α(z) is nonconstant, then ℱ is normal in D.However, there exists a gap in the proof of Theorem C which is Theorem 1.1 in our original paper [11]. We will give the correct proof after the proof of Theorem 1.1 in Section 3.


## 2. Preliminary Lemmas

In order to prove our result, we need the following lemmas. The first one extends a famous result by Zalcman [12] concerning normal families.Lemma 2.1 ( see [13]).
Letℱ be a family of meromorphic functions on the unit disc. Then ℱ is not normal on the unit disc if and only if there exist(a)
a  number0<r<1;(b)
pointszn with |zn|<r;(c)
functionsfn∈ℱ;(d)
positive  numbersρn→0such that gn(ζ)∶=fn(zn+ρnζ) converges locally uniformly to a nonconstant meromorphic function g(ζ), whose order is at most 2.Remark 2.2.
Ifℱ is a family of holomorphic functions on the unit disc in Lemma 2.1, then g(ζ) is a nonconstant entire function.Lemma2.3 is very useful in the proof of our main theorem. In order to state them, we denote by U(z0,r) (or U0(z0,r)) the open (or punctured) disc of radius r around z0, that is, (2.1)U(z0,r):={z∈C:|z-z0|<r},U0(z0,r):={z∈C:0<|z-z0|<r}.Lemma 2.3 (see [9] or [14]).
Let{fn(z)} be a family of analytic functions in U(z0,r). Suppose that {fn(z)} is not normal at z0 but is normal in U0(z0,r).  Then there exists a subsequence {fnk(z)} of {fn(z)} and a sequence of points {znk} tending to z0 such that fnk(znk)=0, but {fnk(z)} tending to infinity locally uniformly on U0(z0,r).


## 3. Proof of the Results

Proof of Theorem 1.1.
Without loss of generality, we assume thatD={z∈C,|z|<1}. Then we consider the following two cases.
Case 1.
P(z0,z)-α(z0) has at least two distinct zeros a and b for any z0∈D.
Suppose thatℱ is not normal in D. Without loss of generality, we assume that ℱ is not normal at z=0.
By Lemma2.1, there exist zn→0, fn∈ℱ, ρn→0+ such that
(3.1)hn(ξ)=fn(zn+ρnξ)⟶h(ξ)
uniformly on any compact subset of C, where h(ξ) is a nonconstant entire function.
Hence(3.2)Pw∘fn(zn+ρnξ)-α(zn+ρnξ)⟶Pw∘h(ξ)-α(0)
uniformly on any compact subset of C.
We claim thatPw∘h(ξ)-α(0) has at least two distinct zeros.
Ifh(ξ) is a nonconstant polynomial, then both of the two equations of h(ξ)=a and h(ξ)=b have roots. So Pw∘h(ξ)-α(0) has at least two distinct zeros.
Ifh(ξ) is a transcendental entire function, then by Picard’s theorem [3] at least one of the two equations h(ξ)=a or h(ξ)=b has infinitely many zeros.
Thus, the claim gives that there existξ1 and ξ2 such that
(3.3)Pw∘h(ξ1)-α(0)=0,Pw∘h(ξ2)-α(0)=0(ξ1≠ξ2).
We choose a positive numberδ small enough such that D1∩D2=∅ and Pw∘h(ξ)-α(0) has no other zeros in D1∪D2 except for ξ1 and ξ2, where
(3.4)D1={ξ∈C;|ξ-ξ1|<δ},D2={ξ∈C;|ξ-ξ2|<δ}.
By (3.2) and Hurwitz’s theorem [14], for sufficiently large n there exist points ξ1n∈D1, ξ2n∈D2 such that
(3.5)Pw∘fn(zn+ρnξ1n)-α(zn+ρnξ1n)=0,Pw∘fn(zn+ρnξ2n)-α(zn+ρnξ2n)=0.
Noting that Pw∘fm(z) and Pw∘fn(z) share α(z) IM, it follows that
(3.6)Pw∘fm(zn+ρnξ1n)-α(zn+ρnξ1n)=0,Pw∘fm(zn+ρnξ2n)-α(zn+ρnξ2n)=0.
Taking n→∞, we obtain
(3.7)Pw∘fm(0)-α(0)=0.
SinceP(z,w) is a polynomial in variable w, we know that the zeros of
(3.8)Pw∘fm(ξ)-α(ξ)
have no accumulation points except for finitely many fm, and then
(3.9)zn+ρnξ1n=0,zn+ρnξ2n=0,
or equivalently
(3.10)ξ1n=-znρn,ξ2n=-znρn.
This contradicts the facts that ξ1n∈D1, ξ2n∈D2, D1∩D2=∅. So ℱ is normal in Case 1.
Case 2.
There existsz0∈D such that P(z0,z)-α(z0) has only one distinct zero and α(z) is nonconstant. Assume that β0 is the zero of P(z0,z)-α(z0) and that the multiplicities l and k of zeros of f(z)-β0 and α(z)-α(z0) at z0, respectively, satisfy k≠lp, for all f(z)∈ℱ.
We will prove thatℱ is normal at any z0∈D. Without loss of generality, we can assume that z0=0.
We can writeP(z0,z)-α(0):=(z-β0)pH(z), where H(z) is a polynomial and H(β0)≠0.
Sinceα(z) is nonconstant, there exists a punctured neighbourhood U0(0,r) such that
(3.11)α(z)≠α(0).
We claim thatℱ is normal at z0′∈U0(0,r) for small enough r. In fact, P(z0,z)-α(z0′) has at least three distinct zeros. Then Case 1 tells us that this claim is true.
Next we prove thatℱ is normal at z=0. For any {fn(z)}⊂ℱ, by the former claim, there exists a subsequence of functions {fnm(z)} such that
(3.12)fnm(z)-β0⟶G(z),
uniformly on a punctured disc U0(0,r).
For the sake of simplicity, we denote{fnm(z)} by {fn(z)} in what follows.
Suppose that{fn(z)} is not normal at z=0. We claim that there exists a sequence of points zn∈U(0,r) (zn→0) such that Pw∘fn(zn)-α(zn)=0.
By Lemma2.3, we have that there exists a sequence of points {zn′} tending to 0 such that fn(zn′)=β0 and G(z)≡∞. Thus, fn(z)→∞ on U0(0,r) and fn(zn′)=β0 for a sequence of points zn′→0. We know that if n is sufficiently large, then
(3.13)|(fn(z)-β)-fn(z)+β0|=|β-β0|≤ρ<|fn(z)-β0|
for |z|=δ and β∈U(β0,ρ). For large n, we also have |zn′|<δ, and thus we deduce from Rouché’s theorem [15] that fn(z) takes the value β∈U(β0,ρ); that is, we have fn(U(0,δ))⊃U(β0,ρ) for large n. Since also fn(∂U(0,δ))∩U(β0,ρ)=∅ for large n, we find a component U of fn-1(U(β0,ρ)) contained in U(0,δ) for such n. Moreover, U is a Jordan domain and fn:U→U(β0,ρ) is a proper map.
Forz∈∂U, we then have fn(z)∈∂U(β0,ρ) and thus |Pw∘fn(z)-α(0)|>ϵ. Hence
(3.14)|Pw∘fn(z)-α(z)-(Pw∘fn(z)-α(0))|=|α(z)-α(0)|<ϵ<|Pw∘fn(z)-α(0)|
for z∈∂U. Noting that Pw∘fn(zn′)-α(0)=0, Rouché’s theorem [15] now shows that our claim holds.
By the claim and almost the same argument as in Case1, we obtain that zn=0 for sufficiently large n. By P(0,z)-α(0)=(z-β0)pH(z), we have
(3.15)Pw∘fn(z)-α(z)=(fn(z)-β0)pH(fn(z))-(α(z)-α(0)),(fn(0)-β0)pH(fn(0))=Pw∘fn(0)-α(0)=0.
Hence
(3.16)Pw∘fn(z)-α(z)=zk[zlp-khn(z)-β(z)],iflp>k,Pw∘fn(z)-α(z)=zlp[hn(z)-zk-lpβ(z)],iflp<k,
where hn(z),β(z) are holomorphic functions and hn(0)≠0,β(0)≠0.
SetHn(z)∶=zlp-khn(z)-β(z), if lp>k, or Hn(z)∶=hn(z)-zk-lpβ(z), if lp<k. Obviously, Hn(0)=-β(0)≠0 or Hn(0)=hn(0)≠0.
By the same argument as in the former case, we see that there exists a sequence of pointszn*∈U(0,r) such that zn*→0 and Hn(zn*)=0. Obviously, zn*≠0 and
(3.17)Pw∘fn(zn*)-α(zn*)=0.
Noting thatPw∘fn(z) and Pw∘fm(z) share α(z) IM, we obtain that
(3.18)Pw∘fm(zn*)-α(zn*)=0
for each m. Thus, taking n→∞,Pw∘fm(0)-α(0)=0. Since the zeros of Pw∘fm(ξ)-α(ξ) have no accumulation points except for finitely many fm, we have zn*=0. This contradicts our supposition.
Theorem 1.1 is proved completely.Proof of the Gap of Theorem C.
In the original proof (3.7) becomes
(3.19)H∘fm(0)-α(0)=0,
where H(z) is a transcendental entire function.
The zeros ofH∘fm(ξ)-α(ξ) maybe have accumulation points, for H∘fm(ξ)-α(ξ)≡0. However, we claim that H∘fm(ξ)-α(ξ)≡0 does not hold for infinitely many m.
In fact, if it is not true, by using Lemma2.1 for these infinitely fm(z), then there exist a sequence of points zm→0, fm∈{fn(z)}, and ρm→0+ such that gm(ξ)∶=fm(zm+ρmξ)→g(ξ) uniformly on any compact subset of C, where g(ξ) is a nonconstant entire function. Thus, H∘g(ξ)-α(0)≡0. This is impossible.



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