# Some Normal Criteria about Shared Values with Their Multiplicity Zeros

Authors: Jianming Qi; Taiying Zhu
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

Letℱ be a family of meromorphic functions in the domain
D, all of whose zeros are multiple. Let n(n≥2) be an integer and let a, b be two nonzero finite complex numbers. If f+a(f')n and g+a(g')n share b in D for every pair of functions f,g∈ℱ, then ℱ is normal in D.


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## Body

## 1. Introduction and Main Results

We useℂ to denote the open complex plane, ℂ̂(=ℂ∪{∞}) to denote the extended complex plane, and D to denote a domain in ℂ. With renewed interest in normal families of analytic and meromorphic functions in plane domains, mainly because of their role in complex dynamics, it has become quite interesting to talk about normal families in their own right.We will be concerned with the analytic maps (i.e., meromorphic functions)(1.1)f:(D,|⋅|R2)⟶(Ĉ,X)
from D (endowed with the Euclidean metric) to the extended complex plane ℂ̂ endowed with the spherically metric X given by (1.2)X(z,z′)=|z-z′|1+|z|21+|z′|2z,z′∈Ĉ.A familyℱ of meromorphic functions defined in D is said to be normal, in the sense of Montel, if for any sequence {fn}⊂ℱ, there exists a subsequence {fnj} such that fnj converges spherically locally and uniformly in D to a meromorphic function or ∞. Clearly, ℱ is normal in D if and only if it is normal at every point of D (see [1, 2]).Letf and g be two nonconstant meromorphic functions in D, and a∈ℂ̂. We say that f and g share the value a in D, if f-a and g-a have the same zeros (ignoring multiplicities). When a=∞ the zeros of f-a means the poles of f (see [3]).Influenced from Bloch's principle [4], every condition which reduces a meromorphic function in the plane ℂ to a constant, makes a family of meromorphic functions in a domain D normal. Although the principle is false in general (see [5]), many authors proved normality criterion for families of meromorphic functions corresponding to Liouville-Picard type theorem (see [1, 2, 6]).It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schiff [1] first proved an interesting result that a family of meromorphic functions in a domian is normal if in which every function shares three distinct finite complex numbers with its first derivative. And later, Sun [7] proved that a family of meromorphic functions in a domian is normal if in which each pair of functions share three fixed distinct valus, which is an improvement of the famous Montel's Normal Criterion [8] by the ideas of shared values. More results about normality criteria concerning shared values can be found, for instance, (see [9–12]) and so on.In 1989, Schwick [13] proved that let ℱ be a family of meromorphic functions in a domain D, if (fn)(k)≠1 for every f∈ℱ, where n, k are two positive integers and n≥k+3, then ℱ is normal in D.Recently, by the ideas of shared values, Li and Gu [14] proved the following.Theorem A.
Letℱ be a family of meromorphic functions in D. If (fn)(k) and (gn)(k) share a in D for each pair of functions f, g in ℱ, where n, k are two positive integers such that n≥k+2 and a is a finite nonzero complex number, then ℱ is normal in D.In 1998, Wang and Fang [15] obtained the following result.Theorem B.
Letℱ be a family of meromorphic functions in D. Let k be a positive integer and b be a nonzero finite complex number. If, for each f∈ℱ, all zeros of f have multiplicity at least k+2, and f(k)(z)≠b on D, then ℱ is normal in D.Remark 1.1.
By a counter-example, Wang and Fang [15] show Theorem B is not valid if all zeros of f have multiplicity less than k+2.It is natural to ask whether Theorem B can be improved by the idea of shared values. In this paper we investigate this problem and obtain the following result.Theorem 1.2.
Letk (k≥2) be an integer and b be a nonzero finite complex number, and let ℱ be a family of meromorphic functions in D, all of whose zeros have multiplicity at least k+2. If, for every pair f,g∈ℱ, all zeros of f(k)(z),g(k)(z) are multiple, f(k)(z) and g(k)(z) share b in D, then ℱ is normal in D.Remark 1.3.
Comparing with Theorem A, Theorem1.2 releases the condition that the poles of f(z) have multiplicity at least k+2, which improves Theorem A in some sense.Example 1.4.
TakeD={z:|z|<1} and
(1.3)F={fm=mzk+2∣m=1,2,…}
or
(1.4)F={fm=emz∣m=1,2,…}.
Obviously, any fm has zeros of multiplicity at least k+2 and fm(k)(z) has zeros of multiplicity at least 2. For any fm and gm in ℱ, we have fm(k)(z) and gm(k)(z) share 0. However the families ℱ is not normal at z=0.Remark 1.5.
Example1.4 shows that the condition b≠0 in Theorem 1.2 is inevitable.In 1959, Hayman [16] proved that let f be a meromorphic function in ℂ, if f′-afn≠b, where n is a positive integer and a, b are two finite complex numbers such that n≥5 and a≠0, then f is a constant. On the other hand, Mues [17] showed that for n=3,4 the conclusion is not valid.The following theorem which confirmed a Hayman's well-known conjecture about normal families in [18].Theorem C.
Letℱ be a family of meromorphic functions in D, n be a positive integer and a, b be two finite complex numbers such that a≠0. If n≥3 and for each function f∈ℱ, f′-afn≠b, then ℱ is normal in D.In 2008, by the ideas of shared values, Zhang [12] proved the following.Theorem D.
Letℱ be a family of meromorphic functions in D, n be a positive integer and a, b be two finite complex numbers such that a≠0, If n≥4 and for every pair of functions f and g in ℱ, If f′-afn and g′-agn share the value b, then ℱ is normal in D.In 1994, Ye [19] considered a similar problem and obtained that if f is a transcendental meromorphic function and a is a nonzero finite complex number, then f+a(f′)n assumes every finite complex value infinitely often for n≥3. Ye [19] also asked whether the conclusion remains valid for n=2.In 2008, Fang and Zalcman [20] solved this problem and obtained the following result.Theorem E.
Letf be a transcendental meromorphic function and a be a nonzero complex number. Then f+a(f′)n assumes every complex value infinitely often for each positive integer n≥2.Remark 1.6.
By a special example, Fang and Zalcman [20] show Theorem E is not valid for n=1.On the basis of the above results, Fang and Zalcman [20] obtained the normality criterion corresponding to Theorem E.Theorem F.
Letℱ be a family of meromorphic functions in D. Let n (n≥2) be an integer, and let a (a≠0), b be two finite complex numbers. If, for each f∈ℱ, all zeros of f are multiple and f+a(f′)n≠b in D, then ℱ is normal in D.Likewise Theorem D, it is natural to ask whether Theorem F can be improved by the ideas of shared values. In this paper we investigate this problem and obtain the following result.Theorem 1.7.
Letℱ be a family of meromorphic functions in D, all of whose zeros are multiple and let n (n≥2) be an integer and a, b be two nonzero finite complex numbers. If f+a(f′)n and g+a(g′)n share b in D for every pair of functions f,g∈ℱ, then ℱ is normal in D.Here we will generalize above results by allowingf(z)+a(f′(z))n-b to have zeros.For the sake of convenience, we give the following notations:(1.5)E(F,n):={z∈D∣f(z)+a(f′(z))n-b=0,f(z)∈F};U(z0,r)¯:={z∈D∣|z-z0|≤r};E(z0,r,F,n):=⋃f∈F{z∈U(z0,r)∣f(z)+a(f′(z))n-b=0,f(z)∈F};♯E denotes the number of the elements in the set E.Corollary 1.8.
Letℱ be a family of meromorphic functions in D, all of whose zeros are multiple and let n (n≥2) be an integer, and let a, b be two nonzero finite complex numbers. If, for each f∈ℱ, there exists a positive constant M such that |f(z)|≥M whenever z∈E(ℱ,n), then ℱ is normal in D.Corollary 1.9.
Letℱ be a family of meromorphic functions in D, all of whose zeros are multiple and let n(n≥2) be an integer, and let a (a≠0), b be two finite complex numbers. Suppose that(i)
there are two distinct complex numbersc,d∈ℂ∪{∞} such that f(z)≠c,d for every f(z)∈ℱ;(ii)
for arbitraryz0∈D, there is r>0 such that U(r,z0)⊂D and ♯E(z0,r,ℱ,n)<∞. Then ℱ is normal in D.Example 1.10 (see [20]).
LetD={z:|z|<1} and ℱ={fm} where fm:=-(1/4)z2+mz+1/2 and m is a positive integer. Then for every pair of functions f,g∈ℱ, fm+(fm′)2≡m2+1/2 and gm+(gm′)2≡m2+1/2 share any point in D. However ℱ is not normal at z=0.Remark 1.11.
Example1.10 shows the condition that all zeros of f∈ℱ are multiple is necessary in Theorem 1.2.Example 1.12.
TakeD={z:|z|<1},
(1.6)F={fm=mz2∣m=1,2,…}.Remark 1.13.
Forn=2, we obtain fm+a(fm′)2=mz2+(2m)2az2. So f+a(f′)2 and g+a(g′)2 share 0 in D for every pair functions f,g∈ℱ. But ℱ is not normal in D.
Forn=2, from mz2+a(2mz)2=b, we deduce (m+4am2)z2=b. Obviously, If m sufficiently large, then m+4am2≠0, |fm(z)|=|mz2|=|mb/(m+4am2)|=|b/(1+4am)| and then |fm(z)|→0. Example 1.12 shows that the condition b≠0 in Theorem 1.7 is inevitable for n=2. Example 1.12 also shows the hypothesis there exists a positive constant M such that |f(z)|≥M for all f∈ℱ whenever z∈E(ℱ,n) cannot be omitted in Corollary 1.8.Remark 1.14.
Some ideas of this paper are based on [9, 12, 21–23].


## 2. Preliminary Lemmas

In order to prove our theorems, we need the following lemmas.First, we need the following well-known Pang-Zalcman lemma, which is the local version of [10, 24].Lemma 2.1.
Letℱ be a family of meromorphic functions in the unit disc Δ with the property that for each f(z)∈ℱ, all zeros of multiplicity at least k. Suppose that there exists a number A≥1 such that |f(k)(z)|≤A whenever f(z)∈ℱ and f(z)=0. If ℱ is not normal at a point z0∈Δ, then for 0≤α≤k, there are(1)
a sequence of complex numberszn∈Δ, zn→z0;(2)
a sequence of functionsfn∈ℱ;(3)
a sequence of positive numbersρn→0+;such that gn(ξ)=ρn-αfn(zn+ρnξ) converge locally uniformly (with respect to the spherical metric) to a nonconstant meromorphic function g(ξ) on ℂ, and moreover, the zeros of g(ξ) are of multiplicity at least k, g♯(ξ)≤g♯(0)=kA+1. In particular, g has order at most 2.In Lemma2.1, the order of g is defined by using the Nevanlinna's characteristic function T(r,g): (2.1)ρ(g)=lim⁡r→∞sup⁡log⁡T(r,g)log⁡r.
Here g♯(ξ) denotes the spherical derivative (2.2)g♯(ξ)=|g′(ξ)|1+|g(ξ)|2.Lemma 2.2 (see [25]).
Letf(z) be a transcendental meromorphic function of finite order in ℂ. If f(z) has no simple zero, then f′(z) assumes every nonzero finite value infinitely often.Lemma 2.3 (see [15]).
Letf(z) be a transcendental meromorphic function in ℂ. If all zeros of f(z) have multiplicity at least 3, for any positive integer k, then f(k)(z) assumes every nonzero finite value infinitely often.Lemma 2.4.
Letk≥2 be an integer and b be a nonzero finite complex number and let f(z) be a nonconstant rational meromorphic function in ℂ, all zeros of f(z) have multiplicity at least k+2. If all zeros of f(k)(z) are multiple, then f(k)(z)-b has at least two distinct zeros.Proof.
In the following, we consider two cases.Case 1.
Assume, to the contrary, thatf(k)(z)-b has at most one zero z0.Subcase 1.1 (f(z) is a nonconstant polynomial).
Setf(k)(z)-b=K(z-z0)l, where K is a nonzero constant, l is a positive integer. Because all zeros of f(k)(z) are multiple, we obtain l≥2 and f(k+1)(z)=Kl(z-z0)l-1 (l-1≥1). Thus, f(k+1)(z) has exactly one zero. Since all zeros of f(z) have multiplicity at least k+2, we derive f(k)(z) has the same zero z0. Hence f(k)(z0)=0, which contradicts with f(k)(z0)=b≠0.Subcase 1.2 (f(z) is rational but not a polynomial).
By the assumption of Lemma2.4, we may set
(2.3)f(k)(z)=A(z-α1)m1(z-α2)m2⋯(z-αs)ms(z-β1)n1(z-β2)n2⋯(z-βt)nt=P(z)Q(z),
where A is a nonzero constant. Since all zeros of f(k)(z) are multiple, we find mi≥2 (i=1,2,…,s), nj≥(k+1) (j=1,2,…,t) (k≥2).
For simplicity, we denote(2.4)m1+m2+⋯+ms=M≥2s,n1+n2+⋯+nt=N≥(k+1)t,
where P(z) and Q(z) are coprime polynomials of degree M, N, respectively, in (2.3).
Sincef(k)(z)-b has just a zero z0, from (2.3) we obtain(2.5)f(k)(z)=b+B(z-z0)l(z-β1)n1(z-β2)n2⋯(z-βt)nt=P(z)Q(z).
By b≠0, we deduce z0≠αi (i=1,…,s), where B is a nonzero constant.
From (2.3), we get(2.6)f(k+1)(z)=(z-α1)m1-1(z-α2)m2-1⋯(z-αs)ms-1g1(z)(z-β1)n1+1⋯(z-βt)nt+1,
where g1(z) is polynomial of degree at most s+t-1.
Differentiating (2.5) yields (2.7)f(k+1)(z)=(z-z0)l-1g2(z)(z-β1)n1+1⋯(z-βt)nt+1,
where g2(z)=B(l-N)zt+bt-1zt-1+⋯+b0, (bt-1⋯b0 are constants).
Next we distinguish two subcases.
Subcase 1.2.1 (l≠N). By using (2.5), we deduce degP(z)≥degQ(z), that is, M≥N. Since z0≠αi, that (2.6) and (2.7) imply M-s≤degg2=t. By using (2.4), we get
(2.8)M≤s+t≤M2+Nk+1≤M2+Mk+1<M(k≥2).
Which is a contradiction since k≥2.Subcase 1.2.2 (l=N). We further distinguish two subcases:
Subcase 1.2.2.1 (M≥N). By using (2.6) and (2.7), we obtain M-s≤degg2≤t-1<t. Similar to Subcase 1.2.1, we obtain a contradiction M<M.Subcase 1.2.2.2 (M<N). By using (2.6) and (2.7) again, we deduce l-1≤degg1≤s+t-1, and hence
(2.9)N=l≤s+t-1+1=s+t≤M2+Nk+1<N
this is impossible for k≥2.Case 2 (f(k)(z)-b≠0 (k≥2)).
By Nevanlinna's second fundamental theorem, we have(2.10)T(r,f(k))≤N¯(r,f(k))+N¯(r,1f(k))+N¯(r,1f(k)-b)+S(r,f(k))≤1k+1N(r,f(k))+12N(r,1f(k))+S(r,f(k))≤(12+1k+1)T(r,f(k))+S(r,f(k)).
It follows that T(r,f(k))=S(r,f(k)), a contradiction.
The  proof of Lemma2.4 is complete.Example 2.5.
Take(2.11)f(z)=(z+1)4z2.Remark 2.6.
By a simple computation, fork=b=2, we deduce f′′(z)-2 has only one zero at z=-3/4. Example 2.5 shows that the condition all of zeros of f(k)(z) are the multiple seems not to be omitted.Lemma 2.7.
Letg(z) be a nonconstant rational meromorphic function in ℂ and let n (n≥2) be an integer. If all zeros of g(z) are multiple, then (g′(z))n-c=0 has at least two distinct zeros (where c is a nonzero constant).Proof.
Suppose that(g′(z))n-c≠0, we set c1,c2,…,cn the distinct solutions of wn=c (n≥2). By Nevanlinna's second fundamental theorem,
(2.12)T(r,g′)≤N¯(r,g′)+N¯(r,1g′-c1)+⋯+N¯(r,1g′-cn)+S(r,g′)≤N¯(r,g′)+S(r,g′)≤12N(r,g′)+S(r,g′)≤12T(r,g′)+S(r,g′).
It follows that T(r,g′)=S(r,g′), a contradiction.
Assume, to the contrary, that(g′(z))n-c has exactly one zero z0.
Ifg(z) is a nonconstant polynomial, then we set (g′(z))n-c=K(z-z0)l, where K is nonzero constant and l is a positive integer. Because all zeros of g(z) are multiple and n≥2, we have l≥2 and ((g′(z))n)′=Kl(z-z0)l-1 (l-1≥1). Thus, ((g′(z))n)′ has exactly one zero. Noting that all zeros of (g′(z))n are multiple, we deduce (g′(z))n has the same zero z0. Hence (g′(z0))n=0, which contradicts with (g′(z0))n=c≠0.
Ifg is a rational function but not a polynomial, by all zeros of g are multiple, then we know g′ is not a constant. For n≥2, we obtain (g′-c1)(g′-c2)⋯(g′-cn)=0 (where c1,…,cn are distinct roots of wn=c). So there exists a z0 such that g′(z0)=cj and g′(z0)≠ck (where 1≤j<k≤n). We have(2.13)g′(z)=cj+A(z-z0)nP(z)≡ck+BP(z)(n≥2,A,B∈C∖{0}).
So we obtain
(2.14)(ck-cj)P(z)+B≡A(z-z0)n,
for some nonconstant polynomial P(z) with P(z0)≠0. Then c0P(z)+B≡A(z-z0)n, where c0=cj-ck is a nonzero constant. Furthermore,
(2.15)c0P′(z)≡An(z-z0)n-1.
Observing that g′ has only multiple poles, and by (2.13), we obtain P(z) has multiple zeros and P′(z0)≠0. Putting z0 into (2.15), we get that P′(z0)=0, which is a contradiction.
The proof of Lemma2.7 is complete.


## 3. Proof of Theorems

Proof of Theorem1.2.
We may assume thatD={|z|<1}. Suppose that ℱ is not normal in D. Without loss of generality, we assume that ℱ is not normal at z0=0. Then, by Lemma 2.1, there are a sequence of complex numbers zj, zj→0 (j→∞); a sequence of functions fj∈ℱ; and a sequence of positive numbers ρj→0+ such that gj(ξ)=ρj-kfj(zj+ρjξ) converges uniformly with respect to the spherical metric to a nonconstant mermorphic function g(ξ) in ℂ and all zeros of g(ξ) have the multiplicity at least k+2. Moreover, g(ξ) is at most  of order 2.
From the above, we getgj(k)(ξ)=fj(k)(zj+ρjξ). Noting that all zeros of fj(k)(z) are multiple, by Hurwitz's theorem, then all zeros of gj(k)(ξ) have the multiplicity at least 2.
On every compact subsets ofℂ which contains no poles of g, we have(3.1)fj(k)(zj+ρjξ)-b=gj(k)(ξ)-b
converges uniformly with respect to the spherical metric to g(k)(ξ)-b(k≥2).
Ifg(k)(ξ)-b≡0, then g is a polynomial of degree k. This contradicts with that all zeros of f have multiplicity at least k+2.
Sinceg is a nonconstant meromorphic function, by Lemmas 2.3 and 2.4, we deduce that g(k)(ξ)-b has at least two distinct zeros.
Next we will prove thatg(k)(ξ)-b has just a unique zero. On the contrary, let ξ0 and ξ0* be two distinct solutions of g(k)(ξ)-b, and choose δ(>0) small enough such that D(ξ0,δ)∩D(ξ0*,δ)=∅ where D(ξ0,δ)={ξ:|ξ-ξ0|<δ} and D(ξ0*,δ)={ξ:|ξ-ξ0*|<δ}. From (3.1) and Hurwitz's theorem, there are points ξj∈D(ξ0,δ), ξj*∈D(ξ0*,δ) such that for sufficiently large j(3.2)fj(k)(zj+ρjξj)-b=0,fj(k)(zj+ρjξj*)-b=0.
By the hypothesis that for each pair of functionsf and g in ℱ, f(k)(ξ) and g(k)(ξ) share b in D, we know that for any positive integer m(3.3)fm(k)(zj+ρjξj)-b=0,fm(k)(zj+ρjξj*)-b=0.
Fix m, take j→∞, and in view of that zj+ρjξj→0, zj+ρjξj*→0, we have
(3.4)fm(k)(0)-b=0.
Since the zeros of fm(k)-b have no accumulation point, we have that zj+ρjξj=0 and zj+ρjξj*=0.
Hence(3.5)ξj=-zjρj,ξj*=-zjρj.
This contradicts with ξj∈D(ξ0,δ), ξj*∈D(ξ0*,δ) and D(ξ0,δ)∩D(ξ0*,δ)=∅ So g(k)(ξ)-b has just a unique zero, which contradicts the fact that g(k)(ξ)-b has at least two distinct zeros.
The proof of Theorem1.2 is complete.Proof of Theorem1.7.
Likewise the proof of Theorem1.2, we assume that ℱ is not normal at z0=0. Then, by Lemma 2.1, there are a sequence of complex numbers zj, zj→0(j→∞), a sequence of functions fj∈ℱ, and a sequence of positive numbers ρj→0+ such that gj(ξ)=ρj-1fj(zj+ρjξ) converges uniformly with respect to the spherical metric to a nonconstant mermorphic functions g(ξ), all zeros of g(ξ) are multiple. Moreover, g(ξ) is of order at most 2.
On every compact subsets ofℂ which contains no poles of g, we have(3.6)fj(zj+ρjξ)+a(fj′(zj+ρjξ))n-b=ρjgj(ξ)+a(gj′(ξ))n-b
converges  uniformly with respect to the spherical metric to a(g′(ξ))n-b(n≥2).
Ifa(g′(ξ))n-b≡0, then g is a polynomial of degree 1, which contradicts with the zeros of g(ξ) are multiple.
By Lemmas2.2 and 2.7, we derive that a(g′(ξ))n-b has at least two distinct zeros.
Next we will prove thata(g′(ξ))n-b has just a unique zero. On the contrary, let ξ0 and ξ0* be two distinct solutions of (g′(ξ))n-c=0 (where c=b/a≠0), and choose δ(>0) small enough such that D(ξ0,δ)∩D(ξ0*,δ)=∅ where D(ξ0,δ)={ξ:|ξ-ξ0|<δ} and D(ξ0*,δ)={ξ:|ξ-ξ0*|<δ}. From (3.6), by Hurwitz's theorem, there are points ξj∈D(ξ0,δ), ξj*∈D(ξ0*,δ) such that for sufficiently large j(3.7)fj(zj+ρjξj)+a(fj′(zj+ρjξj))n-b=0,fj(zj+ρjξj*)+a(fj′(zj+ρjξj*))n-b=0.
By the hypothesis that for each pair of functionsf and g in ℱ, f+a(f′)n and g+a(g′)n share b in D, we know that for any positive integer m(3.8)fm(zj+ρjξj)+a(fm′(zj+ρjξj))n-b=0,fm(zj+ρjξj*)+a(fm′(zj+ρjξj*))n-b=0.
Fix m, take j→∞, and in view of that zj+ρjξj→0, zj+ρjξj*→0, then
(3.9)fm(0)+a(fm′(0))n-b=0.
Since the zeros of fm+a(fm′)n-b have no accumulation point, we have that zj+ρjξj=0, zj+ρjξj*=0.
Hence(3.10)ξj=-zjρj,ξj*=-zjρj.
This contradicts with ξj∈D(ξ0,δ), ξj*∈D(ξ0*,δ) and D(ξ0,δ)∩D(ξ0*,δ)=∅. So (g′(ξ))n-c has just a unique zero which contradicts with the fact (g′(ξ))n-c has at least two distinct zeros.
The proof of Theorem1.7 is complete.Proof of Corollary1.8.
Similar to the proof of Theorem1.2, we assume that ℱ is not normal at z0=0. Then by Lemma 2.1, there exist a sequence of functions fj∈ℱ, a sequence of complex numbers zj→z0 and a sequence of positive numbers ρj→0, such that
(3.11)gj(ξ)=ρj-1fj(zj+ρjξ)⟶g(ξ),
locally uniformly with respect to the spherical metric, where g(ξ) is nonconstant meromorphic function on ℂ, all of whose zeros have multiplicity at least 2.
On every compact subsets ofℂ which contains no poles of g, we have(3.12)fj(zj+ρjξj)+a(fj′(zj+ρjξj))n-b=ρjgj(ξ)+a(gj′(ξ))n-b⟶a(g′(ξ))n-b.
If a(g′)n-b≡0, that is (g′)n≡b/a. Then g is a polynomial with degree 1. It contradicts the fact that all zeros of g have multiplicity at least 2.
By Lemmas2.2 and 2.7, we know there exists some ξ0∈ℂ such that (3.13)a(g′(ξ0))n-b=0.
From the above discussion, we get g(ξ0)≠∞. Since gj(ξ)→g(ξ) uniformly in a closed disk U(ξ0;δ)¯, then by Hurwitz's theorem, there is ξj→ξ0 (j→∞) such that
(3.14)fj(zj+ρjξj)+a(fj′(zj+ρjξj))n-b=ρjgj(ξj)+a(gj′(ξj))n-b⟶a(g′(ξ0))n-b=0.
For j sufficiently large, we obtain fj(zj+ρjξj)+a(fj′(zj+ρjξj))n-b=0. By the assumption, we get |fj(zj+ρjξj)|≥M. This implies
(3.15)|gj(ξj)|≥ρj-1M.
Noting thatg is holomorphic at ξ0, then |g(ξ)|≤C for some positive constant C, and for all ξ∈U(ξ0;η)¯. Again by gj→g and for all ɛ>0, there are some j0 such that for all ξ∈U(ξ0;η)¯, we have (3.16)|gj(ξ)-g(ξ)|<ɛ.
For all j≥j0, by (3.15), we find that
(3.17)C≥|g(ξj)|≥|gj(ξj)|-|gj(ξj)-g(ξj)|≥ρj-1M-ɛ.
That is
(3.18)C≥ρj-1M-ɛ
for all j≥j0, which is impossible.
Thus, the proof of Corollary1.8 is proved.Proof of Corollary1.9 (For any z0∈D, we will prove that ℱ is normal at z0).
From the assumptions in the theorem, we can choose a sufficiently smallr>0 such that E(z0,r,ℱ,n)=∅ or {z0}. Thus we see that f+a(f′)n≠b in U(z0,r)∖{z0} for any f(z)∈ℱ. Theorem F gives that ℱ is normal in U(z0,r)∖{z0}. Next we will prove that ℱ is normal at z0.
Since the Möbius transformation forℱ does not change the normality, without loss of generality, we may assume that (c,d)=(0,∞).
SetCr/2:={z∈U(z0,r):|z-z0|=r/2}. By the given condition, we see that ℱ is normal on Cr/2. Hence for any sequence of functions {fn}n=1∞⊂ℱ, there is a subsequence, say {fnk(z)}⊂{fn(z)}, such that as k→∞,(3.19)fnk(z)⟶g(z)
uniformly convergence on Cr/2.
Ifg(z)≢∞, by fnk(z)≠0,∞, we have g(z) is analytic on Cr/2. Hence for all ɛ>0, there is an integer K such that(3.20)|fnk(z)-fnm(z)|<ɛ
for all z∈Cr/2. For arbitrary numbers k,m>K, by the maximum modular theorem, we have
(3.21)|fnk(z)-fnm(z)|<ɛ
for all |z|≤Cr/2. Thus ℱ is normal at z=z0.
Ifg(z)≡∞, then there is an integer K and a positive number M such that(3.22)|fnk(z)|≥M
for all k≥K, z∈Cr/2. Noting that fnk≠0 in U(z0,r), we obtain
(3.23)|fnk(z)|≥M
for all k≥K, |z|≤Cr/2 by the minimum modular theorem. Hence, we have
(3.24)fnk⟶∞(k⟶∞)
uniformly on |z|≤r/2. Therefore ℱ is normal at z=z0.



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*Source: 147878-2011-01-19.xml*