# Normality Criteria of Meromorphic Functions That Share a Holomorphic Function

Authors: Jie Ding; Jianming Qi; Taiying Zhu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Letℱ be a family of meromorphic functions defined in D, let ψ(≢0), a0,a1,...,ak-1 be holomorphic functions in D, and let k be a positive integer. Suppose that, for every function f∈ℱ, f≠0, P(f)=f(k)+ak-1f(k-1)+⋯+a1f′+a0f≠0 and, for every pair functions (f,g)∈ℱ, P(f), P(g) share ψ, then ℱ is normal in D.


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## Body

## 1. Introduction and Main Results

Letℂ be complex plane. Let D be a domain in ℂ. Let ℱ be a family meromorphic functions defined in the domain D. ℱ is said to be normal in D, in the sense of Montel, if for any sequence {fn}⊂ℱ, there exists a subsequence {fnj} such that fnj converges spherically locally uniformly in D, to a meromorphic function or ∞.Letf(z) and g(z) be two meromorphic functions, let a be a finite complex number. If f(z)-a and g(z)-a have the same zeros, then we say they share a or share a IM (ignoring multiplicity) (see [1–3]).Definition 1.1.
Letai(z),(i=1,2,…,q-1),bj(z),(j=1,2,…,n) be analytic in D, let n0,n1,…,nk be nonnegative integers, set
(1.1)P(ω)=ωq+aq-1(z)ωq-1+⋯+a1(z)ω,M(f,f′,…,f(k))=fn0(f′)n1⋯(f(k))nk,γM=n0+n1+⋯+nk,ΓM=n0+2n1+⋯+(k+1)nk,
where M(f,f′,…,f(k)) is called a differential monomial of f, γM the degree of M(f,f′,…,f(k)), and ΓM the weight of M(f,f′,…,f(k)).
From Definition1.1, we give Definition 1.2.Definition 1.2.
LetMj(f,f′,…,f(k)),(j=1,2,…,n) be differential monomials of f. Set
(1.2)H(f,f′,…,f(k))=b1(z)M1(f,f′,…,f(k))+⋯+bn(z)Mn(f,f′,…,f(k)),γH=max⁡{γM1,γM2,…,γMn},ΓH=max⁡{ΓM1,ΓM2,…,ΓMn},
where H(f,f′,…,f(k)) is called the differential polynomial of f, γH the degree of H(f,f′,…,f(k)), and ΓH the weight of H(f,f′,…,f(k)),(1.3)Γγ|H=max⁡{ΓM1γM1,ΓM2γM2,…,ΓMnγMn},G(f)=P(f(k))+H(f,f′,…,f(k)).
In 1979, Gu [4] proved the following result.Theorem A.
Letℱ be a family of meromorphic functions defined in  D, let  k be a positive integer, and let  a be a nonzero constant. If, for each function  f∈ℱ,  f≠0,  f(k)≠a in  D, then  ℱ is normal in  D.Yang [5] and Schwick [6] proved that Theorem A still holds if a is replaced by a holomorphic function ψ(≢0) in Theorem A.Xu [7] improved Theorem A by the ideas of shared values and obtained the following result.Theorem B.
Letℱ be a family of meromorphic functions defined in  D, let  ψ(≢0) be a holomorphic functions and with only simple zeros in  D, and let   k be a positive integer. Suppose that, for every function  f∈ℱ,  f has all multiple poles and  f≠0. If,  for every pair of functions  f and  g,  f(k) and  g(k) share  ψ in  D, then  ℱ is normal in  D.Recently, Xu [7] did not know whether the condition ψ has only simple zero in D and f has all multiple poles are necessary or not in Theorem B.In 2007, Fang and Chang considered the casea=0 in Theorem A. In this note, Fang and Chang [8] proved the following result.Theorem C.
Letℱ be a family of meromorphic functions defined in  D, and let  k be a positive integer, and let  b be a nonzero complex number. If, for each  f∈ℱ,  f≠0,  f(k)≠0 and the zeros of  f(k)-b have multiplicity at least  (k+2)/k, then  ℱ is normal in  D.Remark 1.3.
The number(k+2)/k is sharp, as is shown by the examples in [8].
In 2009, Xia and Xu [9] replaced the constant 1 by a function ψ(z)≢0 in Theorem C. They obtained the following result.Theorem D.
Letℱ be a family of meromorphic functions defined in  D, let  ψ(≢0),  a0,a1,…,ak-1 be holomorphic functions in  D, and let   k be a positive integer. Suppose that, for every function  f∈ℱ,  f≠0,  f(k)+ak-1f(k-1)+⋯+a1f′+a0f≠0 and all zeros of  f(k)+ak-1f(k-1)+⋯+a1f′+a0f-ψ(z) have multiplicity at least  (k+2)/k. If, for  k=1,  ψ has only zeros with multiplicities at most 2 and, for  k≥2,  ψ has only simple zeros, then  ℱ is normal in  D.It is natural to ask whether Theorem D can be improved by the ideas of shared values. In this paper, we investigate the problem and obtain the following results.Theorem 1.4.
Letℱ be a family of meromorphic functions defined in  D, let  ψ(≢0),  a0,a1,…,ak-1 be holomorphic functions in  D, and let  k be a positive integer. Suppose that, for every function  f∈ℱ,  f≠0,  P(f)=f(k)+ak-1f(k-1)+⋯+a1f′+a0f≠0 and, for every pair functions  (f,g)∈ℱ,  P(f) and  P(g) share  ψ, then  ℱ is normal in  D.By Theorem1.4, we immediately deduce.Corollary 1.5.
Letℱ be a family of meromorphic functions defined in  D, let  ψ(≢0),  a0,a1,,…,ak-1 be holomorphic functions in  D, and  k be a positive integer. Suppose that, for every function  f∈ℱ,  f≠0,  f(k)≠0 and for every pair functions  (f,g)∈ℱ,  f(k) and  g(k) share  ψ, then  ℱ is normal in  D.Remark 1.6.
By the ideas of sharing values, Theorem1.4 and Corollary 1.5 yield the number (k+2)/k can be omitted.Remark 1.7.
Obviously, Corollary1.5 omitted the conditions ψ with only simple zeros, and, for every function f∈ℱ, f has all multiple poles in Theorem D. But the condition for every function f∈ℱ, f(k)≠0 is additional. Hence, Corollary 1.5 improves Theorem B in some sense.The conditionψ≢0 in Theorem 1.4 is necessary. For example, we consider the following families.Example 1.8.
ℱ={fm(z)=emz,m=1,2,…}, obviously, any f∈F satisfies f≠0, f(k)≠0. For distinct positive integers m, l, fm(k), and fl(k) share 0 IM. However, the families ℱ are not normal at z=0.Remark 1.9.
Some ideas of this paper are based on [7, 9, 10].


## 2. Preliminary Lemmas

In order to prove our theorems, we need the following lemmas.The well-known Zalcman’s lemma is a very important tool in the study of normal families. It has also undergone various extensions and improvements. The following is one up-to-date local version, which is due to Pang and Zaclman [11].Lemma 2.1 (see [11, 12]).
Letℱ be a family of meromorphic functions in the unit disc  Δ with the property that, for each  f∈ℱ, all zeros are of multiplicity at least  k. Suppose that there exists a number  A≥1 such that  |f(k)(z)|≤A whenever  f∈ℱ and  f=0. If  ℱ is not normal in  Δ, then, for  0≤α≤k, there exist (1)
a numberr∈(0,1);(2)
a sequence of complex numberszn, |zn|<r;(3)
a sequence of functionsfn∈ℱ;(4)
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.Here, as usual,g♯(ξ)=|g′(ξ)|/(1+|g(ξ)|2) is the spherical derivative.Lemma 2.2 (see [1]).
Letf(z) be a transcendental meromorphic function in  ℂ, let  k(≥1) be a integer, and let  b be a nonzero finite value, then  f or  f(k)-b has infinite zeros.Lemma 2.3 (see [7]).
Letf(z) be a nonconstant rational function. Let k≥1 be an integer, and let b be a non-zero finite value. If f≠0, then f(k)(z)-b has at least two distinct zeros in the plane.Lemma 2.4.
Letf(z) be a nonconstant rational function. Let  k≥1 be an integer, and let   l be a positive integer. If  f≠0,  f(k)≠0, then  f(k)(z)-zl has at least two distinct zeros in the plane.Proof.
Sincef≠0 and f(k)≠0, then f is a nonpolynomial rational function and has the form
(2.1)f(z)=A(z-z1)m1(z-z2)m2⋯(z-zt)mt,
where A≠0 is a constant, and m1,m2,…,mt are positive integers. Set m=m1+m2+⋯+mt. Then,
(2.2)f′(z)=-A(mzt-1+bt-2zt-2+⋯+b0)(z-z1)m1+1(z-z2)m2+1⋯(z-zt)mt+1,
where bt-2,…,b0 are constants. For k≥2, by mathematical induction, we have
(2.3)f(k)(z)=Bzkt-k+ckt-k-1zkt-k-1+⋯+c0(z-z1)m1+k(z-z2)m2+k⋯(z-zt)mt+k,
where B=(-1)km(m+1)(m+2)⋯(m+k-1)A≠0,ckt-k-1,…,c0 are constants. Since f(k)≠0, we deduce that t=1, and thus
(2.4)f(z)=A(z-z1)m1,(2.5)f(k)(z)=B(z-z1)m1+k.Case 1 (iff(k)-zl has exactly one zero z0).
From (2.5), we set
(2.6)f(k)(z)-zl=B(z-z1)m1+k-zl=B′(z-z0)m1+k+l(z-z1)m1+k.
Obviously, B′ is a nonzero constant and l≥1.
From (2.6), we obtain
(2.7)f(k+l+1)(z)=(z-z0)m1+k-1P1(z)(z-z1)m1+k+l+1,
where P1(z)≢0. By (2.4), we deduce
(2.8)f(k+l+1)(z)=A′(z-z1)m1+k+1+1,
where A′ is nonzero constant.
Comparing (2.7) and (2.8), we obtain that degA′=0≥m1+k-1 is impossible.Case 2 (iff(k)(z)-zl≠0).
By (2.5), clearly Case 2 is impossible.
Lemma2.4 is proved.Lemma 2.5 (see [7]).
Letℱ be a family of meromorphic functions defined in  D, let  k be a positive integer, and let  ψ(≠0) be a holomorphic function in  D. If, for any  f∈ℱ satisfying  f≠0 and  if  f(k),g(k) share  ψ IM for every pair of functions  f,g∈ℱ, then  ℱ is normal in  D.In this paper, by the same method of [7], we consider the differential polynomial in Lemma 2.5 and prove a more general result.Lemma 2.6.
Letℱ be a family of meromorphic functions defined in  D, let  k be a positive integer, and let  ψ(≠0) be a holomorphic function in  D. If, for any  f∈ℱ satisfying  f≠0 and  if  G(f),G(g) share  ψ IM for every pair of functions  f,g∈ℱ, where  G(f) is a differential polynomial of  f as the definition 1 satisfying  q≥γH, and  Γ/γ|H<k+1, then  ℱ is normal in  D, where  q,  Γ/γ|H are as in Definitions 1.1 and 1.2.Proof.
We may assume thatD=Δ={|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 exists a number r∈(0,1); a sequence of complex numbers zj, zj→0(j→∞); a sequence of functions fj∈ℱ; 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 meromorphic functions g¯(ξ) in C. Moreover, g¯(ξ) is of order at most 2. Hurwitz’s theorem implies that g¯(ξ)≠0.
We have(2.9)G(fj)(zj+ρjζ)=P(fj(k)(zj+ρjζ))+H(fj,fj′,…,fj(k))(zj+ρjζ),H(fj,fj′,…,fj(k))(zj+ρjζ)=∑i=1nbi(zj+ρjζ)ρj(k+1)γMi-ΓMiMi(gj,gj′,…,gj(k))(ζ).
Considering bi(z) is analytic on D(i=1,2,…,n), we have
(2.10)|bi(zj+ρjζ)|≤M(1+r2,bi(z))<∞,(i=1,2,…,n)
for sufficiently large j.
Hence, we deduce fromΓ/γ|H<k+1 that
(2.11)∑i=1nbi(zj+ρjζ)ρj(k+1)γMi-ΓMiMi(gj,gj′,…,gj(k))(ζ)
converges uniformly to 0 on every compact subset of ℂ which contains no poles of g¯(ξ).
Thus, we have(2.12)G(fj)(zj+ρjζ)⟶P(g¯(k))(ζ),G(fj)(zj+ρjζ)-ψ(zj+ρjζ)⟶P(g¯(k))(ζ)-ψ(z0)
on every compact subset of ℂ which contains no poles of g¯(ζ).
Next, we will prove thatG(fj)(ζ)-ψ(z0) has just a unique zero. By way of contradiction, let ζ0 and ζ0* be two distinct solutions of G(fj)(ζ)-ψ(z0), and choose δ(>0) small enough such that D(ζ0,δ)∩D(ζ0*,δ)=∅ where D(ζ0,δ)={ζ:|ζ-ζ0|<δ} and D(ζ0*,δ)={ζ:|ζ-ζ0*|<δ}. By Hurwitz’s theorem, there exist points ζj∈D(ζ0,δ), ζj*∈D(ζ0*,δ) such that, for sufficiently large j,
(2.13)G(fj)(zj+ρjζj)-ψ(z0)=0,G(fj)(zj+ρjζj*)-ψ(z0)=0.
By the hypothesis that for each pair of functionsf and g in ℱ, G(f) and G(g) share ψ(z0) in D, we know that, for any positive integer m,
(2.14)G(fm)(zj+ρjζj)-ψ(z0)=0,G(fm)(zj+ρjζj*)-ψ(z0)=0.
Fix m, take j→∞, and note zj+ρjζj→0,zj+ρjζj*→0, then
(2.15)G(fm)(0)-ψ(z0)=0.
Since the zeros of G(fm)(0)-ψ(z0)=0 have no accumulation point, so zj+ρjζj=0, zj+ρjζj*=0.
Hence,(2.16)ζj=-zjρj,ζj*=-zjρj.
This contradicts with ζj∈D(ζ0,δ), ζj*∈D(ζ0*,δ), and D(ζ0,δ)∩D(ζ0*,δ)=∅.  So G(fj)-ψ(z0) has just a unique zero. By Hurwitz’s theorem, we know P(g¯(k))(ζ)-ψ(z0) has just a unique zero.
By Lemmas2.2 and 2.3, we know g¯(k)(ζ)-ψ(z0) has at least two distinct zeros. From the definition of P(w), we deduce that P(g¯(k)(ζ))-ψ(z0) has more than two distinct zeros, a contradiction.
Soℱ is normal in D. Lemma 2.6 is proved.By Lemma2.6, we immediately deduce the following lemma.Lemma 2.7.
Letℱ be a family of meromorphic functions defined in  D, let  ψ(≠0),  a0,a1,…,ak-1 be holomorphic functions in  D, and let  k be a positive integer. Suppose that, for every function  f∈ℱ,  f≠0,  f(k)+ak-1f(k-1)+⋯+a1f′+a0f≠0 and, for every pair functions  (f,g)∈ℱ,  f(k)+ak-1f(k-1)+⋯+a1f′+a0f,  g(k)+ak-1g(k-1)+⋯+a1g′+a0g share  ψ, then  ℱ is normal in  D.Lemma 2.8 (see [1]).
Letf(z) be a meromorphic function. Let  k be a positive integer. If  f(z)≠0, then  f(k)(z)≠1,  then  f is a constant.Lemma 2.9 (see [13, 14]).
Letf(z) be a transcendental meromorphic function in  ℂ, and let  P(≢0) be a polynomial. Let  k be a positive integer. If all zeros (except at most finite zeros) of  f(z) have the multiplicity at least 3, then  f(k)(z)-P(z) has infinite zeros.


## 3. Proof of Theorem1.4

Proof.
Since normality is a local property, without loss of generality, we may assumeD=Δ={z:|z|<1}, and
(3.1)ψ(z)=zlφ(z)(z∈Δ),
where l is a positive integer, φ(0)=1, φ(z)≠0 on Δ′={z:0<|z|<1}. By Lemma 2.6, we only need to prove that ℱ is normal at z=0.
Iff∈ℱ, P(f)(0)≠ψ(0), then there exists δ>0 such that P(f)(z)≠ψ(z) on Δδ. By condition of Theorem, for every g∈ℱ, we know P(g)(z)≠ψ(z) on Δδ. By theorem D, ℱ is normal on Δδ, so ℱ is normal on z=0.
Now, we considerP(f)(0)=ψ(0). Suppose P(f)(z)≢ψ(z) on the neighborhood |z|<δ (where δ is a small positive number) (otherwise, P(f)(z)≡ψ(z) on the neighborhood |z|<δ, by condition of theorem, for every g∈ℱ, we also obtain P(g)(z)≡ψ(z). So P(g)(z)≠ψ(z)+1. By Theorem D, ℱ is normal at z=0. So Theorem 1.4 is proved), there exists δ>0 such that P(f)(z)≠ψ(z) on (z∈Δδ′). So, for every g∈ℱ, we obtain
(3.2)P(g)(z)≠ψ(z)(z∈Δδ′).
By Theorem D, ℱ is normal on Δ′.
Next, we will proveℱ is normal at z=0. Suppose, on the contrary, that ℱ is not normal at z=0∈Δ, then there exists a sequence functions (we also denote ℱ) that  has no any normal subsequence on z=0.
Consider the familyℑ={g(z)=(f(z)/ψ(z)):f∈ℱ,z∈Δ}. Since f≠0 for f∈ℱ, we have that g(0)=∞ for each g∈ℑ.
We first prove thatℑ is normal in Δ. Suppose, on the contrary, that ℑ is not normal at z0∈Δ. By Lemma 2.1, there exist a sequence of functions gn∈ℑ, a sequence of complex numbers zn→z0, and a sequence of positive numbers ρn→0, such that
(3.3)Gn(ξ)=gn(zn+ρnξ)ρnk⟶G(ξ)
converges spherically uniformly on compact subsets of ℂ where G(ξ) is a nonconstant meromorphic function on ℂ, and G(ξ)≠0.
We distinguish two cases.
Case 1 (zn/ρn→∞).
By a simple calculation, for0≤i≤k, we have
(3.4)gn(i)(z)=fn(i)(z)ψ(z)-∑j=1iCijgn(i-j)(z)ψ(j)(z)ψ(z)=fn(i)(z)ψ(z)-∑j=1i[Cijgn(i-j)(z)∑t=0jAjt1zj-tφ(t)(z)φ(z)],
where Ajt=l(l-1)⋯(l-j+t+1)Cjt if l<j, for t=0,1,…,j-1 and Ajj=1.
Thus, from (3.4), we have
(3.5)ρnk-iGn(i)(ξ)=gn(i)(zn+ρnξ)=fn(i)(zn+ρnξ)ψ(zn+ρnξ)-∑j=1i[Cijgn(i-j)(zn+ρnξ)∑t=0jAjt1(zn+ρnξ)j-tφ(t)(zn+ρnξ)φ(zn+ρnξ)]=fn(i)(zn+ρnξ)ψ(zn+ρnξ)-∑j=1i[Cijgn(i-j)ρnj(zn+ρnξ)∑t=0jAjt1(zn+ρnξ)j-tρntφ(t)(zn+ρnξ)φ(zn+ρnξ)].
On the other hand, we have
(3.6)lim⁡n→∞1(zn/ρn)+ξ=0,lim⁡n→∞ρntφ(t)(zn+ρnξ)φ(zn+ρnξ)=0,
for t≥1. Noting that gn(i-j)(zn+ρnξ)/ρnj is locally bounded on ℂ minus the set of poles of G(ξ) since gn(zn+ρnξ)/ρnk→G(ξ). Therefore, on every subset of ℂ which contains no poles of G(ξ), we have
(3.7)fn(k)(zn+ρnξ)ψ(zn+ρnξ)⟶G(k)(ξ),fn(i)(zn+ρnξ)ψ(zn+ρnξ)⟶0,
for i=0,1,…,k-1, and thus
(3.8)fn(k)(zn+ρnξ)+Σi=0k-1ai(zn+ρnξ)fn(i)(zn+ρnξ)ψ(zn+ρnξ)⟶G(k)(ξ),fn(k)(zn+ρnξ)+Σi=0k-1ai(zn+ρnξ)fn(i)(zn+ρnξ)-ψ(zn+ρnξ)ψ(zn+ρnξ)⟶G(k)(ξ)-1,
since a0,…,ak-1 are analytic in D.
ByG(ξ)≠0, we know G(k)(ξ)≢1. In fact, if G(k)(ξ0)=1, by Hurwitz’s theorem, then exists ξn→ξ0, for n sufficiently large,
(3.9)P(f)(zn+ρnξn)=ψ(zn+ρnξn).
By the condition of theorem, for every positive number m, we obtain P(fm)(zn+ρnξn)=ψ(zn+ρnξn). We know zn+ρnξn→z0∈Δδ, and, for sufficiently large n, zn+ρnξn∈Δδ. However, zn+ρnξn≠0 (otherwise, zn+ρnξn=0, so ξn=-(zn/ρn)→∞, a contradiction), so for sufficiently large n, zn+ρnξn∈Δδ′. This contradicts with (3.2).
SoG(ξ)≠0 and G(k)(ξ)≠1, by Lemma 2.8, we obtain G is a constant, a contradiction.
Case 2.
zn/ρn→α is a finite complex number. Then,
(3.10)gn(ρnξ)ρnk=gn(zn+ρn(ξ-(zn/ρn)))ρnk=Gn(ξ-znρn)⟶G(ξ-α)=G(ξ).
Obviously, 𝔾(ξ)≠0, and ξ=0 is a pole of 𝔾 with order at least l.
Set(3.11)Hn(ξ)=fn(ρnξ)ρnk+l.
Then,
(3.12)Hn(ξ)=ψ(ρnξ)ρnlfn(ρnξ)ρnkψ(ρnξ)=ψ(ρnξ)ρnlgn(ρnξ)ρnk.
Noting that ψ(ρnξ)/ρnl→ξl, thus
(3.13)Hn(ξ)⟶ξlG(ξ)=H(ξ),
uniformly on compact subsets of ℂ. Since 𝔾 has a pole of order at least at ξ=0, we have H(0)≠0, so that H(ξ)≠0.
From (3.11), we get
(3.14)Hn(i)=fn(i)(ρnξ)ρnk+l-i⟶H(i)(ξ),
spherically uniformly on compact subsets of ℂ minus the set of poles of 𝔾(ξ). As the above, on every compact subset of ℂ minus the set of poles of G(ξ), we have
(3.15)fn(k)(ρnξ)+Σi=0k-1ai(ρnξ)fn(i)(ρnξ)ρnl⟶H(k)(ξ),(3.16)fn(k)(ρnξ)+Σi=0k-1ai(ρnξ)fn(i)(ρnξ)-ψ(ρnξ)ρnl⟶H(k)(ξ)-ξl,
locally uniformly on ℂ.
By the assumption of Theorem and (3.16), Hurwitz’s theorem implies H(k)(ξ)≠0.
Next, we proof that ifξ∈ℂ/{0}, then H(k)(ξ)≠ξl.
First,H(k)(ξ)≢ξl, otherwise H(k)(ξ)≡ξl, which contradicts with H(ξ)≠0. If there exists a ξ0≠0 such that H(k)(ξ0)=ξ0l, by Hurwitz’s theorem and (3.16), there exists ξn→ξ0 such that fn(k)(ρnξn)+Σi=0k-1ai(ρnξn)fn(i)(ρnξn)=ψ(ρnξn). By the assumption of Theorem 1.4, for every positive m such that P(fm)(ρnξn)=ψ(ρnξn). However, for n sufficiently large, ρnξn∈Δδ′, all of these contradict with (3.2). So if ξ∈ℂ/{0}, then H(k)(ξ)≠ξl.
NotingH(ξ)≠0, By Lemma 2.9, we know H must be a rational function. If H is not a constant, By Lemma 2.4, we know H(k)(ξ)-ξl has at least two distinct zeros, a contradiction. So H must be a nonzero constant, also contradicts with H(k)(ξ)≠0. Now, we have proved the ℑ is normal on Δδ.
It remains to show thatℱ is normal at z=0. Since ℑ is normal in Δ, then the family ℑ is equicontinuous on Δ with respect to the spherical distance. On the other hand, g(0)=∞ for each g∈ℑ, so there exists δ>0 such that |g(z)|≥1 for all g∈ℑ and each z∈Δδ={z:|z|<δ}. Suppose that ℱ is not normal at z=0. Since ℱ is normal in 0<|z|<1, the family ℱ1={1/f:f∈ℑ} is normal in Δ={z:0<|z|<1}, but it is not normal at z=0. Then, there exists a sequence {1/fn}⊂ℱ1 which converges locally uniformly in Δ′, but not in Δ. Noting that fn≠0 in Δ, 1/fn is holomorphic in Δ for each n. The maximum modulus principle implies that 1/fn→∞ in Δ′. Thus, fn→0 converges locally uniformly in Δ′, and hence so does {gn}⊂ℑ, where gn=fn/φ. But |gn(z)|≥1 for each z∈Δδ, a contradiction. This finally completes the proof of Theorem 1.4.



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*Source: 582854-2012-01-17.xml*