# On Angular Limits of Normal Meromorphic Functions: A Geometric Aspect

Authors: Zarko Pavicevic
Journal: Journal of Complex Analysis
 (2014)

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## Abstract

We will prove the assertions which give necessary and sufficient conditions for a normal meromorphic function on the open unit disk to have an angular limit. The results obtained show that the conditions from the classical Lindelöf theorem, as well as the theorems of Lehto and Virtanen and Bagemihl and Seidel, concerning angular limit values of meromorphic functions, can be weakened.


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## Body

## 1. Introduction and Preliminaries

Let𝔻={z∈ℂ:|z|<1} be the open unit disk in the complex plane ℂ, with the boundary Γ={z∈ℂ:|z|=1}, and let Ω=ℂ-=ℂ∪{∞} be the Riemann sphere. For any w∈𝔻 and a function f:𝔻→Ω, let fw=f∘φw, where φw(z)=(z+w)/(1+w¯z) (z∈ℂ) is a Möbius map. Further, the pseudohyperbolic distance  dph on 𝔻 is given by
(1)dph(z1,z2)=|z1-z21-z1z2¯|,z1,z2∈𝔻.
The function dh defined on 𝔻 as
(2)dh(z1,z2)=ln⁡⁡1+dph(z1,z2)1-dph(z1,z2),z1,z2∈𝔻,
is the hyperbolic metric on 𝔻. The chordal metric on the Riemann sphere is defined as
(3)ds(w1,w2)={2|w1-w2|1+|w1|21+|w2|2ifw1,w2∈ℂ21+|w1|2ifw1∈ℂ,w2=∞.All the convergence in this paper will be considered with respect to some of the aforementioned metrics. Since the convergence with respect to the hyperbolic and pseudohyperbolic metrics on the disk𝔻 is equivalent, in our proofs we will use one of these metrics that “simplifies” the related proof.For a fixedw∈𝔻, the set 𝔻ph(w,r) defined as
(4)𝔻ph(w,r)={z∈𝔻:dph(z,w)<r},0<r<1,
is called the pseudohyperbolic disk with the pseudohyperbolic center w and the pseudohyperbolic radius r. Notice that for w=0, 𝔻ph(0,r)={z:|z|<r}:=𝔻(0,r), 0<r<1. Similarly, for a fixed w∈𝔻, the set 𝔻h(w,r) defined as
(5)𝔻h(w,r)={z∈𝔻:dh(z,w)<r},0<r<1,
is called the hyperbolic disk with the center w and the hyperbolic radius r.For simplicity, here as always in the sequel, for an arbitrary nonempty setK⊂𝔻 we write fn⇒Kf if a sequence (fn) of complex functions defined on the disk 𝔻 tends uniformly on K to the function f with respect to the pseudohyperbolic (or hyperbolic) metric dph (or dh) and the chordal metric ds of the Riemann sphere Ω.Given a setA⊆𝔻 so that A-∩Γ={eiθ} (A- is the closure of A) and the function f:𝔻→ℂ-, denote by C(f,A,eiθ) the cluster set of the function f at the point eiθ with respect to the set A. Namely, C(f,A,eiθ) is the set of all points ω∈Ω for which there exists a sequence (zn) in A so that (zn)→eiθ and f(zn)→ω as n→∞. It is known that C(f,A,eiθ)=C(f,A,eiθ)-. If
(6)A={z:z∈𝔻,|arg(eiθ-z)|<α,0<α<π2}:=Δ(eiθ,α)
is a Stolz angle of the disk 𝔻 with the vertex at the point eiθ, then the cluster set C(f,Δ(eiθ,α),eiθ) is the limit value of the function f along the angle Δ(eiθ). If for each α with 0<α<π/2 we have C(f,Δ(eiθ,α),eiθ)={ω} for some ω∈Ω, then eiθ is said to be a Fatou point of the function f and ω∈Ω is its angular limit value.Here, as always in the sequel,γ will denote a Jordan arc that ends at a point eiθ. If C(f,γ,eiθ)={ω} for some ω∈Ω then ω is said to be an asymptotic value of the function f at the point eiθ along the curve γ. The classical Lindelöf theorem on boundary values of holomorphic functions asserts that if a bounded analytic function on the disk 𝔻 has an asymptotic value ω, ω∈Ω, at a point eiθ, then ω is its angular limit value and eiθ is a Fatou point of the function f [1].Seidel [2] and Seidel and Walsh [3] investigated the boundary properties of bounded analytic functions and analytic functions that omit two complex values via the convergence of their Heine’s sequences. They proved that the results of Lindelöf’s theorem are also valid for univalent analytic functions.In [4], Lehto and Virtanen extended Lindelöf’s result to the class of normal meromorphic functions in 𝔻.A functionf meromorphic on the disk 𝔻 is said to be normal in 𝔻 if the family ℱf:={fw:fw:=f∘φw:w∈𝔻} is normal in the disk 𝔻 in the sense of Montel, which means that each sequence in ℱf contains a subsequence which converges uniformly on each compact subset of 𝔻. Namely, Lehto and Virtanen proved that if a normal meromorphic function in the disk 𝔻 has an asymptotic value ω, ω∈Ω, at a point eiθ, then ω is its angular limit value and eiθ is a Fatou point of the function f [4, Theorem 2].In the same paper, in terms of the estimate of growth of the spherical derivative, Lehto and Virtanen established a necessary and sufficient conditions for a functionf meromorphic on hyperbolic domains of f to be a normal function. For more information on the properties and applications of normal meromorphic functions in the theory of functions see [5].In [6], Bagemihl and Seidel examined the existence of angular limits of a function f meromorphic on 𝔻 at a point eiθ, eiθ∈Γ, via the existence of its Heine’s sequences (f(zn)), (zn)⊂𝔻 so that zn→eiθ as n→∞.In further considerations, the notion of the rangeR(f,eiθ) will be significant, where f is a function defined in 𝔻. The range R(f,eiθ) is defined as the set of all points c∈Ω so that there exists a sequence (zn)⊂𝔻 for which lim⁡n→∞⁡zn=eiθ and f(zn)=c for all n∈ℕ.Bagemihl and Seidel [6, Theorem 1] showed that if for a normal meromorphic function f on 𝔻 there exists a sequence (zn)⊂𝔻 such that lim⁡n→∞⁡zn=eiθ,
(7)sup⁡⁡{dh(zn,zn+1):n∈ℕ}≤M<+∞
and lim⁡n→∞⁡f(zn)=c for some c∈Ω with c∉R(f,eiθ), then eiθ is a Fatou point of the function f and c∈Ω is its angular limit value. In [6, Examples 1 and 2], the authors gave examples of analytic functions that show that the condition of normality for the function f and the condition
(8)sup⁡⁡{dh(zn,zn+1):n∈ℕ}≤M<+∞
related to the sequence (zn)⊂𝔻 with lim⁡n→∞zn=eiθ cannot be omitted. Example 4 in [6] of a Blaschke product shows the fact that the condition c∉R(f,eiθ) is necessary in that assertion.Further boundary properties of meromorphic functions via their behaviour along sequences(zn)⊂𝔻 with lim⁡n→∞⁡zn=eiθ were investigated by Gavrilov (see, e.g., [5, 7, 8]) and Gauthier ([9, 10]).In this paper, we prove Theorem6 of Section 3 and Theorems 7–10 of Section 4, which give necessary and sufficient conditions for a normal meromorphic function f in 𝔻 at a point eiθ to have an angular limit in terms of a sequence (f(zn)) with (zn)⊂𝔻 and lim⁡n→∞zn=eiθ under the condition sup⁡⁡{dh(zn,zn+1):n∈ℕ}≤M<+∞. Here it is not supposed that c∉R(f,eiθ). The omission of the condition c∉R(f,eiθ) and the retention of the condition sup⁡⁡{dh(zn,zn+1):n∈ℕ}≤M<+∞ related to terms of the sequence (zn) require “good” boundary behaviour of the function f along sufficiently “thick” sets whose elements lie within a “small hyperbolic distance” from the points zn∈𝔻, n∈ℕ, with lim⁡n→∞⁡zn=eiθ and sup⁡⁡{dh(zn,zn+1):n∈ℕ}≤M<+∞. Namely, in Theorem 7 these sets are sufficiently large disks whose pseudohyperbolic (hyperbolic) centers are the terms of the sequence (zn) (see Figure 1); in Theorem 8, these sets are sufficiently large parts of Jordan’s arcs, and in Theorem 10 they are many sequences of points. These theorems give new criteria for the existence of angular limit values of meromorphic functions on 𝔻 at points eiθ of the unit circle Γ.Figure 1In Section2, we prove Lemmas 1, 3, and 4. These lemmas present auxiliary results for proofs of other assertions in this paper.


## 2. Auxiliary Lemmas

In this section, we consider the uniform convergence of function sequences to constant functions. Related classical results, for example, can be found in [11–13], while the uniform convergence of function sequences via properties of boundary sets of functions were investigated in [14, 15]; also see [16].Lemma 1.
Letf:𝔻→Ω be an arbitrary function, let (wn)⊂𝔻 be a sequence such that lim⁡n→∞⁡wn=eiθ, and let G be an arbitrary nonempty subset of 𝔻 with G⊂𝔻(0,r), 0<r<1. Then the following conditions are equivalent:(i)
fn⇒Gc with c∈Ω and fn=f∘φwn, n∈ℕ;(ii)
C(f,∪n=1∞φwn(G),eiθ)={c}.Proof.
From the conditionlim⁡n→∞⁡wn=eiθ it follows that
(9)(⋃n=1∞φwn(G)-)∩Γ={eiθ}.
(i)⇒(ii). It follows from (i) that for each ε>0 there exists a positive integer N=N(ε) such that ds((f∘φwn)(z),c)<ε for all n≥N and z∈G. This yields that f(φwn(G))⊂{ω:ω∈Ω,ds(c,ω)<ɛ} for all n≥N and z∈G, and thus,
(10)f(⋃n=N∞φwn(G))⊂{ω:ω∈Ω,ds(c,ω)<ε},
that is,
(11)ds(c,f(z))<ε,foreachz∈⋃n=N∞φwn(G).
Let (zm) be a sequence contained in ∪n=1∞φwn(G) such that lim⁡m→∞⁡zm=eiθ. We will show that lim⁡m→∞⁡f(zm)=c. For each m∈ℕ there is a nm∈ℕ such that zm∈φwnm(G). From lim⁡m→∞⁡zm=eiθ it follows that lim⁡m→∞⁡nm=∞, and so, for k=N there is a M=M(N)∈ℕ such that nm≥N for every m≥M. As zm∈φwnm(G), by the previous argument it follows that zm∈∪n=N∞φwn(G) for all m≥M. This together with (11) gives
(12)ds(c,f(zm))<εforeachm≥M.
Hence, for each ε>0 there exists M∈ℕ, M=M(N(ε))=M(ε) such that ds(c,f(zm))<ε for each m≥M. Therefore, lim⁡m→∞⁡f(zm)=c and C(f,∪n=1∞φwn(G),eiθ)={c}.
(ii)⇒ (i). Suppose that (i) does not hold. Then we will show that (ii) also is not satisfied. The assumption that (i) is not true implies that there exists ε>0 such that for every n∈ℕ there exist mn∈ℕ and z∈G satisfying ds(c,f(φwmn(z)),c)≥ε. Take un=φwmn(z), n∈ℕ. Then un∈∪k=1∞φwk(G) for each n∈ℕ, and from lim⁡n→∞wn=eiθ it follows that lim⁡n→∞wmn=eiθ, and so, we have lim⁡n→∞⁡un=eiθ. Therefore, we obtain ds(f(un),c)≥ε, whence it follows that there exists a sequence (un) contained in ∪n=1∞φwn(G) such that lim⁡n→∞⁡un=eiθ, but the condition lim⁡n→∞⁡f(un)=c is not satisfied. Hence, the condition (ii) is not satisfied, and the proof is completed.Lemma 2 (Schiff [13, pp. 72–75]).
Let(fn) be a sequence of meromorphic functions fn:U→Ω, where U is a domain in complex plane ℂ. If (fn) forms a normal family of functions on U and there exists a set E with E⊂U which has at least one accumulation point in U, such that a sequence (fn(z)) converges to the limit function f(z) for each z∈E, then fn⇒Kf on each compact subset K of U.Lemma 3.
Let(fn) be a sequence of meromorphic functions fn:U→Ω, where U is a domain in complex plane ℂ. If (fn) forms a normal family of functions on U and there exists a set E with E⊂U which has at least one accumulation point in U, such that lim⁡n→∞⁡fn(z)=c, with some c∈Ω, for each z∈E, then fn⇒Kf on each compact subset K of U and f(z)≡c on U.Proof.
By Lemma2, it follows that fn⇒Gf on each compact subset G of U, where f is a meromorphic function in the domain U. From the conditions of Lemma 3, it follows that f(z)=c for each z∈E. As the set E has an accumulation point in U, by the uniqueness theorem for meromorphic functions, it follows that f(z)≡c on U, and thus, fn⇒Gc on each compact subset G of U.Lemma 4.
Letf:𝔻→Ω be a normal meromorphic function in the disk 𝔻, let (wn)⊂𝔻 be a sequence such that lim⁡n→∞⁡wn=eiθ, and let E be an arbitrary subset of 𝔻 with E⊂𝔻(0,r), 0<r<1, which has at least one accumulation point. Then the following assertions are equivalent:(i)
C(f,∪n=1∞φwn(E),eiθ)={c} with c∈Ω;(ii)
For eachr>0,  C(f,∪n=1∞𝔻h(wn,r),eiθ)={c}.Proof.
(i)⇒ (ii). By (i) and Lemmas 1 and 3 it follows that fn⇒Gc on each compact subset G of 𝔻. Taking G=𝔻(0,r′)-, with 0<r′<1, then applying Lemma 1, we find that C(f,∪n=1∞φwn (𝔻(0,r′)-),eiθ) ={c}. Since φwn(𝔻(0,r′)-)=𝔻ph(wn,r′)-, we have C(f,∪n=1∞φwn (𝔻ph(wn,r′)-),eiθ) ={c} for each r′ with 0<r′<1, and therefore, C(f,𝔻ph(wn,r′)-, eiθ) ={c} for all r′ with 0<r′<1 and n∈ℕ. Since for each r′ with 0<r′<1 and n∈ℕ, it holds 𝔻ph(wn,r′)-=𝔻h(wn,r)- with r=ln⁡⁡((1+r′)/(1-r′)), we have C(f,𝔻h(wn,r)¯,eiθ)={c} for each r>0 and n∈ℕ.
For any givenr>0, there exists r1 with 0<r1<r for which 𝔻h(wn,r1)-⊂𝔻h(wn,r)⊂𝔻h(wn,r)¯. Then
(13){c}=C(f,𝔻h(wn,r1)-,eiθ)⊆C(f,𝔻h(wn,r),eiθ)⊆C(f,𝔻h(wn,r)¯,eiθ)={c},
whence we see that C(f,𝔻h(wn,r),eiθ)={c} for all n∈ℕ. This shows that
(14)C(f,⋃n=1∞𝔻h(wn,r),eiθ)={c}foreachr>0,
as desired.
(ii)⇒ (i). Suppose that r>0 and C(f,∪n=1∞𝔻h(wn,r),eiθ)={c}. Then
(15)C(f,⋃n=1∞𝔻h(wn,r)-,eiθ)={c}foreachr>0,
and hence, C(f,𝔻h(wn,r)-,eiθ)={c} for all r>0 and n∈ℕ. Since a sequence (fn) defined as fn=f∘φwn forms a normal family of functions on 𝔻, it follows by Lemma 3 that fn⇒Gc, with c∈Ω, for each compact subset G of 𝔻. Therefore, for each compact subset K of 𝔻 which has at least one accumulation point, we have fn⇒Kc. Then by Lemma 1, C(f,∪n=1∞φwn(K),eiθ)={c}, as required.Lemma 5 (Schiff [13, p. 98]).
Letℱ be a normal family of meromorphic functions f:U→Ω, where U is a domain of the complex plane ℂ, and suppose that f(z)≠c, c∈Ω, for all f∈ℱ and z∈U. Then for every meromorphic function φ for which there exists a sequence (fn)⊂ℱ such that fn⇒Gφ on each compact subset G of U either there holds φ(z)≠c for each z∈U, or φ(z)≡c on U.


## 3. The Main Result

Theorem 6.
Letf:𝔻→Ω be a normal meromorphic function in 𝔻. Then the following assertions are equivalent.(i)
There exists a sequence(wn)⊂𝔻 such that lim⁡n→∞⁡wn=eiθ,
(16)sup⁡⁡{dh(wn,wn+1):n∈ℕ}≤M<+∞,
and let E be an arbitrary subset of 𝔻 with E⊂𝔻(0,r), 0<r<1, which has at least one accumulation point for which C(f,∪n=1∞φwn(E),eiθ)={c}, c∈Ω;(ii)
A pointeiθ is a Fatou point of a function f, and c∈Ω is its angular limit value at eiθ.Proof.
(i)⇒ (ii). From (i) and Lemma 4 it follows that for any r>0,
(17)C(f,⋃n=1∞𝔻h(wn,r),eiθ)={c}.
Suppose that r>M. Then ∪n=1∞𝔻h(wn,r) is a simply connected domain. If Ln denotes the arc of geodesic curve in hyperbolic metric that joins points wn and wn+1, then γ=∪n=1∞Ln is a Jordan curve that lies in ∪n=1∞𝔻h(wn,r) and ends at the point eiθ. Therefore, C(f,γ,eiθ)⊂C(f,∪n=1∞𝔻h(wn,r),eiθ)={c}, whence it follows that c is an asymptotic value of a function f at point eiθ. Then by [4, Theorem 2], it follows that eiθ is a Fatou point of a function f, and c∈Ω is its angular limit.
(ii)⇒ (i). Let ρθ=[0,eiθ] be the radius of the disk 𝔻 that joins the center 0 of 𝔻 and a point eiθ on the unit circle Γ. Then the set ∪w∈ρθ𝔻h(w,r), r>0, is a domain bounded by parts of two hypercircles Hθ+ and Hθ- which join points eiθ and -eiθ and by arc of the circle Γr={z:|z|=r}. Hypercircles Hθ+ and Hθ- form the angle α=(π/2)-arctge-rr with the radius ρθ at the point eiθ. Therefore, a domain ∪w∈ρθ𝔻h(w,r) lies in a Stolz angle Δ(eiθ,2α). Then assuming that the terms of a sequence (wn) are points of the radius ρθ satisfying lim⁡n→∞⁡wn=eiθ and (16) and taking the disk 𝔻(0,r1) with 0<r1<r as the set E, we have
(18)⋃n=1∞φwn(𝔻h(0,r1))=⋃w∈ρθ𝔻h(w,r)⊂Δ(eiθ,α).
Since eiθ is a Fatou point of a function f and c∈Ω is its angular limit, we conclude that C(f,∪n=1∞φwn(𝔻(0,r)),eiθ)={c}, as desired. This completes the proof.


## 4. Applications

Theorem 7.
Letf:𝔻→Ω be a normal meromorphic function in the disk 𝔻, and let (wn) be a sequence contained in 𝔻 satisfying lim⁡n→∞⁡wn=eiθ and (16). If C(f,∪n=1∞𝔻h(wn,r),eiθ)={c}, c∈Ω, for some r>0, then eiθ is a Fatou point of a function f, and c∈Ω is its angular limit at eiθ.Proof.
The set∪n=1∞𝔻h(wn,r) is presented in Figure 1. As φwn-1(𝔻h(wn,r))=𝔻(0,r′), with r′=(er-1)/(er+1), that is, 𝔻h(wn,r)=φwn(𝔻(0,r′)) for all n∈ℕ, we have
(19)C(f,⋃n=1∞φwn(𝔻(0,r′)),eiθ)=C(f,⋃n=1∞𝔻h(wn,r),eiθ)={c},
and taking E=𝔻(0,r′) in Theorem 6, from this theorem it follows that eiθ is a Fatou point of a function f, and c∈Ω is its angular limit.Theorem 8.
Letf:𝔻→Ω be a normal meromorphic function in the disk 𝔻, and let γ be a Jordan arc contained in 𝔻 such that γ∩Γ={eiθ}. Further, let (wn)⊂γ be a sequence satisfying lim⁡n→∞⁡wn=eiθ and (16), and let γn=γ∩𝔻h(wn,r) with n∈ℕ and r>0. If the equality C(f,∪n=1∞γn,eiθ)={c} holds for some c∈Ω and for some r>0, then eiθ is a Fatou point of the function f, and c∈Ω is its angular limit at eiθ.Proof.
From the conditions of the theorem it follows that a sequence(fn):=(f∘φwn) forms a normal family of functions meromorphic on the disk 𝔻. This means that (fn) contains a subsequence (fnk) that converges uniformly on compact subsets of 𝔻 to a function ψ meromorphic on 𝔻. For simplicity, in the sequel we will write (fn) instead of (fnk), and wk instead of wnk, k∈ℕ. The proof will be deduced using the pseudohyperbolic metric. Then 𝔻ph(wn,r)=φwn(𝔻(0,r)) for any fixed r with 0<r<1. Set γk=γ∩𝔻ph(wk,r) and Γk=φwk-1(γk)⊂𝔻(0,r),  k∈ℕ. For any fixed m∈ℕ and each k∈ℕ, let zk(m)∈Γk∩{z:|z|=rm}, with 0<r0<rm+1<rm<r<1. Points zk(m) may be chosen so that zk(m)≠zj(i) when k≠j or m≠i. In such a way for any fixed m∈ℕ, we find a sequence (zk(m))k=1∞:=(zk(m)). It follows that a sequence (zk(m)) can be chosen so that lim⁡k→∞⁡zk(m)=z(m)∈𝔻(0,r)- because of (zk(m))⊂𝔻(0,r). We will prove that ψ(z(m))=c for each m∈ℕ.
Letε be an arbitrary positive number. Then
(20)ds(ψ(z(m)),c)≤ds(ψ(z(m)),ψ(zk(m)))+ds(ψ(zk(m)),fk(zk(m)))+ds(fk(zk(m)),c).
Since ψ is a function meromorphic on 𝔻, then using a standard “<ε/3” estimate of each term on the right hand side of (20), we find that ds(ψ(z(m)),c)<ε for any given ε>0, and thus, ψ(z(m))=c for each m∈ℕ. Because (z(m))⊂𝔻(0,r)- and since a sequence (z(m)) has an accumulation point in 𝔻(0,r)-, by using the uniqueness theorem for meromorphic functions it follows that ψ(z)≡c on 𝔻(0,r)-.
From the previous part of the proof we see that every subsequence(fk) of a sequence (fn) which uniformly converges on compact subsets of the disk 𝔻 converges to a constant c. Suppose that there exists a subsequence (fk) of the sequence (fn) that does not converge uniformly on compact subsets of the disk 𝔻 to a constant c. Then there exists ε>0 such that for each p∈ℕ there is a kp∈ℕ and a point zkp∈𝔻(0,r)¯ satisfying
(21)ds(fkp(zkp),c)≥ε.
As a sequence (fn) forms a normal family of meromorphic functions, it follows that a sequence (fkp) has a subsequence that converges uniformly on compacts of the disk 𝔻. As was previously proved, it follows that this subsequence converges uniformly on compacts of the disk 𝔻 to a constant c. This shows that (21) does not hold for all p∈ℕ. This contradiction proves our assertion.
Hence, as proved above, a sequence(fn) satisfies fn⇒𝔻(0,r)-c. Since
(22)𝔻h(wn,r′)=φwn(𝔻(0,r)),r′=ln⁡⁡1+r1-r,
this together with Lemma 1 yields C(f,∪n=1∞𝔻h(wn,r′),eiθ)={c}, and so by Theorem 7 it follows that c is an angular limit of a function f at a point eiθ which is its Fatou point.Remark 9.
Theorem8 shows that the condition on the existence of asymptotic values of normal meromorphic function from Theorem 2 in [4] by Lehto and Virtanen can be replaced by a weaker condition on the existence of its boundary value along arbitrarily “small” arcs of a Jordan curve, where the hyperbolic distance between the consecutive arcs is uniformly bounded from above.Theorem 10.
Letf:𝔻→Ω be a normal meromorphic function in the disk 𝔻, and let (wn) be a sequence in 𝔻 satisfying lim⁡n→∞⁡wn=eiθ and (16). If for any r with 0<r<1 there exist countably many sequences (zn(m))n=1∞=(zn(m)), with zn(m)∈Γn(m),  Γn(m)={z:dph(z,wn)=r(m)},  0<r(m)<r(m+1) <r<1, n,m∈ℕ, for which lim⁡n→∞⁡f(zn(m))=c for each m∈ℕ, then eiθ is a Fatou point of a function f, and c is its angular limit value.Proof.
Putφwn-1(zn(m))=ωn(m), with n,m∈ℕ. Since φwn-1(Γn(m))=Γ(0,r(m))={z:|z|=r(m)}, we have ωnm∈Γ(0,rm) with n∈ℕ and any fixed m∈ℕ. Each of sequences (ωn(m))n=1∞,  m∈ℕ, has a subsequence which tends to a limit value z(m) which lies on the circle Γ(0,r(m)). This shows that z(m)≠z(m+1) and z(m)∈𝔻(0,r), with m∈ℕ and 0<r<1.
Since a function sequence(f∘φwn):=(fn) forms a normal family of meromorphic functions on the disk 𝔻, it follows that (fn) has a subsequence (fnk) that converges uniformly on compact subsets of the disk 𝔻 to a meromorphic function ψ. The remainder of the proof is the same as that of Theorem 8, where terms of a sequence (z(m)) and a function ψ have the same notations and roles as the corresponding notions related to the proof of Theorem 8.Using the results obtained, we can prove Theorem 1 in [6, p. 4] by Bagemihl and Seidel.Theorem 11 (Bagemihl and Seidel [6, Theorem 1]).
Letf:𝔻→Ω be a normal meromorphic function in the disk 𝔻, and let (wn) be a sequence in 𝔻 satisfying lim⁡n→∞⁡wn=eiθ, (16) and lim⁡n→∞⁡f(wn)=c with c∉R(f,eiθ). Then eiθ is a Fatou point of a function f, and c is its angular limit value.Proof.
We use the same notations as those in the proof of Theorem8. By the condition c∉R(f,eiθ) it follows that f(z)≠c on O={z:|z-eiθ|<r}∩𝔻 for some r with 0<r<1 (see Figure 1). This shows that, for each term of a subsequence (fk) with k>K of the sequence (fn) such that (f∘φwn)=(fn) which converges uniformly on compacts of the disk 𝔻 to a meromorphic function φ(z), there holds fk(z)≠c for all z∈𝔻(0,r). Since φ(0)=lim⁡k→∞⁡fk(0)=lim⁡k→∞⁡f(wk)=c, it follows from Lemma 5 that fk⇒Gc on each compact subset G of 𝔻. Further, it follows that fn⇒Gc on each compact subset G of 𝔻 (see proof of Theorem 8). This together with Lemma 1 yields C(f,∪n=1∞φwn(G),eiθ)={c}, and so by Theorem 6, we conclude that c is an angular limit value of a function f at a point eiθ, and eiθ is its Fatou point. This completes the proof.



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*Source: 216398-2014-01-20.xml*