# Families of Differential Equations in the Unit Disk

Authors: Kari E. Fowler
Journal: Journal of Mathematics
 (2019)

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## Abstract

We study the interaction between coefficient and solution conditions for complex linear differential equations in the unit disk within the context of normal families and corresponding families of differential equations. In addition, we consider this interaction within the context of normal functions in terms of Noshiro. Consideration of families of differential equations introduces a new perspective for studying normality. Consequently, sharper results are found than in previous studies involving normal functions within the context of one differential equation.


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## Body

## 1. Introduction

The interaction between coefficient conditions and solution conditions for linear differential equations in the unit diskD={z∈C:|z|<1} has been a topic of many investigations, including [1–7]. Instead of studying this interaction within the context of one differential equation as in previous works, we look at this interaction within the setting of a family of differential equations with a corresponding family of coefficients and family of solutions to the differential equations. For example, families of differential equations have been studied in the real setting in relation to spherical surfaces and wave equations [8], vector fields [9], and both 2-iterated Appell [10] and Hermite-based [11] polynomials in addition to studies in the complex plane involving asymptotic existence [12] and resolution of singularities [13]. By looking at the interaction between coefficients and solutions within the setting of families of differential equations in the unit disk, we obtain sharper results than were found in the setting of one differential equation. Specifically, we consider normal families, within both analytic and meromorphic settings. In order to motivate the meromorphic results involving normal families, we present a sharp improvement of a result by Fowler [14] concerning normal meromorphic functions in the context of one differential equation.


## 2. Preliminaries

The idea of a normal function in the unit diskD originated with Noshiro [15].Definition 1.
A meromorphic functionf is called normal in D if the set of functions f∘S is a normal family in D, where S ranges over the conformal mappings of D onto itself.LetG be an open subset of C and let C(G,C) be the set of all continuous functions from G to C.Definition 2 (see [16]).
A set of functionsF⊂C(G,C) is called a normal family if each sequence in F has a subsequence which converges to a function f∈C(G,C).LetH(G) be the set of analytic functions on G, and let M(G) represent the set of meromorphic functions on G.Definition 3 (see [6]).
A functionf∈M(D) is called α-normal for α>0 if(1)supz∈D1-z2αf#z<∞,where f#(z)=|f′(z)|/(1+|f(z)|2) is the spherical derivative of f.We note that meromorphic functionsf in Definition 3 with α=1 were classified by Noshiro asnormal.Letk≥1 be an integer. In [17], Fowler and Sons showed that if a coefficient A(z) of the differential equation(2)fk+Azf=0is normal, then a solution f may not be normal. In addition, they showed that if a solution f of (2) is normal, a coefficient A(z) may not be normal. However, they showed in [17, Theorem 1.1(i)] that if f is an α-normal solution of (2), then the coefficient A(z) is “almost" α(k+1)-normal in the unit disk D, except for a factor of |f|/(1+|f|k+2).When considering normalfamilies of analytic functions in C(D,C), the normality of families of solutions implies the normality of families of coefficients. We additionally get a strong normality implication in the other direction. Note that we follow the convention that if a sequence of functions {fn} in a family F tends to f≡∞ in G, then the sequence is not regarded as convergent. A useful characterization of normal families of analytic functions is given by the following two results.Theorem 4 (Montel’s theorem).
A familyF in H(G) is normal if and only if F is locally bounded.Theorem 5 (see [16, Chapter VII, Lemma 2.8]).
A setF in H(G) is locally bounded if and only if for each compact set K⊂G there is a constant M such that(3)fz≤Mfor all f in F and z in K.The catalyst for this study of the interaction between families of solutions and coefficients of differential equations was a result in [16] which connects families of normal functions to corresponding families of derivatives of functions. Theorem 6(i) below is mentioned in Exercise 6, p. 154 of [16] for k=1 and for an open set G⊂C but not proven. Part (ii) is alluded to in the same exercise for k=1 but also not proven. We include a proof in both directions for k≥1.Theorem 6.
Letk≥1 be an integer, and let F⊂H(D) be a family of functions.
(i) If the familyF is normal, then the family(4)Fk=fk:f∈F⊂HDis also normal.
(ii) If the family(5)Fk=fk:f∈F⊂HDis normal, then the family F is also normal.We next examine families of differential equations. LetΓ⊂R and let k≥1 be an integer. We consider the family of differential equations(6)fγk+Aγzfγ=0γ∈Γwith the corresponding family of coefficients(7)A=Aγzγ∈Γ=-fγkfγγ∈Γ and family of solutions(8)F=fγγ∈Γ.Note that, in the family of solutions F={fγ}γ∈Γ of (6), we include all solutions fγ corresponding to Aγ(z) for any γ∈Γ.


## 3. Main Results

The next theorem relates normal families of coefficients and solutions.Theorem 7.
(i) Letk≥1 be an integer. Suppose(9)A=-fγkfγγ∈Γ⊂HD is a family of coefficients of (6), and suppose the family of solutions F⊂H(D) is a normal family. Then, A is also a normal family.
(ii) Suppose(10)A=-fγ′fγγ∈Γ⊂HD is a normal family of coefficients of (6). Then, the family of solutions F⊂H(D) is also a normal family.
(iii) Letk≥2 be an integer. Suppose(11)A=-fγkfγγ∈Γ⊂HD is a normal family of coefficients of (6) for which there exists a positive constant M such that ∑n=0k-1fγn0≤M for any γ∈Γ. Then, the family of solutions F⊂H(D) is also a normal family.Observe that, by Theorem6(i), we conclude in Theorem 7, parts (ii) and (iii), that Fk={fγ(k):fγ∈F} is also a normal family.In [14], Fowler considered a subset of the set of normal functions called normal meromorphic functions of the first category.Definition 8 (see [15]).
A meromorphic functionf in D is called a normal function of the first category if and only if f is a normal function and any sequence fn which is a subset of the family f∘S, where S ranges over the conformal mappings of D onto itself, cannot admit a constant as a limiting function.We designate the set of normal functions of the first category byN1. Fowler showed in [14, Theorem 4] that if f∈N1 is a solution of (2), then we can eliminate the factor |f|/(1+|f|k+2) in [17, Theorem 1.1(i)], and it follows that the coefficient A(z) of (2) is α(k+1)-normal in the unit disk D, except for disks of positive radius as small as we like within D containing the poles and zeros of a solution f. In the same theorem, Fowler showed that if the coefficient A(z) of (2) is a normal meromorphic function of the first category, then a solution f of (2) behaves like an α-normal function within the context of [17, Lemma 3.1] but within the same subset of D as above, for α>0.We present a sharp improvement of this result concerning normal functions of the first category for the entire unit disk. We later extend these ideas to normal meromorphic families of functions.Theorem 9.
(i) Supposef is a meromorphic solution of (2), with coefficient A meromorphic in D. If f∈N1, then A is α(k+1)-normal in D for α≥1.
(ii) LetA be the meromorphic coefficient of (2) in D satisfying A∈N1, and suppose that f is a meromorphic solution of (2) in D such that any pole of f has order m>1 and any zero of f has order m>k≥1. Then, for any α>0, there exists a constant C(A,ρ) such that(12)fkz1-z2αk1+fzk+1≤CA,ρ, for some ρ>0 and for each z∈D.Observe that, by [17, Lemma 3.1], we conclude in Theorem 9(ii) that f behaves as an α-normal function would in D.The natural bound on(1-|z|2)α(k+1)A#(z) when f∈N1 motivates a similar type of bound for a family of meromorphic functions. We consider normal meromorphic families in C(D,C∞).Letf be a meromorphic function defined on D. Following [16], we let μ(f):D→R be defined by(13)μfz=2f′z1+fz2 if z is not a pole of f, and(14)μfa=limz→a2f′z1+fz2 if a is a pole of f.We can characterize normal meromorphic families with the following result.Theorem 10 (see [16, Chapter VII, Theorem 3.8]).
A familyF⊂M(G) is normal in C(G,C∞) if and only if μ(F)={μ(f):f∈F} is locally bounded.We extend the following property of functions inN1 to normal meromorphic families to get a bound in line with the bound for functions in N1. Let χ denote chordal distance, let zν,f(α) be the set of α-points of f, and let U(z,δ)={ζ:d(z,ζ)<δ}, where d designates the hyperbolic metric.Theorem 11 (see [18, Theorem 1(iii)]).
Iff(z) belongs to N1, then for any value α and any positive number ρ, there exists a positive number mρ<1 such that χ(f(z),α)>mρ whenever z∈D-⋃ν=1∞U(zν,f(α),ρ).The following result of Lappan gives a useful property of a normal family of meromorphic functions that is used to characterize a family of solutions of (6) and to prove Theorem 13.Theorem 12 (see [19, Theorem 5]).
LetF be a normal family of meromorphic functions on D and let K be a compact subset of D. For each positive integer n there exists a constant Pn(K,F) such that(15)fnz1-z2n1+fzn+1≤PnK,F for each z∈K and each f∈F.Theorem 13.
(i) Letk≥1 be an integer. Suppose A⊂M(D) is a family of coefficients of (6), and suppose the set of solutions F is a normal family of meromorphic functions with the property that, for the values α=0,∞ and any positive number ρ, there exists a positive number mρ<1 such that χ(fγ(z),α)>mρ whenever z∈D-⋃ν=1∞U(zν,fγ(α),ρ) and fγ∈F. Then, A is a normal family.
(ii) Letk≥1 be an integer and let K be a compact subset of D. Suppose A⊂M(D) is a normal family of coefficients of (6) with the property that, for any positive number ρ, there exists a positive number mρ<1 such that χ(Aγ(z),∞)>mρ whenever z∈D-⋃ν=1∞U(zν,Aγ(∞),ρ) and Aγ∈A. Further suppose that the family of solutions F⊂M(D) has the property that any pole of fγ has order m>1 and any zero of fγ has order m>k. Then, for each positive integer k there exists a constant Ck(K,A,ρ) such that the set of solutions F satisfies(16)fγkz1-z2k1+fγzk+1≤CkK,A,ρ,for some ρ>0, for each z∈K, and for each fγ∈F.Thus, we conclude in Theorem13(ii) that F behaves as a normal family would on D within the context of Theorem 12.


## 4. Examples

Example 14.
We present an example of a familyF that satisfies the conditions of Theorem 13(i). Let f(z) belong to N1 and let an/dnn=1∞ be a sequence of distinct complex fractions such that, for all integers n≥1, we have that an/dn>1 and andn≠0. Further suppose that there is a finite constant ϵ such that an/dn→ϵ and |ϵ|>1. Then, by [18, Lemma 1], each member of the family F=(an/dn)f(z)n=1∞ belongs to N1.
We will show thatF is a normal family. Let K be a compact subset of D. Since f(z)∈N1 and is thus a normal function, we get that(17)supz∈D1-z2f′z1+fz2<∞. It follows that for all a∈K there is some positive constant M such that(18)limz→a2f′z1+fz2<M<∞. Since an/dn→ϵ, there is some finite number P such that(19)andn≤P,for all n∈N. Then, by inequality (19) and since an/dn>1 for all integers n≥1, we get(20)μandnfa=limz→a2an/dnf′z1+an/dnfz2=andnlimz→a2f′z1+an/dn2fz2≤P·M<∞,for all a∈K and all integers n≥1. Therefore, F is locally bounded and by Theorem 10 is thus a normal family.
We will next show that the normal familyF satisfies the additional conditions of Theorem 13(i). Let ρ>0. Then, since f(z)∈N1, there is a positive number mρ<1 such that χ(f(z),0)>mρ whenever z∈D-⋃ν=1∞U(zν,f(0,ρ)) and χ(f(z),∞)>mρ whenever z∈D-⋃ν=1∞U(zν,f(∞,ρ)). Then, for all z∈D,(21)χandnfz,0=an/dnfz-01+an/dnfz21+02=an/dnfzan/dn21/an/dn2+fz2.By (21) and since an/dn>1 for all integers n≥1, it follows that(22)χandnfz,0=fzdn/an2+fz2≥fz1+fz2=χfz,0>mρ, whenever z∈D-⋃ν=1∞U(zν,(an/dn)f(0,ρ))=D-⋃ν=1∞U(zν,f(0,ρ)), for all integers n≥1.
In addition, we have that, for allz∈D,(23)χandnfz,∞=11+an/dnfz2=1an/dn21/an/dn2+fz2.Then, by inequality (19) and (23), and since an/dn>1 for all integers n≥1, it follows that(24)χandnfz,∞=dn/andn/an2+fz2≥dnan11+fz2=dnanχfz,∞>dnanmρ≥1Pmρ, whenever z∈D-⋃ν=1∞U(zν,(an/dn)f(∞,ρ))=D-⋃ν=1∞U(zν,f(∞,ρ)), for all integers n≥1, where (1/P)mρ<1.
It then clearly follows by Theorems9(i) and 10 that the corresponding family of coefficients of (6),(25)A=Anzn=1∞=-an/dnfkzan/dnfzn=1∞,is a normal family in D. This verifies the conclusion in Theorem 13(i).Example 15.
For a second example of a familyF that satisfies the conditions of Theorem 13(i), let f(z) belong to N1 and let {bn}n=1∞ be a sequence of distinct complex numbers such that bn→0 and such that, for all integers n≥1, we have that bn≠0. Then, by [18, Lemma 1], each member of the family F={f(z)+bn}n=1∞ belongs to N1.
We will show thatF is a normal family. Let K be a compact subset of D. Since f(z)∈N1 and is thus a normal function, we get that(26)supz∈D1-z2f′z1+fz2<∞. It follows that for all a∈K there is some positive constant M where M is the smallest such number such that(27)limz→a2f′z1+fz2≤M<∞.Similarly, for each n∈N, since f(z)+bn∈N1, there exists a positive constant Mn where Mn is the smallest such number such that(28)limz→a2f′z1+fz+bn2≤Mn<∞,for all a∈K. Then, since bn→0, we get that Mn→M. Thus, there exists a finite positive constant P such that(29)Mn≤P,for all n∈N. Then, by inequality (29), it follows that(30)μf+bna=limz→a2f′z1+fz+bn2≤P<∞, for all a∈K and all integers n≥1. Therefore, F is locally bounded and by Theorem 10 is thus a normal family.
We will next show that the normal familyF satisfies the additional conditions of Theorem 13(i). Let ρ>0. Then, since f(z)∈N1, there is a positive number mρ≤1, where mρ is the largest such number such that χ(f(z),0)≥mρ whenever z∈D-⋃ν=1∞U(zν,f(0,ρ)) and χ(f(z),∞)≥mρ whenever z∈D-⋃ν=1∞U(zν,f(∞,ρ)). In addition, for each n∈N, since f(z)+bn∈N1, there is a positive number mρ,n≤1, where mρ,n is the largest such number such that(31)χfz+bn,0≥mρ,n,whenever z∈D-⋃ν=1∞U(zν,f+bn(0,ρ)), and(32)χfz+bn,∞≥mρ,n,whenever z∈D-⋃ν=1∞U(zν,f+bn(∞,ρ)). Then, since bn→0, it follows that mρ,n→mρ. Thus, there is a positive constant S<1 such that(33)mρ,n>S,for all n∈N. Then, for all integers n≥1, it follows by inequalities (31), (32), and (33) that(34)χfz+bn,0>S>0,whenever z∈D-⋃ν=1∞U(zν,f(z)+bn(0,ρ)), and(35)χfz+bn,∞>S>0,whenever z∈D-⋃ν=1∞U(zν,f(z)+bn(∞,ρ)).
Next, by Theorem9(i), for each n∈N, An(z)=-f(k)(z)/(f(z)+bn) is α(k+1)-normal in D for α≥1. Let K be a compact subset of D. Then, for each n∈N, there is a finite positive constant Wn where Wn is the smallest such number such that(36)limz→a2An′z1+Anz2≤Wn<∞,for all a∈K. It also follows by Theorem 9(i) that there is a finite positive constant W where W is the smallest such number such that(37)limz→a2A′z1+Az2≤W<∞,for all a∈K, where A(z)=-f(k)(z)/f(z). Since bn→0, it follows that Wn→W. Thus, there is a finite positive constant C such that(38)Wn<C,for all n∈N. It follows by inequality (38) that, for all n∈N,(39)μAna=limz→a2An′z1+Anz2≤C<∞,for all a∈K. Thus, the family of coefficients of (6),(40)A=Anzn=1∞=-fkzfz+bnn=1∞,is locally bounded and by Theorem 10 is therefore a normal family in D. This verifies the conclusion in Theorem 13(i).Example 16.
In Examples14 and 15, normal families of meromorphic functions were created using functions f(z) in N1. An example of a function in N1 is any Schwarzian triangle function, as long as the closure of one of the triangle functions’ fundamental domains is located entirely inside D (see [15]). We can construct Schwarzian triangle functions with a prescribed integer order for all poles and all zeros. See Nehari [20, Chapter VI, Section 5] for additional details. This gives a natural example of functions which satisfy conditions in both Theorems 9(ii) and 13(ii).


## 5. Proofs

### 5.1. Proof of Theorem6

Proof of Theorem6(i).
SupposeF⊂H(D) is a normal family. Then, F is locally bounded. Let f be in F and let K be a compact subset of D. Since D is open, the shortest distance between the boundary of K and the boundary of D is equal to some ϵ>0. Thus, there is some compact set K0⊂D such that K⊂K0 and such that the shortest distance between the boundary of K and the boundary of K0 equals ϵ/2. Then, by Theorem 5 and Cauchy’s Estimate, there is a constant M such that the open cover of K, ⋃a∈KB(a;ϵ/2)⊂K0, gives us that(41)fka≤k!Mϵ/2k, for all a in K and for each f in F. Thus, Fk={f(k):f∈F} is a normal family.Proof of Theorem6(ii).
We start withk=1. Suppose the family F1={f′:f∈F}⊂H(D) is normal. Let f′∈F1, suppose K is a compact subset of D, and let γ⊂K be the shortest rectifiable curve from a fixed point a∈K to z∈K. Then, f(z)=∫γf′(z)dz. Let V(γ) be the total variation of a curve γ. Since K is compact, it follows that there exists a positive constant M1 such that(42)Vγ≤M1,for all rectifiable curves γ⊂K as described above. In addition, since K is compact and F1 is a normal family, it follows that there exists a positive constant M2 such that(43)supf′z:z∈γ≤M2,for all f in F and any γ⊂K. Since f′ is analytic on D and thus continuous on γ, we have by inequalities (42) and (43) and [16, Chapter IV, Proposition 1.17(b)] that(44)fz=∫γf′zdz≤∫γf′zdz≤Vγsupf′z:z∈γ≤M1·M2<∞,for all z in K and all f in F. Therefore, by Theorem 5, F is a normal family. It clearly follows by induction that if Fk={f(k):f∈F} is a normal family, then F is a normal family.


### 5.2. Proof of Theorem7

Proof of Theorem7(i).
We follow a method of proof analogous to the proof of Montel’s theorem in [16], extended to the setting of families of differential equations. Towards a contradiction, suppose F is a normal family but that the family A is not locally bounded. Then, there exists a compact set K⊂D such that(45)sup-fγkfγ:z∈K,-fγkfγ∈A=∞.Thus, there is a sequence {-fγn(k)/fγn} in A such that(46)sup-fγnkfγn:z∈K≥n.Since F is a normal family, there is an f∈H(D) and a subsequence {fγn(j)}, with n(1)<n(2)<…, such that fγn(j)→f. It follows that fγn(j)(k)→f(k)∈H(D) for each integer k≥1. Since by assumption -fγn(j)(k)/fγn(j) is analytic for each n(j), since C(D,C) is a complete metric space, and since f and f(k) are analytic,(47)-fγnjkfγnj→-fkf∈HD.Thus, there is a positive constant M such that(48)-fkf≤M,for all z in K. Therefore, by inequalities (46) and (48),(49)nj≤sup-fγnjkfγnj--fkf-fkf:z∈K≤sup-fγnjkfγnj+fkf+fkf:z∈K≤sup⁡-fγnjkfγnj+fkf:z∈K+M.Since, as j→∞, we have that both n(j)→∞ and(50)sup⁡-fγnjkfγnj+fkf:z∈K+M→M, we have a contradiction. Therefore, the family A is locally bounded and is thus a normal family.Proof of Theorem7(ii).
As noted in [6, Section 2], for k=1, if fγ is a solution of (2), where the coefficient Aγ(z) is analytic in D, then by solving (2) explicitly, it follows that(51)fγz=fγreiφ≤C·exp⁡∫0rAγteiφdt,for 0≤t≤r<1, where C is an arbitrary complex constant.
LetK⊂D be a compact set. Then, since A is a normal family, there is a positive constant P such that |Aγ(z)|=-fγ′/fγ≤P, for all z in K and all γ∈Γ. Let(52)r0=supr:z=reiφ∈K.Then, we also have that there exists a positive constant P0 such that Aγz≤P0, for all z in B(0,r0)¯={z∈D:z≤r0} and all γ∈Γ. Let P1=max{P,P0}. Then, by inequality (51), it follows that(53)fγreiφ≤C·exp⁡P1<∞,for all z in K and all γ∈Γ, which gives the desired result.Proof of Theorem7(iii).
The proof of part (iii) proceeds just as the proof of part (ii), with the exception that, by assumption and by [6, Theorem 4.2], we get(54)fγreiφ≤∑n=0k-1fγn0n!rnexp⁡1k-1!∫0r1-tk-1Aγteiφdt≤M·exp⁡P1<∞, for all z in K and all γ∈Γ. This gives the desired result.


### 5.3. Proof of Theorem9

Proof of Theorem9(i) (let α≥1).
We start by exploring poles and zeros off in D.
Supposea is a pole of f∈M(D) of order m≥1. Then, there is a function g(z) analytic on some disk containing a for which we can express(55)fz=gz+Amz-am+⋯+A1z-a, for z in that disk and for some constants A1,…,Am. Using simple calculation as per [16, pp. 157-158], it follows that(56)limz→af′z1+fz2=0,if m≥2, and(57)limz→af′z1+fz2=1A1, if m=1.
For the differential equationf(k)+A(z)f=0, any pole of A(z) arises from zeros and poles of a solution f. If a is a zero of f of order m≥1, then there is a disk containing a and a function g(z) analytic on that disk for which g(a)≠0 and such that we can write(58)fz=z-amgz,for z in that disk. Through simple calculation, we see that a is a pole of A(z)=-f(k)/f of order k. If a is a pole of f of order m≥1, then there is a disk containing a and a function g(z) analytic on that disk for which g(a)≠0 and such that we can write(59)fz=gzz-am, for z in that disk. Through calculation, it follows that a is a pole of A(z)=-f(k)/f of order k.
Thus, fork≥2, if a is a zero or pole of f, then(60)limz→aA′z1+Az2=0,by (56), and so(61)limz→a1-z2αk+1A′z1+Az2=0.We next consider k=1. We get that(62)A′z1+Az2=-f·f′′+f′·f′/f21+-f′/f2=-f′′/f+f′/f21+f′/f2≤f′′/f+f′/f21+f′/f2.If a is a zero of f of order m≥1, then for all z≠a on some disk containing a we get that(63)f′f=z-ag′z+gz·mgz·1z-a,and(64)f′′f=z-a2g′′z+2m·g′zz-a+mm-1gzgz·1z-a2.It follows by inequality (62) and (63) and (64) that(65)A′z1+Az2≤f′′/f+f′/f21+f′/f2=z-a2g′′z+2m·g′zz-a+mm-1gz/gz+z-ag′z+gz·m/gz2z-a2+z-ag′z+gz·m/gz2.Then, by inequality (65), we get that(66)limz→aA′z1+Az2≤mm-1+m2m2=2-1m<2,for m≥1. It follows that(67)limz→a1-z2αk+1A′z1+Az2<2.If a is a pole of f of order m≥1, then for all z≠a on some disk containing a we get that(68)f′f=z-ag′z-gz·mgz·1z-a,and(69)f′′f=z-a2g′′z-3m2-mgzgz·1z-a2.So by inequality (62) and (68) and (69), we have that(70)A′z1+Az2≤f′′/f+f′/f21+f′/f2=z-a2g′′z-3m2-mgz/gz+z-ag′z-gz·m/gz2z-a2+z-ag′z-gz·m/gz2.Thus, by inequality (70) it follows that(71)limz→aA′z1+Az2≤3m2-m+m2m2=4-1m<4,for m≥1. And thus(72)limz→a1-z2αk+1A′z1+Az2<4.Towards a contradiction, suppose that there is a sequence zn∈D such that zn is not a pole or zero of f for all n and a corresponding set of poles and/or zeros of f, {zν,f(α)n}, such that(73)limn→∞zn-zν,fαn=0,and(74)limn→∞1-zn2αk+1A′zn1+Azn2=∞.Then, by (73) and (74),(75)limn→∞limz→zν,fαn1-z2αk+1A′z1+Az2=limn→∞limz→zν,fαn1-z-zn+zn2αk+1·A′z-zn+zn1+Az-zn+zn2=limn→∞1-zn2αk+1A′zn1+Azn2=∞. This contradicts the bound of 4 on limz→a(1-|z|2)αk+1(A′z/(1+|Az|2)) for all poles and zeros a of f.
Next letρ>0, and define(76)B∞,f,ρ=D-⋃ν=1∞Uzν,f∞,ρ, and(77)B0,f,ρ=D-⋃ν=1∞Uzν,f0,ρ.Then, by [14, Theorem 4(i)] there exists a constant C(f,ρ) such that the coefficient A of (2) satisfies(78)1-z2αk+1A′z1+Az2≤Cf,ρ,for each z∈B∞,f,ρ⋂B0,f,ρ, for k≥1. Then, by (61) and inequalities (67), (72), and (78), there must be some ρ0>0 such that(79)maxρ∈R+Cf,ρ=Cf,ρ0<∞.Let M=max⁡{C(f,ρ0),4}. It follows that(80)limz→a1-z2αk+1A′z1+Az2≤M, for all a∈D. Therefore, A(z) is α(k+1)-normal in D, for k≥1.Proof of Theorem9(ii) (let α>0).
Poles of a coefficientA arise from zeros and poles of a solution f. Suppose a is a zero of f of order m≥1. Then, there is a disk containing a on which(81)fz=z-amgz,where g(z) is analytic on that disk and g(a)≠0 there. We further get that(82)fkzfz=jzgz·1z-ak,where j(z) is also analytic in that disk and j(a)≠0. Then, by (81) and (82) we get(83)fkz1-z2αk1+fzk+1=jz/gz·z-amgz/z-ak1-z2αk1+z-amgzk+1=jzz-am-k1-z2αk1+z-amgzk+1,for z≠a in D. It follows by (83) that(84)limz→afkz1-z2αk1+fzk+1=0,if m>k.
Next, supposea is a pole of f of order m≥1. Then, there is a disk containing a on which(85)fz=gzz-am,where g(z) is analytic on that disk and g(a)≠0 there. We further get that(86)fkzfz=pzgz·1z-ak,where p(z) is also analytic in that disk and p(a)≠0. Then, by (85) and (86) we get(87)fkz1-z2αk1+fzk+1=pz/gz·gz/z-am/z-ak1-z2αk1+gz/z-amk+1=z-akm-1·pz1-z2αkz-amk+1+gzk+1,for z≠a in D. It follows by (87) that(88)limz→afkz1-z2αk1+fzk+1=0,if m>1.
By [14, Theorem 4(ii)], there exists a constant C(A,ρ) such that(89)fkz1-z2αk1+fzk+1≤CA,ρ,for each z∈B∞,A,ρ. Then, by (84) and (88) and inequality (89), there must be some ρ0>0 such that(90)maxρ∈R+CA,ρ=CA,ρ0<∞.Thus,(91)limz→afkz1-z2αk1+fzk+1≤CA,ρ0,for all a∈D, which is the desired result.


### 5.4. Proof of Theorem13

Proof of Theorem13(i).
The first part of our proof of Theorem13(i) uses methods analogous to those in the proof of [14, Theorem 4(i)] but extended for families of functions. Let K be a compact subset of D and let ρ>0. For all z∈D and γ∈Γ, we have that(92)1-z2k+1Aγ′1+Aγ2=1-z2k+1-fγ·fγk+1+fγk·fγ′/fγ21+Aγ2.Thus, by (92), it follows that(93)1-z2k+1Aγ′1+Aγ2≤1-z2k+1fγ·fγk+1+fγk·fγ′fγ2=1-z2k+1fγk+1fγ+1-z2kfγk1-z2fγ′fγ2, for all z∈(B∞,fγ,ρ⋂B0,fγ,ρ)⋂K and γ∈Γ. Then, by inequality (93) and Theorem 12, there exist constants Pk+1(K,F), Pk(K,F), and P1(K,F) such that(94)1-z2k+1Aγ′1+Aγ2≤Pk+1K,F1+fγk+2fγ+PkK,F1+fγk+1P1K,F1+fγ2fγ2,for each z∈(B∞,fγ,ρ⋂B0,fγ,ρ)⋂K and each fγ∈F.
Next, we have by assumption that there exists a positive numbermρ<1 such that(95)χfγz,∞=11+fγz2>mρ, for z∈B∞,fγ,ρ and each fγ∈F, and(96)χfγz,0=fγz-01+fγz21+02>mρ, for z∈B0,fγ,ρ and each fγ∈F. It follows that(97)fγz<1mρ2-1,for z∈B∞,fγ,ρ and each fγ∈F, and(98)fγz>mρ21-mρ2>0,for z∈B0,fγ,ρ and each fγ∈F.
Then, by inequalities (94), (97), and (98), we get that(99)1-z2k+1Aγ′1+Aγ2≤Pk+1K,F1+1/mρ2-1k+2mρ2/1-mρ2+PkK,F1+1mρ2-1k+1·P1K,F1+1/mρ2-12mρ2/1-mρ22=CK,F,ρ<∞,for each z∈(B∞,fγ,ρ⋂B0,fγ,ρ)⋂K and each γ∈Γ.
It follows that there is a finite constantJ such that(100)Aγ′1+Aγ2≤CK,F,ρ·11-z2k+1≤CK,F,ρ·J,for each z∈(B∞,fγ,ρ⋂B0,fγ,ρ)⋂K and each γ∈Γ.
By (60) and inequalities (66) and (71), we have that(101)limz→aAγ′1+Aγ2<4,for all poles and zeros a of fγ, for each γ∈Γ. It follows by inequalities (100) and (101) that there must be some ρ0>0 such that(102)maxρ∈R+CK,F,ρ·J=CK,F,ρ0·J<∞.Let M=max{C(K,F,ρ0)·J,4}. Then, it follows that(103)limz→aAγ′1+Aγ2≤M,for all z∈K and each γ∈Γ. Therefore, A is a normal family.Proof of Theorem13(ii).
The first part of our proof of Theorem13(ii) uses methods comparable to those used in the proof of [14, Theorem 4(ii)] but extended for the setting of families of functions. Let ρ>0 and let k≥1 be an integer. Then, by assumption, there exists a positive number mρ<1 such that(104)χAγz,∞=11+Aγz2>mρ,whenever z∈B∞,Aγ,ρ⋂K and Aγ∈A. Thus,(105)Aγz<1mρ2-1,whenever z∈B∞,Aγ,ρ⋂K and Aγ∈A.
The family (6) gives fγ(k)(z)=-Aγ(z)fγ(z) for all z∈D and γ∈Γ. It then follows for all z∈D and γ∈Γ that(106)fγkz1-z2k1+fγzk+1=Aγzfγz1-z2k1+fγzk+1.Then, by inequality (105) and (106), we get(107)fγkz1-z2k1+fγzk+1≤Aγz<1mρ2-1=CkK,A,ρ,for each z∈B∞,Aγ,ρ⋂K and each γ∈Γ.
Poles of a coefficientAγ arise from zeros and poles of solutions fγ. Let γ∈Γ. Suppose a is a zero of fγ of order m≥1. Then, there is a disk containing a on which fγ(z)=(z-a)mgγ(z), where gγ(z) is analytic on that disk and gγ(a)≠0 there. By (82), (83), and (84), it follows that(108)limz→afγkz1-z2k1+fγzk+1=limz→ajγzz-am-k1-z2k1+z-amgγzk+1=0,if m>k, where jγ(z) is a function analytic on a disk containing a and jγ(a)≠0 there, for each γ∈Γ.
Next, supposea is a pole of fγ of order m≥1. Then, there is a disk containing a on which fγ(z)=gγ(z)/(z-a)m, where gγ(z) is analytic on that disk and gγ(a)≠0 there. It then follows by (86), (87), and (88) that(109)limz→afγkz1-z2k1+fγzk+1=limz→az-akm-1pγz1-z2kz-amk+1+gγzk+1=0,if m>1, where pγ(z) is a function analytic on a disk containing a and pγ(a)≠0 there, for each γ∈Γ.
Thus, by inequality (107) and (108) and (109), there must be some ρ0>0 such that (110)maxρ∈R+CkK,A,ρ=CkK,A,ρ0<∞.Therefore,(111)limz→afγkz1-z2k1+fγzk+1≤CkK,A,ρ0, for all a∈K and for each fγ∈F, which is the desired result.


## 5.1. Proof of Theorem6

Proof of Theorem6(i).
SupposeF⊂H(D) is a normal family. Then, F is locally bounded. Let f be in F and let K be a compact subset of D. Since D is open, the shortest distance between the boundary of K and the boundary of D is equal to some ϵ>0. Thus, there is some compact set K0⊂D such that K⊂K0 and such that the shortest distance between the boundary of K and the boundary of K0 equals ϵ/2. Then, by Theorem 5 and Cauchy’s Estimate, there is a constant M such that the open cover of K, ⋃a∈KB(a;ϵ/2)⊂K0, gives us that(41)fka≤k!Mϵ/2k, for all a in K and for each f in F. Thus, Fk={f(k):f∈F} is a normal family.Proof of Theorem6(ii).
We start withk=1. Suppose the family F1={f′:f∈F}⊂H(D) is normal. Let f′∈F1, suppose K is a compact subset of D, and let γ⊂K be the shortest rectifiable curve from a fixed point a∈K to z∈K. Then, f(z)=∫γf′(z)dz. Let V(γ) be the total variation of a curve γ. Since K is compact, it follows that there exists a positive constant M1 such that(42)Vγ≤M1,for all rectifiable curves γ⊂K as described above. In addition, since K is compact and F1 is a normal family, it follows that there exists a positive constant M2 such that(43)supf′z:z∈γ≤M2,for all f in F and any γ⊂K. Since f′ is analytic on D and thus continuous on γ, we have by inequalities (42) and (43) and [16, Chapter IV, Proposition 1.17(b)] that(44)fz=∫γf′zdz≤∫γf′zdz≤Vγsupf′z:z∈γ≤M1·M2<∞,for all z in K and all f in F. Therefore, by Theorem 5, F is a normal family. It clearly follows by induction that if Fk={f(k):f∈F} is a normal family, then F is a normal family.


## 5.2. Proof of Theorem7

Proof of Theorem7(i).
We follow a method of proof analogous to the proof of Montel’s theorem in [16], extended to the setting of families of differential equations. Towards a contradiction, suppose F is a normal family but that the family A is not locally bounded. Then, there exists a compact set K⊂D such that(45)sup-fγkfγ:z∈K,-fγkfγ∈A=∞.Thus, there is a sequence {-fγn(k)/fγn} in A such that(46)sup-fγnkfγn:z∈K≥n.Since F is a normal family, there is an f∈H(D) and a subsequence {fγn(j)}, with n(1)<n(2)<…, such that fγn(j)→f. It follows that fγn(j)(k)→f(k)∈H(D) for each integer k≥1. Since by assumption -fγn(j)(k)/fγn(j) is analytic for each n(j), since C(D,C) is a complete metric space, and since f and f(k) are analytic,(47)-fγnjkfγnj→-fkf∈HD.Thus, there is a positive constant M such that(48)-fkf≤M,for all z in K. Therefore, by inequalities (46) and (48),(49)nj≤sup-fγnjkfγnj--fkf-fkf:z∈K≤sup-fγnjkfγnj+fkf+fkf:z∈K≤sup⁡-fγnjkfγnj+fkf:z∈K+M.Since, as j→∞, we have that both n(j)→∞ and(50)sup⁡-fγnjkfγnj+fkf:z∈K+M→M, we have a contradiction. Therefore, the family A is locally bounded and is thus a normal family.Proof of Theorem7(ii).
As noted in [6, Section 2], for k=1, if fγ is a solution of (2), where the coefficient Aγ(z) is analytic in D, then by solving (2) explicitly, it follows that(51)fγz=fγreiφ≤C·exp⁡∫0rAγteiφdt,for 0≤t≤r<1, where C is an arbitrary complex constant.
LetK⊂D be a compact set. Then, since A is a normal family, there is a positive constant P such that |Aγ(z)|=-fγ′/fγ≤P, for all z in K and all γ∈Γ. Let(52)r0=supr:z=reiφ∈K.Then, we also have that there exists a positive constant P0 such that Aγz≤P0, for all z in B(0,r0)¯={z∈D:z≤r0} and all γ∈Γ. Let P1=max{P,P0}. Then, by inequality (51), it follows that(53)fγreiφ≤C·exp⁡P1<∞,for all z in K and all γ∈Γ, which gives the desired result.Proof of Theorem7(iii).
The proof of part (iii) proceeds just as the proof of part (ii), with the exception that, by assumption and by [6, Theorem 4.2], we get(54)fγreiφ≤∑n=0k-1fγn0n!rnexp⁡1k-1!∫0r1-tk-1Aγteiφdt≤M·exp⁡P1<∞, for all z in K and all γ∈Γ. This gives the desired result.


## 5.3. Proof of Theorem9

Proof of Theorem9(i) (let α≥1).
We start by exploring poles and zeros off in D.
Supposea is a pole of f∈M(D) of order m≥1. Then, there is a function g(z) analytic on some disk containing a for which we can express(55)fz=gz+Amz-am+⋯+A1z-a, for z in that disk and for some constants A1,…,Am. Using simple calculation as per [16, pp. 157-158], it follows that(56)limz→af′z1+fz2=0,if m≥2, and(57)limz→af′z1+fz2=1A1, if m=1.
For the differential equationf(k)+A(z)f=0, any pole of A(z) arises from zeros and poles of a solution f. If a is a zero of f of order m≥1, then there is a disk containing a and a function g(z) analytic on that disk for which g(a)≠0 and such that we can write(58)fz=z-amgz,for z in that disk. Through simple calculation, we see that a is a pole of A(z)=-f(k)/f of order k. If a is a pole of f of order m≥1, then there is a disk containing a and a function g(z) analytic on that disk for which g(a)≠0 and such that we can write(59)fz=gzz-am, for z in that disk. Through calculation, it follows that a is a pole of A(z)=-f(k)/f of order k.
Thus, fork≥2, if a is a zero or pole of f, then(60)limz→aA′z1+Az2=0,by (56), and so(61)limz→a1-z2αk+1A′z1+Az2=0.We next consider k=1. We get that(62)A′z1+Az2=-f·f′′+f′·f′/f21+-f′/f2=-f′′/f+f′/f21+f′/f2≤f′′/f+f′/f21+f′/f2.If a is a zero of f of order m≥1, then for all z≠a on some disk containing a we get that(63)f′f=z-ag′z+gz·mgz·1z-a,and(64)f′′f=z-a2g′′z+2m·g′zz-a+mm-1gzgz·1z-a2.It follows by inequality (62) and (63) and (64) that(65)A′z1+Az2≤f′′/f+f′/f21+f′/f2=z-a2g′′z+2m·g′zz-a+mm-1gz/gz+z-ag′z+gz·m/gz2z-a2+z-ag′z+gz·m/gz2.Then, by inequality (65), we get that(66)limz→aA′z1+Az2≤mm-1+m2m2=2-1m<2,for m≥1. It follows that(67)limz→a1-z2αk+1A′z1+Az2<2.If a is a pole of f of order m≥1, then for all z≠a on some disk containing a we get that(68)f′f=z-ag′z-gz·mgz·1z-a,and(69)f′′f=z-a2g′′z-3m2-mgzgz·1z-a2.So by inequality (62) and (68) and (69), we have that(70)A′z1+Az2≤f′′/f+f′/f21+f′/f2=z-a2g′′z-3m2-mgz/gz+z-ag′z-gz·m/gz2z-a2+z-ag′z-gz·m/gz2.Thus, by inequality (70) it follows that(71)limz→aA′z1+Az2≤3m2-m+m2m2=4-1m<4,for m≥1. And thus(72)limz→a1-z2αk+1A′z1+Az2<4.Towards a contradiction, suppose that there is a sequence zn∈D such that zn is not a pole or zero of f for all n and a corresponding set of poles and/or zeros of f, {zν,f(α)n}, such that(73)limn→∞zn-zν,fαn=0,and(74)limn→∞1-zn2αk+1A′zn1+Azn2=∞.Then, by (73) and (74),(75)limn→∞limz→zν,fαn1-z2αk+1A′z1+Az2=limn→∞limz→zν,fαn1-z-zn+zn2αk+1·A′z-zn+zn1+Az-zn+zn2=limn→∞1-zn2αk+1A′zn1+Azn2=∞. This contradicts the bound of 4 on limz→a(1-|z|2)αk+1(A′z/(1+|Az|2)) for all poles and zeros a of f.
Next letρ>0, and define(76)B∞,f,ρ=D-⋃ν=1∞Uzν,f∞,ρ, and(77)B0,f,ρ=D-⋃ν=1∞Uzν,f0,ρ.Then, by [14, Theorem 4(i)] there exists a constant C(f,ρ) such that the coefficient A of (2) satisfies(78)1-z2αk+1A′z1+Az2≤Cf,ρ,for each z∈B∞,f,ρ⋂B0,f,ρ, for k≥1. Then, by (61) and inequalities (67), (72), and (78), there must be some ρ0>0 such that(79)maxρ∈R+Cf,ρ=Cf,ρ0<∞.Let M=max⁡{C(f,ρ0),4}. It follows that(80)limz→a1-z2αk+1A′z1+Az2≤M, for all a∈D. Therefore, A(z) is α(k+1)-normal in D, for k≥1.Proof of Theorem9(ii) (let α>0).
Poles of a coefficientA arise from zeros and poles of a solution f. Suppose a is a zero of f of order m≥1. Then, there is a disk containing a on which(81)fz=z-amgz,where g(z) is analytic on that disk and g(a)≠0 there. We further get that(82)fkzfz=jzgz·1z-ak,where j(z) is also analytic in that disk and j(a)≠0. Then, by (81) and (82) we get(83)fkz1-z2αk1+fzk+1=jz/gz·z-amgz/z-ak1-z2αk1+z-amgzk+1=jzz-am-k1-z2αk1+z-amgzk+1,for z≠a in D. It follows by (83) that(84)limz→afkz1-z2αk1+fzk+1=0,if m>k.
Next, supposea is a pole of f of order m≥1. Then, there is a disk containing a on which(85)fz=gzz-am,where g(z) is analytic on that disk and g(a)≠0 there. We further get that(86)fkzfz=pzgz·1z-ak,where p(z) is also analytic in that disk and p(a)≠0. Then, by (85) and (86) we get(87)fkz1-z2αk1+fzk+1=pz/gz·gz/z-am/z-ak1-z2αk1+gz/z-amk+1=z-akm-1·pz1-z2αkz-amk+1+gzk+1,for z≠a in D. It follows by (87) that(88)limz→afkz1-z2αk1+fzk+1=0,if m>1.
By [14, Theorem 4(ii)], there exists a constant C(A,ρ) such that(89)fkz1-z2αk1+fzk+1≤CA,ρ,for each z∈B∞,A,ρ. Then, by (84) and (88) and inequality (89), there must be some ρ0>0 such that(90)maxρ∈R+CA,ρ=CA,ρ0<∞.Thus,(91)limz→afkz1-z2αk1+fzk+1≤CA,ρ0,for all a∈D, which is the desired result.


## 5.4. Proof of Theorem13

Proof of Theorem13(i).
The first part of our proof of Theorem13(i) uses methods analogous to those in the proof of [14, Theorem 4(i)] but extended for families of functions. Let K be a compact subset of D and let ρ>0. For all z∈D and γ∈Γ, we have that(92)1-z2k+1Aγ′1+Aγ2=1-z2k+1-fγ·fγk+1+fγk·fγ′/fγ21+Aγ2.Thus, by (92), it follows that(93)1-z2k+1Aγ′1+Aγ2≤1-z2k+1fγ·fγk+1+fγk·fγ′fγ2=1-z2k+1fγk+1fγ+1-z2kfγk1-z2fγ′fγ2, for all z∈(B∞,fγ,ρ⋂B0,fγ,ρ)⋂K and γ∈Γ. Then, by inequality (93) and Theorem 12, there exist constants Pk+1(K,F), Pk(K,F), and P1(K,F) such that(94)1-z2k+1Aγ′1+Aγ2≤Pk+1K,F1+fγk+2fγ+PkK,F1+fγk+1P1K,F1+fγ2fγ2,for each z∈(B∞,fγ,ρ⋂B0,fγ,ρ)⋂K and each fγ∈F.
Next, we have by assumption that there exists a positive numbermρ<1 such that(95)χfγz,∞=11+fγz2>mρ, for z∈B∞,fγ,ρ and each fγ∈F, and(96)χfγz,0=fγz-01+fγz21+02>mρ, for z∈B0,fγ,ρ and each fγ∈F. It follows that(97)fγz<1mρ2-1,for z∈B∞,fγ,ρ and each fγ∈F, and(98)fγz>mρ21-mρ2>0,for z∈B0,fγ,ρ and each fγ∈F.
Then, by inequalities (94), (97), and (98), we get that(99)1-z2k+1Aγ′1+Aγ2≤Pk+1K,F1+1/mρ2-1k+2mρ2/1-mρ2+PkK,F1+1mρ2-1k+1·P1K,F1+1/mρ2-12mρ2/1-mρ22=CK,F,ρ<∞,for each z∈(B∞,fγ,ρ⋂B0,fγ,ρ)⋂K and each γ∈Γ.
It follows that there is a finite constantJ such that(100)Aγ′1+Aγ2≤CK,F,ρ·11-z2k+1≤CK,F,ρ·J,for each z∈(B∞,fγ,ρ⋂B0,fγ,ρ)⋂K and each γ∈Γ.
By (60) and inequalities (66) and (71), we have that(101)limz→aAγ′1+Aγ2<4,for all poles and zeros a of fγ, for each γ∈Γ. It follows by inequalities (100) and (101) that there must be some ρ0>0 such that(102)maxρ∈R+CK,F,ρ·J=CK,F,ρ0·J<∞.Let M=max{C(K,F,ρ0)·J,4}. Then, it follows that(103)limz→aAγ′1+Aγ2≤M,for all z∈K and each γ∈Γ. Therefore, A is a normal family.Proof of Theorem13(ii).
The first part of our proof of Theorem13(ii) uses methods comparable to those used in the proof of [14, Theorem 4(ii)] but extended for the setting of families of functions. Let ρ>0 and let k≥1 be an integer. Then, by assumption, there exists a positive number mρ<1 such that(104)χAγz,∞=11+Aγz2>mρ,whenever z∈B∞,Aγ,ρ⋂K and Aγ∈A. Thus,(105)Aγz<1mρ2-1,whenever z∈B∞,Aγ,ρ⋂K and Aγ∈A.
The family (6) gives fγ(k)(z)=-Aγ(z)fγ(z) for all z∈D and γ∈Γ. It then follows for all z∈D and γ∈Γ that(106)fγkz1-z2k1+fγzk+1=Aγzfγz1-z2k1+fγzk+1.Then, by inequality (105) and (106), we get(107)fγkz1-z2k1+fγzk+1≤Aγz<1mρ2-1=CkK,A,ρ,for each z∈B∞,Aγ,ρ⋂K and each γ∈Γ.
Poles of a coefficientAγ arise from zeros and poles of solutions fγ. Let γ∈Γ. Suppose a is a zero of fγ of order m≥1. Then, there is a disk containing a on which fγ(z)=(z-a)mgγ(z), where gγ(z) is analytic on that disk and gγ(a)≠0 there. By (82), (83), and (84), it follows that(108)limz→afγkz1-z2k1+fγzk+1=limz→ajγzz-am-k1-z2k1+z-amgγzk+1=0,if m>k, where jγ(z) is a function analytic on a disk containing a and jγ(a)≠0 there, for each γ∈Γ.
Next, supposea is a pole of fγ of order m≥1. Then, there is a disk containing a on which fγ(z)=gγ(z)/(z-a)m, where gγ(z) is analytic on that disk and gγ(a)≠0 there. It then follows by (86), (87), and (88) that(109)limz→afγkz1-z2k1+fγzk+1=limz→az-akm-1pγz1-z2kz-amk+1+gγzk+1=0,if m>1, where pγ(z) is a function analytic on a disk containing a and pγ(a)≠0 there, for each γ∈Γ.
Thus, by inequality (107) and (108) and (109), there must be some ρ0>0 such that (110)maxρ∈R+CkK,A,ρ=CkK,A,ρ0<∞.Therefore,(111)limz→afγkz1-z2k1+fγzk+1≤CkK,A,ρ0, for all a∈K and for each fγ∈F, which is the desired result.


## 6. Concluding Remarks

Although not as widely studied as normal families, a generalization of normal families called quasinormal families has been of interest since its introduction by Montel in 1922 in [21]. Further generalizations of normal families called Qm-normal families and Qω-normal families were described by Chuang in [22], and an additional generalization called hyponormal families was investigated by Bloch in [23]. A potential line of further inquiry would be the possibility of results similar to those in this paper involving these and also other families of functions within the context of the correspondence between families of solutions and families of coefficients of a family of differential equations (6). We plan to study such considerations in future work.



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