Families of Differential Equations in the Unit Disk

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.


Introduction
The interaction between coefficient conditions and solution conditions for linear differential equations in the unit disk D = { ∈ C : || < 1} has been a topic of many investigations, including [1][2][3][4][5][6][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.

Preliminaries
The idea of a normal function in the unit disk D originated with Noshiro [15].Definition 1.A meromorphic function  is called normal in D if the set of functions  ∘  is a normal family in D, where  ranges over the conformal mappings of D onto itself.
Let  be an open subset of C and let (, C) be the set of all continuous functions from  to C. Definition 2 (see [16]).A set of functions F ⊂ (,C) is called a normal family if each sequence in F has a subsequence which converges to a function  ∈ (, C).
We note that meromorphic functions  in Definition 3 with  = 1 were classified by Noshiro as normal.
Let  ≥ 1 be an integer.In [17], Fowler and Sons showed that if a coefficient () of the differential equation is normal, then a solution  may not be normal.In addition, they showed that if a solution  of ( 2) is normal, a coefficient () may not be normal.However, they showed in [17,Theorem 1.1(i)] that if  is an -normal solution of (2), then the coefficient () is "almost" ( + 1)-normal in the unit disk D, except for a factor of ||/(1 + || +2 ).
When considering normal families of analytic functions in (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 {  } in a family F tends to  ≡ ∞ in , 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 ( for all  in F and  in . 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  = 1 and for an open set  ⊂ C but not proven.Part (ii) is alluded to in the same exercise for  = 1 but also not proven.We include a proof in both directions for  ≥ 1. Theorem 6.Let  ≥ 1 be an integer, and let F ⊂ (D) be a family of functions.
(i) If the family F is normal, then the family is also normal.
(ii) If the family is normal, then the family F is also normal.
We next examine families of differential equations.Let Γ ⊂ R and let  ≥ 1 be an integer.We consider the family of differential equations with the corresponding family of coefficients and family of solutions Note that, in the family of solutions F = {  } ∈Γ of (6), we include all solutions   corresponding to   () for any  ∈ Γ.

Main Results
The next theorem relates normal families of coefficients and solutions.
Theorem 7. (i) Let  ≥ 1 be an integer.Suppose is a family of coefficients of (6), and suppose the family of solutions F ⊂ (D) is a normal family.Then, A is also a normal family.
(ii) Suppose is a normal family of coefficients of (6).Then, the family of solutions F ⊂ (D) is also a normal family.
(iii) Let  ≥ 2 be an integer.Suppose is a normal family of coefficients of (6) for which there exists a positive constant  such that ∑ −1 =0 | ()  (0)| ≤  for any  ∈ Γ.Then, the family of solutions F ⊂ (D) 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 function  in D is called a normal function of the first category if and only if  is a normal function and any sequence   which is a subset of the family  ∘ , where  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 by N 1 .Fowler showed in [14,Theorem 4] that if  ∈ N 1 is a solution of (2), then we can eliminate the factor ||/(1 + || +2 ) in [17, Theorem 1.1(i)], and it follows that the coefficient () of ( 2) is ( + 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 .In the same theorem, Fowler showed that if the coefficient () of ( 2) is a normal meromorphic function of the first category, then a solution  of (2) behaves like an -normal function within the context of [17,Lemma 3.1] but within the same subset of D as above, for  > 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.
(ii) Let  be the meromorphic coefficient of (2) in D satisfying  ∈ N 1 , and suppose that  is a meromorphic solution of ( 2) in D such that any pole of  has order  > 1 and any zero of  has order  >  ≥ 1.Then, for any  > 0, there exists a constant (, ) such that for some  > 0 and for each  ∈ D.
Observe that, by [17, Lemma 3.1], we conclude in Theorem 9(ii) that  behaves as an -normal function would in D.
The natural bound on (1 − || 2 ) (+1)  # () when  ∈ N 1 motivates a similar type of bound for a family of meromorphic functions.We consider normal meromorphic families in (D, C ∞ ).
Let  be a meromorphic function defined on D. Following [16], we let () : D → R be defined by We extend the following property of functions in N 1 to normal meromorphic families to get a bound in line with the bound for functions in N 1 .Let  denote chordal distance, let  ], () be the set of -points of , and let (, ) = { : (, ) < }, where  designates the hyperbolic metric.
Theorem 11 (see [18, Theorem 1(iii)]).If () belongs to N 1 , then for any value  and any positive number , there exists a positive number   < 1 such that ((), ) >   whenever 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]).Let F be a normal family of meromorphic functions on D and let  be a compact subset of D. For each positive integer  there exists a constant   (, F) such that       ()  (ii) Let  ≥ 1 be an integer and let  be a compact subset of D. Suppose A ⊂ (D) is a normal family of coefficients of (6) with the property that, for any positive number , there exists a positive number (∞), ) and   ∈ A. Further suppose that the family of solutions F ⊂ (D) has the property that any pole of   has order  > 1 and any zero of   has order  > .Then, for each positive integer  there exists a constant   (, A, ) such that the set of solutions F satisfies for some  > 0, for each  ∈ , and for each   ∈ F.
Thus, we conclude in Theorem 13(ii) that F behaves as a normal family would on D within the context of Theorem 12.

Examples
Example 14.We present an example of a family F that satisfies the conditions of Theorem 13(i).Let () belong to N 1 and let {  /  } ∞ =1 be a sequence of distinct complex fractions such that, for all integers  ≥ 1, we have that |  /  | > 1 and     ̸ = 0. Further suppose that there is a finite constant  such that   /  →  and || > 1.Then, by [18,Lemma 1], each member of the family We will show that F is a normal family.Let  be a compact subset of D. Since () ∈ N 1 and is thus a normal function, we get that sup for all  ∈  and all integers  ≥ 1.Therefore, F is locally bounded and by Theorem 10 is thus a normal family.
We will next show that the normal family F satisfies the additional conditions of Theorem 13(i).Let  > 0.Then, since By (21) and since |  /  | > 1 for all integers  ≥ 1, it follows that whenever , for all integers  ≥ 1.
It then clearly follows by Theorems 9(i) and 10 that the corresponding family of coefficients of ( 6), is a normal family in D. This verifies the conclusion in Theorem 13(i).
Example 15.For a second example of a family F that satisfies the conditions of Theorem 13(i), let () belong to N 1 and let {  } ∞ =1 be a sequence of distinct complex numbers such that   → 0 and such that, for all integers  ≥ 1, we have that   ̸ = 0.Then, by [18, Lemma 1], each member of the family F = {() +   } ∞ =1 belongs to N 1 .We will show that F is a normal family.Let for all  ∈ .Then, since   → 0, we get that   → .Thus, there exists a finite positive constant  such that for all  ∈ N.Then, by inequality (29), it follows that for all  ∈  and all integers  ≥ 1.Therefore, F is locally bounded and by Theorem 10 is thus a normal family.We will next show that the normal family F satisfies the additional conditions of Theorem 13(i).Let  > 0.Then, since () ∈ N 1 , there is a positive number   ≤ 1, where   is the largest such number such that ((), 0) ≥   whenever In addition, for each  ∈ N, since () +   ∈ N 1 , there is a positive number  , ≤ 1, where  , is the largest such number such that whenever whenever Then, since   → 0, it follows that  , →   .Thus, there is a positive constant  < 1 such that for all  ∈ N.Then, for all integers  ≥ 1, it follows by inequalities (31), (32), and (33) that whenever  ∈ D − ⋃ ∞ ]=1 ( ],()+  (0, )), and whenever for all  ∈ , where () = − () ()/().Since   → 0, it follows that   → .Thus, there is a finite positive constant  such that for all  ∈ N. It follows by inequality (38) that, for all  ∈ N, for all  ∈ .Thus, the family of coefficients of ( 6), 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 Examples 14 and 15, normal families of meromorphic functions were created using functions () in N 1 .An example of a function in N 1 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).

Proofs
for all  in  and for each  in F. Thus, F  = { () :  ∈ F} is a normal family.

Proof of Theorem 6(ii).
We start with  = 1.Suppose the family suppose  is a compact subset of D, and let  ⊂  be the shortest rectifiable curve from a fixed point  ∈  to  ∈ .
Then, () = ∫    () .Let () be the total variation of a curve .Since  is compact, it follows that there exists a positive constant  1 such that for all rectifiable curves  ⊂  as described above.In addition, since  is compact and F for all  in  and all  in F. Therefore, by Theorem 5, F is a normal family.It clearly follows by induction that if F  = { () :  ∈ F} is a normal family, then F is a normal family.

Proof of Theorem 7
Proof of Theorem 7(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.

Proof of Theorem 7(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 for all  in  and all  ∈ Γ.This gives the desired result.

Proof of Theorem 9
Proof of Theorem 9(i) (let  ≥ 1).We start by exploring poles and zeros of  in D.
Suppose  is a pole of  ∈ (D) of order  ≥ 1.Then, there is a function () analytic on some disk containing  for which we can express for  in that disk and for some constants  1 , .
For the differential equation  () + () = 0, any pole of () arises from zeros and poles of a solution .If  is a zero of  of order  ≥ 1, then there is a disk containing  and a function () analytic on that disk for which () ̸ = 0 and such that we can write for  in that disk.Through simple calculation, we see that  is a pole of () = − () / of order .If  is a pole of  of order  ≥ 1, then there is a disk containing  and a function () analytic on that disk for which () ̸ = 0 and such that we can write for  in that disk.Through calculation, it follows that  is a pole of () = − () / of order . ) . ( If  is a zero of  of order  ≥ 1, then for all  ̸ =  on some disk containing  we get that and Then, by inequality (65), we get that lim for  ≥ 1.It follows that lim If  is a pole of  of order  ≥ 1, then for all  ̸ =  on some disk containing  we get that and So by inequality (62) and ( 68) and (69), we have that Thus, by inequality (70) it follows that lim for  ≥ 1.And thus lim Towards a contradiction, suppose that there is a sequence   ∈ D such that   is not a pole or zero of  for all  and a corresponding set of poles and/or zeros of , { ], ()  }, such that lim Then, by ( 73) and (74), lim This contradicts the bound of 4 on lim )) for all poles and zeros  of .
Proof of Theorem 9(ii) (let  > 0).Poles of a coefficient  arise from zeros and poles of a solution .Suppose  is a zero of  of order  ≥ 1.Then, there is a disk containing  on which where () is analytic on that disk and () ̸ = 0 there.We further get that where () is also analytic in that disk and () ̸ = 0.Then, by (81) and ( 82 if  > .
Next, suppose  is a pole of  of order  ≥ 1.Then, there is a disk containing  on which where () is analytic on that disk and () ̸ = 0 there.We further get that where () is also analytic in that disk and () ̸ = 0.Then, by (85) and (86) we get if  > 1, where   () is a function analytic on a disk containing  and   () ̸ = 0 there, for each  ∈ Γ.Thus, by inequality (107) and ( 108) and ( 109), there must be some  0 > 0 such that max ∈R + {  (, A, )} =   (, A,  0 ) < ∞.
for all  ∈  and for each   ∈ F, which is the desired result.

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   -normal families and   -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.
(23), by inequality(19)and(23), and since |  /  | > 1 for all integers  ≥ 1, it follows that 1/       /      2 +      ()     2 .(23) be a compact subset of D. Then, for each  ∈ N, there is a finite positive constant   where   is the smallest such number such that Since   is analytic on D and thus continuous on , we have by inequalities (42) and (43) and [16, Chapter IV, Proposition 1.17(b)] that 1 is a normal family, it follows that there exists a positive constant  2 such that sup [        ()      :  ∈ {}] ≤  2 , (43) for all  in F and any  ⊂ .