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Journal of Mathematics
Volume 2019, Article ID 8541705, 12 pages
https://doi.org/10.1155/2019/8541705
Research Article

Families of Differential Equations in the Unit Disk

Department of Mathematics, University of Tampa, Tampa, FL 33606, USA

Correspondence should be addressed to Kari E. Fowler; ude.tu@relwofk

Received 26 July 2018; Accepted 19 December 2018; Published 20 January 2019

Academic Editor: Ming-Sheng Liu

Copyright © 2019 Kari E. Fowler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

The interaction between coefficient conditions and solution conditions for linear differential equations in the unit disk has been a topic of many investigations, including [17]. 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 disk originated with Noshiro [15].

Definition 1. A meromorphic function is called normal in if the set of functions is a normal family in , where ranges over the conformal mappings of onto itself.

Let be an open subset of and let be the set of all continuous functions from to .

Definition 2 (see [16]). A set of functions is called a normal family if each sequence in has a subsequence which converges to a function .

Let be the set of analytic functions on , and let represent the set of meromorphic functions on .

Definition 3 (see [6]). A function is called -normal for ifwhere is the spherical derivative of .

We note that meromorphic functions in Definition 3 with were classified by Noshiro as normal.

Let be an integer. In [17], Fowler and Sons showed that if a coefficient of the differential equationis 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" -normal in the unit disk , except for a factor of .

When considering normal families of analytic functions in , 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 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 (Montel’s theorem). A family in is normal if and only if is locally bounded.

Theorem 5 (see [16, Chapter VII, Lemma 2.8]). A set in is locally bounded if and only if for each compact set there is a constant such thatfor all in 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 and for an open set but not proven. Part (ii) is alluded to in the same exercise for but also not proven. We include a proof in both directions for .

Theorem 6. Let be an integer, and let be a family of functions.
(i) If the family is normal, then the familyis also normal.
(ii) If the familyis normal, then the family is also normal.

We next examine families of differential equations. Let and let be an integer. We consider the family of differential equationswith the corresponding family of coefficients and family of solutionsNote that, in the family of solutions of (6), we include all solutions corresponding to for any .

3. Main Results

The next theorem relates normal families of coefficients and solutions.

Theorem 7. (i) Let be an integer. Suppose is a family of coefficients of (6), and suppose the family of solutions is a normal family. Then, is also a normal family.
(ii) Suppose is a normal family of coefficients of (6). Then, the family of solutions is also a normal family.
(iii) Let be an integer. Suppose is a normal family of coefficients of (6) for which there exists a positive constant such that for any . Then, the family of solutions is also a normal family.

Observe that, by Theorem 6(i), we conclude in Theorem 7, parts (ii) and (iii), that 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 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 onto itself, cannot admit a constant as a limiting function.

We designate the set of normal functions of the first category by . Fowler showed in [14, Theorem 4] that if is a solution of (2), then we can eliminate the factor in [17, Theorem 1.1(i)], and it follows that the coefficient of (2) is -normal in the unit disk , except for disks of positive radius as small as we like within 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 as above, for .

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) Suppose is a meromorphic solution of (2), with coefficient meromorphic in . If , then is -normal in for .
(ii) Let be the meromorphic coefficient of (2) in satisfying , and suppose that is a meromorphic solution of (2) in such that any pole of has order and any zero of has order . Then, for any , there exists a constant such that for some and for each .

Observe that, by [17, Lemma 3.1], we conclude in Theorem 9(ii) that behaves as an -normal function would in .

The natural bound on when motivates a similar type of bound for a family of meromorphic functions. We consider normal meromorphic families in .

Let be a meromorphic function defined on . Following [16], we let be defined by if is not a pole of , and if is a pole of .

We can characterize normal meromorphic families with the following result.

Theorem 10 (see [16, Chapter VII, Theorem 3.8]). A family is normal in if and only if is locally bounded.

We extend the following property of functions in to normal meromorphic families to get a bound in line with the bound for functions in . 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 , then for any value and any positive number , there exists a positive number 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 be a normal family of meromorphic functions on and let be a compact subset of . For each positive integer there exists a constant such that for each and each .

Theorem 13. (i) Let be an integer. Suppose is a family of coefficients of (6), and suppose the set of solutions is a normal family of meromorphic functions with the property that, for the values and any positive number , there exists a positive number such that whenever and . Then, is a normal family.
(ii) Let be an integer and let be a compact subset of . Suppose is a normal family of coefficients of (6) with the property that, for any positive number , there exists a positive number such that whenever and . Further suppose that the family of solutions has the property that any pole of has order and any zero of has order . Then, for each positive integer there exists a constant such that the set of solutions satisfiesfor some , for each , and for each .

Thus, we conclude in Theorem 13(ii) that behaves as a normal family would on within the context of Theorem 12.

4. Examples

Example 14. We present an example of a family that satisfies the conditions of Theorem 13(i). Let belong to and let be a sequence of distinct complex fractions such that, for all integers , we have that and . Further suppose that there is a finite constant such that and . Then, by [18, Lemma 1], each member of the family belongs to .
We will show that is a normal family. Let be a compact subset of . Since and is thus a normal function, we get that It follows that for all there is some positive constant such that Since , there is some finite number such thatfor all . Then, by inequality (19) and since for all integers , we getfor all and all integers . Therefore, is locally bounded and by Theorem 10 is thus a normal family.
We will next show that the normal family satisfies the additional conditions of Theorem 13(i). Let . Then, since , there is a positive number such that whenever and whenever . Then, for all ,By (21) and since for all integers , it follows that whenever , for all integers .
In addition, we have that, for all ,Then, by inequality (19) and (23), and since for all integers , it follows that whenever , for all integers , where .
It then clearly follows by Theorems 9(i) and 10 that the corresponding family of coefficients of (6),is a normal family in . This verifies the conclusion in Theorem 13(i).

Example 15. For a second example of a family that satisfies the conditions of Theorem 13(i), let belong to and let be a sequence of distinct complex numbers such that and such that, for all integers , we have that . Then, by [18, Lemma 1], each member of the family belongs to .
We will show that is a normal family. Let be a compact subset of . Since and is thus a normal function, we get that It follows that for all there is some positive constant where is the smallest such number such thatSimilarly, for each , since , there exists a positive constant where is the smallest such number such thatfor all . Then, since , we get that . Thus, there exists a finite positive constant such thatfor all . Then, by inequality (29), it follows that for all and all integers . Therefore, is locally bounded and by Theorem 10 is thus a normal family.
We will next show that the normal family satisfies the additional conditions of Theorem 13(i). Let . Then, since , there is a positive number , where is the largest such number such that whenever and whenever . In addition, for each , since , there is a positive number , where is the largest such number such thatwhenever , andwhenever . Then, since , it follows that . Thus, there is a positive constant such thatfor all . Then, for all integers , it follows by inequalities (31), (32), and (33) thatwhenever , andwhenever .
Next, by Theorem 9(i), for each , is -normal in for . Let be a compact subset of . Then, for each , there is a finite positive constant where is the smallest such number such thatfor all . It also follows by Theorem 9(i) that there is a finite positive constant where is the smallest such number such thatfor all , where . Since , it follows that . Thus, there is a finite positive constant such thatfor all . It follows by inequality (38) that, for all ,for all . Thus, the family of coefficients of (6),is locally bounded and by Theorem 10 is therefore a normal family in . 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 . An example of a function in is any Schwarzian triangle function, as long as the closure of one of the triangle functions’ fundamental domains is located entirely inside (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 Theorem 6

Proof of Theorem 6(i). Suppose is a normal family. Then, is locally bounded. Let be in and let be a compact subset of . Since is open, the shortest distance between the boundary of and the boundary of is equal to some . Thus, there is some compact set such that and such that the shortest distance between the boundary of K and the boundary of equals . Then, by Theorem 5 and Cauchy’s Estimate, there is a constant such that the open cover of , , gives us that for all in and for each in . Thus, is a normal family.

Proof of Theorem 6(ii). We start with . Suppose the family is normal. Let , suppose is a compact subset of , 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 such thatfor all rectifiable curves as described above. In addition, since is compact and is a normal family, it follows that there exists a positive constant such thatfor all in and any . Since is analytic on and thus continuous on , we have by inequalities (42) and (43) and [16, Chapter IV, Proposition 1.17(b)] thatfor all in and all in . Therefore, by Theorem 5, is a normal family. It clearly follows by induction that if is a normal family, then is a normal family.

5.2. 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 is a normal family but that the family is not locally bounded. Then, there exists a compact set such thatThus, there is a sequence in such thatSince is a normal family, there is an and a subsequence , with , such that . It follows that for each integer . Since by assumption is analytic for each , since is a complete metric space, and since and are analytic,Thus, there is a positive constant such thatfor all in . Therefore, by inequalities (46) and (48),Since, as , we have that both and we have a contradiction. Therefore, the family is locally bounded and is thus a normal family.

Proof of Theorem 7(ii). As noted in [6, Section 2], for , if is a solution of (2), where the coefficient is analytic in , then by solving (2) explicitly, it follows thatfor , where is an arbitrary complex constant.
Let be a compact set. Then, since is a normal family, there is a positive constant such that , for all in and all . LetThen, we also have that there exists a positive constant such that , for all in and all . Let . Then, by inequality (51), it follows thatfor all in and all , which gives the desired result.

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 . This gives the desired result.

5.3. Proof of Theorem 9

Proof of Theorem 9(i) (let ). We start by exploring poles and zeros of in .
Suppose is a pole of of order . Then, there is a function analytic on some disk containing for which we can express for in that disk and for some constants . Using simple calculation as per [16, pp. 157-158], it follows thatif , and if .
For the differential equation , any pole of arises from zeros and poles of a solution . If is a zero of of order , then there is a disk containing and a function analytic on that disk for which and such that we can writefor in that disk. Through simple calculation, we see that is a pole of of order . If is a pole of of order , then there is a disk containing and a function analytic on that disk for which and such that we can write for in that disk. Through calculation, it follows that is a pole of of order .
Thus, for , if is a zero or pole of , thenby (56), and soWe next consider . We get thatIf is a zero of of order , then for all on some disk containing we get thatandIt follows by inequality (62) and (63) and (64) thatThen, by inequality (65), we get thatfor . It follows thatIf is a pole of of order , then for all on some disk containing we get thatandSo by inequality (62) and (68) and (69), we have thatThus, by inequality (70) it follows thatfor . And thusTowards a contradiction, suppose that there is a sequence such that is not a pole or zero of for all and a corresponding set of poles and/or zeros of , , such thatandThen, by (73) and (74), This contradicts the bound of 4 on for all poles and zeros of .
Next let , and define andThen, by [14, Theorem 4(i)] there exists a constant such that the coefficient of (2) satisfiesfor each , for . Then, by (61) and inequalities (67), (72), and (78), there must be some such thatLet . It follows that for all . Therefore, is -normal in , for .

Proof of Theorem 9(ii) (let ). Poles of a coefficient arise from zeros and poles of a solution . Suppose is a zero of of order . Then, there is a disk containing on whichwhere is analytic on that disk and there. We further get thatwhere is also analytic in that disk and . Then, by (81) and (82) we getfor in . It follows by (83) thatif .
Next, suppose is a pole of of order . Then, there is a disk containing on whichwhere is analytic on that disk and there. We further get thatwhere is also analytic in that disk and . Then, by (85) and (86) we getfor in . It follows by (87) thatif .
By [14, Theorem 4(ii)], there exists a constant such thatfor each . Then, by (84) and (88) and inequality (89), there must be some such that