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Abstract and Applied Analysis
Volume 2011, Article ID 928194, 25 pages
http://dx.doi.org/10.1155/2011/928194
Research Article

Possible Intervals for - and -Orders of Solutions of Linear Differential Equations in the Unit Disc

1Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22 Santiago, 6904411 Macual Santiago, Chile
2Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Received 19 May 2011; Accepted 5 July 2011

Academic Editor: Jean Michel Combes

Copyright © 2011 Martin Chuaqui et al. 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

In the case of the complex plane, it is known that there exists a finite set of rational numbers containing all possible growth orders of solutions of with polynomial coefficients. In the present paper, it is shown by an example that a unit disc counterpart of such finite set does not contain all possible - and -orders of solutions, with respect to Nevanlinna characteristic and maximum modulus, if the coefficients are analytic functions belonging either to weighted Bergman spaces or to weighted Hardy spaces. In contrast to a finite set, possible intervals for - and -orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums of - and -orders of functions in the solution bases.

1. Introduction

This research is a continuation of recent activity in the field of complex differential equations. In particular, the present paper concerns linear differential equations of the type where the coefficients are analytic functions in the unit disc of the complex plane . A variety of publications in the existing literature illustrate that the connection between the growth of coefficient functions and the growth of solutions is relatively well understood. On the one hand, the growth estimates in [1] have been proven to be instrumental tools in estimating the growth of solutions when the growth of coefficients is known. On the other hand, proofs of the converse direction have taken advantage of the method of order reduction as well as different types of logarithmic derivative estimates.

For an analytic function in , it is known that - and -orders of growth, with respect to Nevanlinna characteristic and maximum modulus, are not equal in general. This is in contrast to the corresponding case in . Hence, there are two distinct cases in to work with. First, if the growth of solutions is measured by using the -order, then it is natural to express the other growth aspects by means of integration as well. In particular, it is reasonable to consider coefficient functions belonging to some weighted Bergman spaces and use integrated estimates for logarithmic derivatives [2]. Second, if the growth of solutions is measured by using the -order, then it is natural to express the other growth aspects by means of the maximum modulus function. In particular, it is sensible to restrict the growth of the maximum modulus of coefficient functions, which leads to weighted Hardy spaces, and work with estimates for the maximum modulus of logarithmic derivatives involving exceptional sets [3].

The main focus of this paper is in improving the lower bounds for the growth of solutions of (1.1) given in [2, 3] and explore some consequences, which are motivated by the following observations.

By the classical results in making use of Newton-Puiseux diagram, there is a finite set containing the possible growth orders of solutions of (1.1) assuming that coefficients are polynomials. In particular, Gundersen-Steinbart-Wang showed that this finite set consists of rational numbers obtained from simple arithmetic with the degrees of the polynomial coefficients in (1.1) [4, Theorem  1]. Their proof relies on classical Wiman-Valiron theory in . Even though a recent unit disc counterpart of Wiman-Valiron theory [5] has been successfully applied to differential equations, the possible orders of solutions of (1.1) in have been obtained only by assuming that coefficients are -polynomial regular. These -polynomial regular functions have similar growth properties than polynomials in the sense that maximal growth is attained in every direction. However, they appear to be only a relatively small subset of the Korenblum space, which characterizes finite-order solutions of (1.1) in [6, Theorem  6.1]. Note that in the case of , all solutions of (1.1) are of finite order if and only if coefficients are polynomials [7, Satz  1].

In the present paper, it is shown by an example that a unit disc counterpart of the finite set constructed by Gundersen-Steinbart-Wang does not contain all possible orders of solutions of (1.1), provided that the coefficients belong either to weighted Bergman spaces or to weighted Hardy spaces. In contrast to a finite set, we introduce possible intervals for -orders and -orders, giving detailed information about the growth of solutions. Finally, these findings are applied to estimate the sums of - and -orders of functions in the solution bases of (1.1) from below.

2. Results and Motivation

The results concerning - and -orders of solutions of (1.1) are given, respectively, in Sections 2.1-2.2 and 2.3-2.4. Due to the similarities of the assertions, we omit the proofs of results regarding -orders of solutions of (1.1), excluding the sketched proof of Theorem 2.5 in Section 7.

Let and denote the sets of all meromorphic and analytic functions in . For simplicity, we write for any , if there exists a constant independent of such that , and if there exist constants and independent of such that .

2.1. Growth of Solutions with Respect to Nevanlinna Characteristic

The -order of growth of is defined as where is the Nevanlinna characteristic of . For and , the weighted Bergman space consists of those for which Functions of maximal growth in are distinguished by denoting , if .

If the growth of the coefficients is expressed by means of integration, then it is natural to consider the growth of solutions of (1.1) with respect to -order.

Theorem A (see [2, Theorems  1 and  2]). Suppose that , where for , and denote . (i)Let . Then all solutions of (1.1) satisfy if and only if .(ii)All nontrivial solutions of (1.1) satisfy (iii)If is the smallest index for which , then each solution base of (1.1) contains at least linearly independent solutions such that .

The assumption in Theorem A(i) cannot be replaced by ; see [8]. We refine Theorem A and then further underscore its consequences.

Theorem 2.1. Suppose that , where for , and let be the smallest index for which . If , then each solution base of (1.1) contains at least linearly independent solutions such that where .

The case clearly reduces to Theorem A(ii). If , then the condition implies that where the minimum is attained for . Hence the assertion of Theorem 2.1 for is contained in Theorem A(iii). Our contribution is to extend the first inequality in (2.4) for . Theorem 2.1 is proved in Section 4, and the sharpness and the special cases and are further discussed in Section 3.1.

Let be the smallest index for which . If , then all solutions in each solution base of (1.1) are of zero -order by Theorem A(ii). Suppose that . In order to state the following corollaries of Theorem 2.1, we denote where . Moreover, we define Remark that , since (2.5) implies .

Corollary 2.2. Suppose that , where for , and let be the smallest index for which . Then each solution base of (1.1) admits at most solutions satisfying . In particular, there are at most solutions satisfying .

To estimate the quantity by using Theorem 2.1, we set Evidently for , and for .

Corollary 2.3. Suppose that , where for , and let be the smallest index for which . Let be a solution base of (1.1). If , then , while if , then

Note that the sum in (2.9) is considered to be empty, if .

2.2. Gundersen-Steinbart-Wang Method for -Order

We proceed to give an alternative statement of Theorem 2.1 and its corollaries by modifying the key steps in [4]. This yields a natural way to define possible intervals for -orders of solutions of (1.1). As a consequence, we get a useful estimate following from Corollary 2.3.

Set for all . Let be the smallest index satisfying , which is equivalent to If cannot be found, then all solutions of (1.1) are of zero -order by Theorem A(ii). Otherwise, for a given , , let be the smallest index satisfying Eventually this process will stop, yielding a finite list of indices such that and . Further, set where and . Due to resemblance between (2.12) and [4, Equation (2.4)], it seems plausible that the possible nonzero -orders of solutions of (1.1) in the unit disc case could be found among the numbers , where . However, Example 3.1 shows that this is not the case.

The following lemma allows us to view the results in Section 2.1 in a new perspective. In particular, Lemma 2.4 emphasizes the connection between and .

Lemma 2.4. One has the following: (i) ; (ii) for all ; (iii) , for all and , and for all . In particular, .

By relying on Lemma 2.4, Theorem 2.1, and Corollary 2.2, we proceed to state possible intervals for -orders of functions in solution bases of (1.1) in the case , where for . In fact, each solution base of (1.1) contains the following: (i)at least solutions satisfying ; (ii)at least solutions satisfying for ; (iii)at most solutions satisfying . For the following application, let be a solution base of (1.1). Knowing the possible intervals for -orders, we get In view of Lemma 2.4, the lower estimates in (2.9) and (2.13) are equal.

Finally, we point out a useful consequence of (2.13). If , then . If , then by (2.11), and . Hence, in both cases we can state that where the equalities hold, if .

2.3. Growth of Solutions with Respect to Maximum Modulus

Alongside of the -order, we may also define the -order of growth of by where is the maximum modulus of . It is well known that the inequalities are satisfied for all , and all possibilities allowed by (2.16) can be assumed [9, Theorems  3.5–3.7]. A function is said to belong to the weighted Hardy space , if there exists such that Functions of maximal growth in are distinguished by denoting , if . Remark that is the space of all bounded analytic functions in . The union is also known as the Korenblum space [10], and since [11] is also known as .

If the growth of coefficients is measured by means of maximum modulus estimates, then it is natural to consider the growth of solutions with respect to -order.

Theorem B (see [3, Theorem  1.4]). Suppose that , where for , and denote . (i)Suppose that and let . Then all solutions of (1.1) satisfy if and only if .(ii)All nontrivial solutions of (1.1) satisfy , and (iii)Suppose that (2.18) holds. If is the smallest index for which , then each solution base of (1.1) contains at least linearly independent solutions such that .

To conclude [3, Equation  (4.17)] in the proof of Theorem B, the inequality [3, Equation  (1.9)], corresponding to (2.18), must be strict. By a simple modification of the proof of Theorem B, the assumption (2.18) can be relaxed to which allows us to apply Theorem B(iii) also in the case that there are solutions satisfying . To see that (2.20) is in fact a weaker assumption than (2.18), we refer to [2, Example  10], which is further considered in Section 3.2. In this case Note that by taking in (2.18), we obtain . Hence (2.18) implies (2.20).

Theorem 2.5 corresponds to Theorem 2.1.

Theorem 2.5. Suppose that , where for , and let be the smallest index for which . If , then each solution base of (1.1) contains at least linearly independent solutions such that

Note that (2.22) gives information on only in the case when the minimum in (2.22) is strictly greater than 1. The case in Theorem 2.5 reduces to Theorem B(ii), and the case reduces to Theorem B(iii) with the assumption (2.20), since now where the minimum is attained for . For a similar argumentation, see (2.5). Our contribution is to extend (2.22) for . The proof of Theorem 2.5 is sketched in Section 7, and the sharpness and the the special cases and are further discussed in Section 3.2.

Let be the smallest index for which . If , then all solutions in each solution base of (1.1) satisfy by Theorem B(ii). Suppose that . In order to state the following corollaries of Theorem 2.5, we denote where . Moreover, we define Remark that , since (2.23) implies .

Corollary 2.6. Suppose that , where for , and let be the smallest index for which . Then each solution base of (1.1) admits at most solutions satisfying . In particular, there are at most solutions satisfying .

To estimate the quantity by using Theorem 2.5, we set Evidently for , and for .

Corollary 2.7. Suppose that , where for , and let be the smallest index for which . Let be a solution base of (1.1). If , then , while if , then

Note that the sum in (2.27) is considered to be empty, if .

2.4. Gundersen-Steinbart-Wang Method for -Order

We proceed to give an alternative statement of Theorem 2.5 and its corollaries by modifying the key steps in [4]. This yields a natural way to define the possible intervals for -orders of solutions of (1.1). As a consequence, we get a useful estimate following from Corollary 2.7.

Set for all . Let be the smallest index satisfying , which is equivalent to If cannot be found, then all solutions of (1.1) satisfy by Theorem B(ii). Otherwise, for a given , , let be the smallest index satisfying Eventually this process will stop, yielding a finite list of indices such that and . Further, set where and . By Example 3.1, it is possible that (1.1) possesses a solution of -order strictly greater than one such that for all .

The following lemma, which can be proved similarly than Lemma 2.4, allows us to view the results in Section 2.3 in a new perspective.

Lemma 2.8. One has the following: (i) ; (ii) for all ; (iii) , for all and , and for all . In particular, .

By relying on Lemma 2.8, Theorem 2.5, and Corollary 2.6, we proceed to state possible intervals for -orders of functions in solution bases of (1.1) in the case , where for . In fact, each solution base of (1.1) contains the following: (i)at least solutions satisfying ; (ii)at least solutions satisfying for ; (iii)at most solutions satisfying .

For results of the same type, we refer to [12, Theorem  1] and [13,  Corollary   1]. To compare (i) and (ii) to the estimates given in [13, Corollary  1], note that there is −1 in (2.30) instead of −2 in [13, Equation  (1.3)]. Evidently, assertions (i) and (ii) improve the estimates given for the -orders of solutions in [13, Corollary    1]. Moreover, by means of (2.16) we see that (i) and (ii) reduce to [13, Corollary    1], if we consider the growth of solutions of (1.1) with respect to -order.

For the following application, let be a solution base of (1.1). Knowing the possible intervals for -orders, we get Corresponding to the case in Section 2.2, by means of Lemma 2.8 we see that the lower estimates in (2.27) and (2.31) are equal.

Finally, we point out a practical estimate, which is a consequence of (2.31). If , then . If , then by (2.29), and . Hence, in both cases we can state that We conclude that if , then the equalities hold in (2.32), since in this case . Note that if (2.18) holds, then we can conclude that .

3. Sharpness Discussion

3.1. Sharpness of Theorem 2.1

We may assume that , for otherwise all solutions are of zero -order. If , then the statement of Theorem 2.1 is contained in Theorem A, and all the assertions are sharp [2, Examples  3 and 6].

If , then we have three different cases to consider. (A1) If , then all nontrivial solutions of (1.1) satisfy . In this case .(A2) If and , then in every solution base of (1.1) there are at least two solutions and such that for both , and all solutions satisfy In this case or .(A3) If , then in every solution base of (1.1) there is at least one solution such that , two solutions and such that and all solutions satisfy (3.1). In this case , , or .

It is clear that the assertion in (A1) is sharp, and so are the ones in (A2) by [2, Example   10]. Moreover, [2,  Example   9] shows that the assertions in (A3) are sharp for and . Example 3.2 shows the sharpness of the assertions in (A3) for . That is, in all cases there exists a solution for which the lower bound for the -order of growth is attained.

3.2. Sharpness of Theorem 2.5

We may assume that , for otherwise all solutions of (1.1) satisfy , and we cannot conclude anything from (2.22). If , then the statement of Theorem 2.5 is contained in Theorem B, and all the assertions are sharp by [2, Examples  3 and 6]. In the case of [2, Examples  3 and 6], for , linearly independent solutions and satisfy and . Moreover, , where and . Note that , and hence . An easy computation shows the sharpness for and for . In the case of [2, Example  6], for , linearly independent solutions and satisfy and . Moreover, , where and . Note now that , and hence . This example shows the sharpness for . For another example, see [3, Example  2].

If , then we have three different cases to consider. (B1) If , then , and all nontrivial solutions of (1.1) satisfy by (2.23).(B2) If , and , then in every solution base of (1.1) there are at least two solutions and such that for both , and all solutions satisfy In this case or .(B3)If , then in every solution base of (1.1) there is at least one solution such that , two solutions and such that and all solutions satisfy (3.3). In this case , , or .

It is clear that the assertion in (B1) is sharp. By [2, Example  10], we see that the assertion in (B2) corresponding to is sharp. In this case, for , linearly independent solutions , and satisfy and . Now , where , , and . Moreover, by [2, Example   9], we see that the assertions in (B3) are sharp for and . In this case for , linearly independent solutions , , and satisfy and . Moreover, , where , , and . Example 3.2 shows the sharpness of the assertions in (B3) for . That is, in all cases there exists a solution for which equality holds in (2.22).

3.3. Examples

Example 3.1 shows that a unit disc counterpart of the finite set constructed by Gundersen-Steinbart-Wang does not contain growth orders of solutions of if coefficients belong either to weighted Bergman spaces or to weighted Hardy spaces.

Example 3.1. Let be constants satisfying . Then the functions are linearly independent analytic solutions of (3.5), where belong to , and .
It is clear that , where and . We calculate that , , , and . Hence is the only possible interval for -orders of solutions of (3.5). Since , we conclude that the -order of a solution does not have to be one of the endpoints.
On the other hand, it is also clear that , where and . We calculate that , , , and . Hence is the only possible interval for -orders of solutions of (3.5). Since , we conclude that the -order of a solution does not have to be one of the endpoints.
The following example demonstrates the sharpness of Theorems 2.1 and 2.5 in the case that they do not reduce to known results.

Example 3.2   2. Let , and denote . Then the functions are linearly independent solutions of , where are such that The zeros of lie in the open disc of radius centered at the origin by [14, Lemma  1.3.2]. Since for all , we conclude that . In fact, the coefficients satisfy in a neighborhood of , while they are bounded in a neighborhood of any boundary point in .
Note that , where , , and . Evidently for . We deduce that there is one solution such that , two solutions and such that and three solutions , , and such that That is, in all cases there exists a solution for which the lower bound in (2.4) is attained. Further, this example is in line with Corollary 2.2, since all solutions , , and are of strictly positive -order, and in this case .
Now , , and . It follows that for the solution base equality holds in the first inequality in (2.9), and for the solution base equality holds in the last inequality in (2.9). This shows the sharpness of Corollary 2.3.
On the other hand, , where , , and . Evidently for . We deduce that there is one solution such that , two solutions and such that and three solutions , , and such that That is, in all cases there exists a solution for which the lower bound in (2.22) is attained. Further, this example is in line with Corollary 2.6, since all solutions , , and are of -order strictly greater than 1, and in this case .
Now , , and . It follows that for the solution base equality holds in (2.27), and for the solution base upper bound for the sum of -orders is attained. This shows the sharpness of Corollary 2.7.

4. Proof of Theorem 2.1

The following lemma on the order reduction procedure originates from .

Lemma C (see [4, Lemma  6.4]). Let be linearly independent meromorphic solutions of where are meromorphic functions in . For , set Then are linearly independent meromorphic solutions of where for . Here for all .

Lemma D (see [15, Theorem  E(b)]). Let and be integers satisfying , and let . If is meromorphic in such that , and , then

4.1. Case

Let , , , and , since otherwise there is nothing to prove. In particular, if , then (2.4) is trivial, since by taking in (2.6), we obtain . Let be a solution base of (1.1), and assume on the contrary to the assertion that there exist linearly independent solutions and such that . Then the meromorphic function satisfies . Moreover, Lemma C implies that satisfies where for , and . Therefore where since satisfies (4.6). Putting the last two inequalities together, we obtain Let . Raising both sides to the power and integrating over the disc of radius with respect to , we obtain

To deal with the second sum in (4.11), consider By Lemma D, for being small enough since . Moreover, by Hölder’s inequality, with indices and , we have for all , where The first member in the product is finite since for all by the assumption, and so is the second one for small enough since by the antithesis. Thus is finite for being small enough.

To deal with the first sum in (4.11), denote Lemma D implies that for being small enough since . Moreover, by (4.7) we have for all . Since , we deduce that behaves like and hence for being small enough. Moreover, by Hölder’s inequality, with indices and , and Lemma D we have for all when is sufficiently small. It remains to consider the double sum . By Hölder’s inequality, with indices and , we have where is defined in (4.15). The first term in the product is bounded for all since for all by the assumption. One more application of Hölder's inequality, with indices and , together with Lemma D and the antithesis shows that also the second term in the product is bounded for being small enough, and thus for being small enough.

We have proved that the right-hand side of (4.11) is uniformly bounded for all , if is small enough. However, by the assumption, and hence the left-hand side of (4.11) diverges as . Contradiction follows.

4.2. Case

Let , , , and , since otherwise there is nothing to prove. In particular, it follows that . Let be a solution base of (1.1), and assume on the contrary to the assertion that there exist linearly independent solutions such that Then the meromorphic functions satisfy for all . This in turn implies that satisfy for all . In general, for all and . Moreover, as in the case , Lemma C implies where Putting these inequalities together, we obtain Let . Raising both sides to the power and integrating over the disc with respect to