POSSIBLE INTERVALS FOR T-AND M-ORDERS OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS IN THE UNIT DISC

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 f (k) + ak−1(z)f (k−1) + · · · + a1(z)f ′ + a0(z)f = 0 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 T and M -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 T and M -orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums of T and M -orders of functions in the solution bases.


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 (1.1) where the coefficients a 0 (z), . . ., a k−1 (z) are analytic functions in the unit disc D := {z : |z| < 1} of the complex plane C. A variety of publications in the existing literature illustrates 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 [8] 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 D, it is known that T -and M -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 C. Hence, there are two distinct cases in D to work with.First, if the growth of solutions is measured by using the T -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 [12].Second, if the growth of solutions is measured by using the M -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 [4].
The main focus of this paper is in improving the lower bounds for the growth of solutions of (1.1) given in [4,12], and explore some consequences, which are motivated by the following observations.By the classical results in C 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) [6, Theorem 1].Their proof relies on classical Wiman-Valiron theory in C.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 D have been obtained only by assuming that coefficients are α-polynomial regular.These α-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 D [7,Theorem 6.1].Note that in the case of C, all solutions of (1.1) are of finite order if and only if coefficients are polynomials [15, 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 T -orders and M -orders, giving detailed information about the growth of solutions.Finally, these findings are applied to estimate the sums of T -and M -orders of functions in the solution bases of (1.1) from below.

Results and motivation
The results concerning T -and M -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 M -orders of solutions of (1.1), excluding the sketched proof of Theorem 5 in Section 7.
Let M(D) and H(D) denote the sets of all meromorphic and analytic functions in D. For simplicity, we write α

2.1.
Growth of solutions with respect to Nevanlinna characteristic.The T -order of growth of f ∈ M(D) is defined as where T (r, f ) is the Nevanlinna characteristic of f .For p > 0 and α > −1, the weighted Bergman space A p α consists of those f ∈ H(D) for which

Functions of maximal growth in
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 T -order.
The assumption a j ∈ A 1 k−j αj in Theorem A(i) cannot be replaced by a j ∈ A 1 k−j αj , see [9].We refine Theorem A, and then further underscore its consequences.
2.2.Gunderssen-Steinbart-Wang method for T -order.We proceed to state the assertions of Theorem 1 and its corollaries by using a technique introduced in [6].This yields a natural way to define possible intervals for T -orders of solutions of (1.1).As a consequence, we get a useful estimate following from Corollary 3. Set δ j := (α j + 1)(k − j) for all j = 0, . . ., k − 1.Let s 1 ∈ {0, . . ., k − 1} be the smallest index satisfying α s1 = max j=0,...,k−1 {α j } > 0, which is equivalent to If s 1 cannot be found then all solutions of (1.1) are of zero T -order by Theorem A(ii).
Otherwise, for a given s m , m ∈ N, let s m+1 ∈ {0, . . ., s m − 1} be the smallest index satisfying (2.5) Eventually this process will stop, yielding a finite list of indices s 1 , . . ., s p such that p ≤ k and where s 0 := k and δ k := 0. Due to resemblance between (2.6) and [6, Eq. (2.4)], it seems plausible that the possible non-zero T -orders of solutions of (1.1) in the unit disc case could be found among the numbers B T (t), where t = 1, . . ., p.However, Example 1 below 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 4 emphasizes the connection between B T and γ T .Lemma 4. We have (ii) β T (s t ) = B T (t) for all t ∈ {1, . . ., p}; (iii) γ T (q) = B T (1), γ T (j) = B T (t) for all s t ≤ j < s t−1 and t ∈ {2, . . ., p}, and γ T (j) ≤ 0 for all j < s p .In particular, s p = s .
By relying on Lemma 4, Theorem 1 and Corollary 2, we proceed to state possible intervals for T -orders of functions in solution bases of (1.1) in the case a j ∈ A 1 k−j αj , where α j ≥ −1 for j = 0, . . ., k − 1.In fact, each solution base of (1.1) contains For the following application, let {f 1 , . . ., f k } be a solution base of (1.1).Knowing the possible intervals for T -orders, we get (2.7) In view of Lemma 4, the lower estimates in (2.4) and (2.7) are equal.
2.3.Growth of solutions with respect to maximum modulus.Alongside of the T -order, we may also define the M -order of growth of f ∈ H(D) by σ M (f ) := lim sup where M (r, f ) := max |z|=r |f (z)| is the maximum modulus of f .It is well known that the inequalities are satisfied for all f ∈ H(D), and all possibilities allowed by (2.8) can be assumed [14, Theorems 3.5-3 p is also known as G p .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 M -order.
To conclude [4, Eq. (4.17)] in the proof of Theorem B, the inequality [4, Eq. (1.9)], corresponding to (2.9), must be strict.By a simple modification of the proof of Theorem B, the assumption (2.9) can be relaxed to which allows us to apply Theorem B(iii) also in the case that there are solutions f satisfying σ M (f ) ≤ 1.To see that (2.10) is in fact a weaker assumption than (2.9), we refer to [12, Example 10], which is further considered in Section 3.2.In this case min j=1,...,k Note that by taking j = k in (2.9), we obtain p 0 > 1. Hence (2.9) implies (2.10).Theorem 5 below corresponds to Theorem 1.
2.4.Gunderssen-Steinbart-Wang method for M -order.We proceed to state the assertions of Theorem 5 and its corollaries by using a technique introduced in [6].This yields a natural way to define the possible intervals for M -orders of solutions of (1.1).As a consequence, we get a useful estimate following from Corollary 7. Set δ j := (p j + 1)(k − j) for all j = 0, . . ., k − 1.Let s 1 ∈ {0, . . ., k − 1} be the smallest index satisfying p s1 = max j=0,...,k−1 {p j } > 1, which is equivalent to If s 1 cannot be found then all solutions f of (1.1) satisfy σ M (f ) ≤ 1 by Theorem B(ii).
Otherwise, for a given s m , m ∈ N, let s m+1 ∈ {0, . . ., s m − 1} be the smallest index satisfying (2.15) Eventually this process will stop, yielding a finite list of indices s 1 , . . ., s p such that p ≤ k and where s 0 := k and δ k := 0. By Example 1 below, it is possible that (1.1) possesses a solution f of non-zero M -order such that σ M (f ) = B M (t) for all t = 1, . . ., p.
The following lemma, which can be proved similarly than Lemma 4, allows us to view the results in Section 2.3 in a new perspective.
By relying on Lemma 8, Theorem 5 and Corollary 6, we proceed to state possible intervals for M -orders of functions in solution bases of (1.1) in the case a j ∈ H ∞ (pj +1)(k−j) , where p j ≥ −1 for j = 0, . . ., k − 1.In fact, each solution base of (1.1) contains For results of the same type, we refer to [1, Theorem 1] and [2, Corollary 1].To compare (i) and (ii) to the estimates given in [2, Corollary 1], note that there is −1 in (2.16) instead of −2 in [2, Eq. (1.3)].Evidently, assertions (i) and (ii) improve the estimates given for the M -orders of solutions in [2, Corollary 1].Moreover, by means of (2.8) we see that (i) and (ii) reduce to [2, Corollary 1], if we consider the growth of solutions of (1.1) with respect to T -order.
For the following application, let {f 1 , . . ., f k } be a solution base of (1.1).Knowing the possible intervals for M -orders, we get (2.17) Correspondingly to the case in Section 2.2, by means of Lemma 8 we see that the lower estimates in (2.14) and (2.17) are equal.
Finally, we point out a practical estimate, which is a consequence of (2.17).If s p = 0 then δ sp + s p = δ 0 − s p .If s p > 0 then (δ 0 − δ sp )/s p ≤ 2 by (2.15), and δ sp + s p ≥ δ 0 − s p .Hence, in both cases we can state that (2.18) We conclude that, if s 1 = 0 then the equalities hold in (2.18), since in this case s p = s 1 = 0. Note that, if (2.9) holds then we can conclude that s p = 0.

Sharpness discussion
3.1.Sharpness of Theorem 1.We may assume that max j=0,...,k−1 {α j } > 0, for otherwise all solutions are of zero T -order.If k = 2 then the statement of Theorem 1 is contained in Theorem A, and all the assertions are sharp [12,Examples 3 and 6].
If k = 3 then we have three different cases to consider.
It is clear that the assertion in (A1) is sharp, and so are the ones in (A2) by [12,Example 10].Moreover, [12,Example 9] shows that the assertions in (A3) are sharp for s = 0 and s = 2. Example 2 below shows the sharpness of the assertions in (A3) for s = 0, 1, 2. That is, in all cases there exists a solution for which the lower bound for the T -order of growth is attained.
If k = 3 then we have three different cases to consider.
(B1) If p 1 , p 2 ≤ p 0 then s = 0 = q, and all nontrivial solutions f of (1.1) satisfy σ M (f ) = p 0 by (2.12).(B2) If p 0 < p 1 and p 2 ≤ p 1 then in every solution base {f 1 , f 2 , f 3 } of (1.1) there are at least two solutions f 1 and f 2 such that σ M (f j ) = p 1 for both j = 1, 2, and all solutions f j satisfy In this case s = 0 or s = 1 = q.(B3) If p 0 , p 1 < p 2 then in every solution base {f 1 , f 2 , f 3 } of (1.1) there is at least one solution and all solutions f j satisfy (3.2).In this case s = 0, s = 1 or s = 2 = q.

Examples. Example 1 below shows that a unit disc counterpart of the finite set constructed by Gundersen-Steinbart-Wang does not contain growth orders of solutions of (3.3)
f + a 1 (z)f + a 0 (z)f = 0, if coefficients belong either to weighted Bergman spaces or to weighted Hardy spaces.
Example 1.Let α, β ∈ R be constants satisfying 1 < β < α < 2β −1.Then the functions are linearly independent analytic solutions of (3.3), where is the only possible interval for T -orders of solutions of (3.3).Since σ T (f 2 ) = β − 1, we conclude that the T -order of a solution does not have to be one of the endpoints.
On the other hand, it is also clear that a j ∈ H ∞ (pj +1)(2−j) , where p 0 = β and p 1 = α.We calculate that s 1 = 1, s 2 = 0, B M (1) = α and B M (2) = 2β − α.Hence [2β − α, α] is the only possible interval for M -orders of solutions of (3.3).Since σ M (f 2 ) = β, we conclude that the M -order of a solution does not have to be one of the endpoints.
The following example demonstrates the sharpness of Theorems 1 and 5 in the case that they do not reduce to known results.
Note that a j ∈ A 1 3−j αj , where α 2 = 3β − 1, α 1 = 5 2 β − 1 and α 0 = 2β − 1. Evidently σ T (f j ) = βj − 1 for j = 1, 2, 3. We deduce that there is one solution and three solutions f 1 , f 2 and f 3 such that That is, in all cases s = 0, 1, 2 there exists a solution for which the lower bound in (2.1) is attained.Further, this example is in line with Corollary 2, since all solutions f 1 , f 2 and f 3 are of strictly positive T -order, and in this case It follows that for the solution base {f 1 , f 2 , f 3 } equality holds in the first inequality in (2.4), and for the solution base {f 1 + f 3 , f 2 + f 3 , f 3 } equality holds in the last inequality in (2.4).This shows the sharpness of Corollary 3.
On the other hand, a j ∈ H ∞ (pj +1)(3−j) , where p 2 = 3β, p 1 = 5β 2 and p 0 = 2β.Evidently σ M (f j ) = βj for j = 1, 2, 3. We deduce that there is one solution and three solutions f 1 , f 2 and f 3 such that That is, in all cases s = 0, 1, 2 there exists a solution for which the lower bound in (2.11) is attained.Further, this example is in line with Corollary 6, since all solutions f 1 , f 2 and f 3 are of M -order strictly greater than 1, and in this case It follows that for the solution base {f 1 , f 2 , f 3 } equality holds in (2.14), and for the solution base {f 1 + f 3 , f 2 + f 3 , f 3 } upper bound for the sum of M -orders is attained.This shows the sharpness of Corollary 7.

Proof of Theorem 1
The following lemma on the order reduction procedure originates from C.

By Lemma D,
Moreover, by Hölder's inequality, with indices k−1 k−n and k−1 n−1 , we have The first member in the product is finite since a 0,n ∈ A αn for all n = 2, . . ., k − 1 by the assumption, and so is the second one for ε > 0 small enough since by the antithesis.Thus k n=2 I n is finite for ε > 0 small enough.To deal with the first sum in (4.3), denote Lemma D implies that for all j = 1, . . ., k − 2. Since max{σ T (g), σ T (f 0,1 )} ≤ σ < β T (1), we deduce that K j behaves like I j+1 and hence k−2 j=1 K j < ∞ for ε > 0 small enough.Moreover, by Hölder's inequality, with indices k−1 k−j−1 and k−1 j , and Lemma D we have for all j = 1, . . ., k − 2 when ε > 0 is sufficiently small.It remains to consider the double sum k−3 j=1 k−1 n=j+2 L j,n .By Hölder's inequality, with indices k−1 k−n and k−1 n−1 , we have where ω 1 (n) is defined in (4.4).The first term in the product is bounded for all ε > 0 since αn for all n = 3, . . ., k − 1 by the assumption.One more application of Hölder's inequality, with indices n−1 n−j−1 and n−1 j , together with Lemma D and the antithesis shows that also the second term in the product is bounded for ε > 0 small enough, and thus We have proved that the right-hand side of (4.3) is uniformly bounded for all r ∈ (0, 1), if ε > 0 is small enough.However, a 0,1 ∈ A α1 by the assumption, and hence the left-hand side of (4.3) diverges as r → 1 − .Contradiction follows.
If n m < k then an appropriate application of Hölder's inequality with indices k−s k−nm and k−s nm−s separates the coefficient from the solutions.The first term is finite by the assumption, and the second term can seen to be finite by another application of general form of Hölder's inequality with indices (4.6).This gives the desired contradiction, since the left-hand side of (4.5) diverges as r → 1 − and the right-hand side of (4.5) is uniformly bounded for all r ∈ (0, 1).
To complete the proof of (6.4) we argue as follows.First, we show that β T (s t ) ≥ B T (t).If s t < j ≤ s t−1 then (6.3) holds for m = t.If j > s t−1 then let m ∈ {1, . . ., t − 1} be the smallest index such that s m < j.From (6.1), (6.2) and (6. Second, we note that equality in (6.4) follows by taking j = s t−1 .
Let ε, δ ∈ (0, 1).Now [4,Lemma 4.3] for m = s + 1 implies that there exists a solution f s,1 ≡ 0 of Here m(Ω) is the Lebesque measure of the set Ω. We note that set E may not be the same at each occurrence, however, it always satisfies (7.This shows that each solution base of (1.1) contains at least k − s solutions f satisfying σ M (f ) ≥ β M (s).