Stability properties of ultraholomorphic classes of Roumieu-type defined by weight matrices

We characterize several stability properties, such as inverse or composition closedness, for ultraholomorphic function classes of Roumieu type defined in terms of a weight matrix. In this way we transfer and extend known results from J. Siddiqi and M. Ider, from the weight sequence setting and in sectors not wider than a half-plane, to the weight matrix framework and for sectors in the Riemann surface of the logarithm with arbitrary opening. The key argument rests on the construction, under suitable hypotheses, of characteristic functions in these classes for unrestricted sectors. As a by-product, we obtain new stability results when the growth control in these classes is expressed in terms of a weight sequence, or of a weight function in the sense of Braun–Meise–Taylor.


Introduction
When dealing with function spaces (usually called classes) it is very interesting to decide whether the usual operations (pointwise product, composition, algebraic inversion, differentiation, integration, etc.) on the functions of the space provide new functions inside it.These stability properties play a crucial role in the setting and the solution of, for example, algebraic, differential or integro-differential equations in the class.
In the literature one can frequently find the so-called ultradifferentiable classes, both in the Carleman and the Braun-Meise-Taylor sense, whose elements are smooth functions defined on open subsets of R n (or possibly germs at a point) such that the rate of growth of their successive derivatives is controlled (except for a geometric factor) in terms of a given sequence of positive real numbers in the first case, or of a given weight function in the second one.Moreover, depending on the choice of a universal or existential quantifier for the geometric factor in the estimates, one can consider Beurlingor Roumieu-like classes in both situations.The study of stability under inversion (or division) in these frameworks has a long history, see the works of Rudin [17], Bruna [3] and Siddiqi [24], and also composition has been studied in Fernández and Galbis [4].Recently, the introduction of classes associated with a weight matrix, by the fourth author of this paper [19,20], which strictly encompass those classes mentioned before, has led him and Rainer [13,14] to the characterization of stability under different operations in terms of conditions for the weight matrix under consideration, so giving a satisfactory general solution to these problems.
In connection with the asymptotic theory of solutions for differential and difference equations around singular points in the complex domain, it is natural to consider the complex analogue of such classes, usually called ultraholomorphic classes.They consist of holomorphic functions in sectorial regions in the Riemann surface of the logarithm (the singular point is assumed to be at 0, the vertex of the region) whose derivatives admit again suitable estimates of Roumieu type in terms of a sequence of positive real numbers, which in the applications is typically a Gevrey sequence (p! a ) p∈N 0 for some a > 1.The study of stability properties in such classes is well-known for the Gevrey ones, see [1], but already in 1987 Ider and Siddiqi [25] studied stability under composition with analytic functions and under inversion for general Carleman-Roumieu classes in unbounded sectors not wider than a half-plane.Our aim is to extend their results in several senses: (1) we consider Roumieu classes defined by weight matrices, so including in our considerations those of Carleman type and those defined by a weight function, as in the ultradifferentiable setting; (2) we are able to deal with classes defined in sectors of arbitrary opening in the Riemann surface of the logarithm, and (3) we extend the list of stability properties, including that of composition closedness.It is important to note that, in the case of classes given by a weight function, a fundamental role in the stability properties is played by the condition that this function is equivalent to a concave weight function, what amounts to the root almost increasing property for the associated weight matrix.
The main novelties arise from two different sources.On the one hand, the techniques coming with the weight matrix structure allow for a better understanding of the conditions usually appearing in such stability results, and provide a clear way to establish results for the weight sequence and weight function approach.Indeed, our results extend the known ones for Carleman classes, and they match, in the limit when the opening of the sector tends to 0, with the ones for ultradifferentiable classes on a half-line.On the other hand, the main statements heavily rest on the construction of so-called characteristic functions in Carleman-Roumieu ultraholomorphic classes in sectors of arbitrary opening.These functions are those in a class which cannot belong to a class strictly contained in the original one, and so are in a sense maximal within the class.While Ider and Siddiqi only got such functions in narrow sectors, the work of Rodríguez Salinas [16] provides indeed the key facts for working in general sectors, and this is in turn crucial for our purposes.
The paper is organized as follows.Section 2 contains all the preliminary information about sequences, weight functions and weight matrices.For the ultraholomorphic classes introduced in Section 3 we show how to construct characteristic functions in Section 4. The stability results for classes associated with weight matrices are given in Section 5, If M is log-convex, then M is called strongly log-convex, denoted by (slc).We say that a sequence M is a weight sequence if it is (lc) and lim j→∞ m j = ∞.We see that M is a normalized weight sequence if and only if 1 ≤ m 0 ≤ m 1 ≤ . . ., lim j→+∞ m j = +∞ (e.g.see [13, p. 104]) and there is a one-to-one correspondence between M and m by taking M j := j−1 i=0 m i .For a ∈ R we set G a := (j! a ) j∈N 0 , G a := (j ja ) j∈N 0 , i.e. for a > 0 the sequence G a is the Gevrey-sequence of index a.Clearly G a and G a are normalized weight sequences for any a > 0 (by the convention 0 0 := 1).M satisfies the condition of moderate growth, denoted by (mg), if In the classical work of Komatsu [9] this condition is named (M.2) and also known in the literature under the name stability under ultradifferential operators.M satisfies the weaker requirement of derivation closedness, denoted by (dc), if In [9] this is condition (M.2 ′ ).Both (mg) and (dc) are preserved when multiplying or dividing M by any sequence G a .In particular, both conditions hold simultaneously true or false for M and M.
We say M has the root almost increasing property, denoted by (rai), if the sequence of roots ( M 1/j j ) j∈N is almost increasing, that is, M has the Faà-di-Bruno property, denoted by (FdB), if where Let M, L ∈ R N 0 >0 be given with arbitrary M 0 , L 0 > 0, we write M L if sup j∈N (M j /L j ) 1/j < +∞ or, equivalently, if there exist A, B > 0 such that M j ≤ AB j L j for every j ∈ N 0 .We say M and L are equivalent, denoted by M ≈ L, if M L and L M. Note that, in case M 0 = L 0 = 1, equivalence amounts to B j M j ≤ L j ≤ C j M j for every j ∈ N 0 and suitable B, C > 0. Properties (mg) and (dc) are clearly preserved under ≈.
Let us write M ≤ L if M j ≤ L j for all j ∈ N 0 .
Finally, we recall some useful elementary estimates, which immediately imply that G a ≈G a for any a ∈ R.

Associated weight function
For an abstract introduction of the associated function we refer to [12, Chapitre I], see also [9,Definition 3.1].
If lim inf j→+∞ (M j ) 1/j > 0, then ω M (t) = 0 for sufficiently small t > 0, since t 0 /M 0 = 1 and ln t j M j < 0 precisely if t < (M j ) 1/j , j ∈ N (in particular, if M j ≥ 1 for all j ∈ N 0 , then ω M vanishes on [0, 1]).Moreover, under this assumption t → ω M (t) is a continuous nondecreasing function, which is convex in the variable ln(t) and tends faster to infinity than any ln(t j ), j ≥ 1, as t → +∞.If lim j→+∞ (M j ) 1/j = +∞, then ω M (t) < +∞ for each finite t, so this will be a basic assumption for defining ω M .
If M is a weight sequence, then we can compute M by involving ω M as follows, see [12,Chapitre I,1.4,1.8] and also [9,Prop. 3.2]: Moreover, in this case one has by the known integral representation formula for ω M , see [12, 1.8.III] and also [9, (3.11)].
If M ∈ R N 0 >0 satisfies lim j→+∞ (M j ) 1/j = +∞, then the right-hand side of formula (3) yields the j-th term of the log-convex minorant M lc of M, i.e. the log-convex sequence such that each log-convex sequence L with L ≤ M satisfies L ≤ M lc (moreover, M lc ≡ M if and only if M is log-convex).By the results from [12, Chapitre I] it also follows that Finally, if for β > 0 we write M 1/β := (M 1/β j ) j∈N 0 , we recall the following immediate equality, e.g.see [7, (2.7)]:
Note that m ≃ ℓ implies M≈L.
If (P γ ) holds true for M, then (P γ ′ ) also holds for any γ ′ ≤ γ.It is then natural to define the growth index γ(M) by ).For a comprehensive study of this index we refer to [6,Sect. 3], especially to the characterizing result [6,Thm. 3.11].This growth index was originally defined and considered for so-called strongly regular sequences by V. Thilliez in [26, Sect.1].
For any a > 0 we put ω a for the function given by ω a (t) := ω(t a ), i.e. composing with a so-called Gevrey weight t → t a .
We consider the following (standard) conditions, this list of properties has already been used in [20].
For any ω ∈ W 0 we define the Legendre-Fenchel-Young-conjugate of ϕ ω by are nondecreasing on [0, +∞).Note that by normalization we can extend the supremum in (4) from y ≥ 0 to y ∈ R without changing the value of ϕ * ω (x) for given x ≥ 0.
Finally, let us introduce and recall the following crucial growth assumption on ω: In the literature this condition is frequently denoted by (α 0 ).It is known that a weight function ω is equivalent to a subadditive weight function σ (i.e., σ(s + t) ≤ σ(s) + σ(t) for every s, t ≥ 0), or even to a concave weight function, if and only if (5) holds true, we refer to [22,Sect. 4.1] and the introduction of [22] with the citations therein.In [22,Thm. 4.5] this condition for ω M has been characterized in terms of M.
It is also known that (α 0 ) characterizes some desired stability properties for ultradifferentiable classes E [ω] , e.g.closedness under composition, inverse closedness and closedness under solving ODE's.The definition of such classes (which will not be used in this paper) and these results can be found in [13], [14, Thm. 1, Thm. 3] and [5,Thm. 4.8] and in the references therein (see also [4] for closedness under composition).

Weight matrices
For the following definitions and conditions see also [13,Sect. 4].
Let I = R >0 denote the index set (equipped with the natural order), a weight matrix M associated with I is a (one parameter) family of sequences We call a weight matrix M log-convex, denoted by (M lc ), if M (α) is a log-convex sequence for all α ∈ I.Moreover, we say that a weight matrix M is standard log-convex, abbreviated by (M sc ), if M (α) is a normalized weight sequence for all α ∈ I.We put If M is a weight matrix with lim j→∞ (M (α) j ) 1/j = +∞ for all α, then let us set and call M and L R-equivalent, if M{ }L and L{ }M.
Let us consider the following crucial assumptions (of Roumieu-type) on a given weight matrix M, see [13,Sect. 4.1] and [20,Sect. 7.2]: , where ( ) • is the sequence defined by (1).Moreover, let us consider k , and the weaker requirement Let us gather now some relevant information needed in the forthcoming sections.
Note that the indices α and α ′ are related by property (M {rai} ).
Proof.If j 1 , . . ., j k ≥ 1 we estimate by and the remaining cases follow by (α) : α ∈ I} be a weight matrix.Then we have the following: i.e. each sequence ((M In particular, (7) holds true (with H = 1 for any α) provided that M is log-convex.
Proof.(i) By the order of the sequences we can assume w.l.o.g.β ≥ α and for each C > 0 for some C ≥ 1 and all j ≥ 1 (see also [22,Lemma 3.6 (ii)]).Since w.l.o.g.we can restrict in the Roumieu case to all β(α) (yielding an R-equivalent matrix) we are done.

Weight matrices associated with weight functions
We summarize some facts which are shown in [13,Section 5] and are needed in this work.All properties listed below are valid for ω ∈ W 0 , except (9) for which (ω 1 ) is necessary.
(i) The idea was that to each ω ∈ W 0 we can associate a standard log-convex weight matrix so both (M {mg} ) and (M {dc} ) are satisfied.
(iv) If ω ∈ W 0 is given with associated weight matrix

Ultraholomorphic classes
We introduce now the classes under consideration in this paper, see also [7, Sect.2.5] and [8, Sect.2.5].For the following definitions, notation and more details we refer to [18,Section 2].Let R be the Riemann surface of the logarithm.We wish to work in general unbounded sectors in R with vertex at 0, but all our results will be unchanged under rotation, so we will only consider sectors bisected by direction 0: For α > 0 we set i.e. the unbounded sector of opening απ, bisected by direction 0.
Let M be a sequence, S ⊆ R an (unbounded) sector and h > 0. We define ) is a Banach space and we put A {M} (S) is called the Denjoy-Carleman ultraholomorphic class (of Roumieu type) associated with M in the sector S (it is an (LB) space).By definition it is immediate that M≈L implies A {M} (S) = A {L} (S) (as locally convex vector spaces) for any sector S.
Similarly as for the ultradifferentiable case, we now define ultraholomorphic classes associated with ω ∈ W 0 .Given an unbounded sector S, and for every ℓ > 0, we first define ) is a Banach space and we put A {ω} (S) is called the Denjoy-Carleman ultraholomorphic class (of Roumieu type) associated with ω in the sector S (it is an (LB) space).Again, equivalent weight functions provide equal associated ultraholomorphic classes.Finally, we define ultraholomorphic classes of Roumieu type defined by a weight matrix M analogously as the ultradifferentiable counterparts introduced in [20, Section 7] and also in [13,Section 4.2].
Given a weight matrix M = {M (α) ∈ R N 0 >0 : α ∈ I} and a sector S we may introduce the class A {M} (S) of Roumieu type as R-equivalent weight matrices yield (as locally convex vector spaces) the same function class on each sector S.
Let now ω ∈ W be given and let M ω be the associated weight matrix defined in Subsection 2.6, then holds as locally convex vector spaces.This equality is an easy consequence of (9) and the way the seminorms are defined in these spaces.
On the other hand, by (iii) in Subsection 2.6 we get the following: Let ω ∈ W be given and assume that ω has (ω 6 ).Then, for all sectors S we get that as locally convex vector spaces.
If f belongs to any of such classes, we may define the complex numbers

Characteristic functions in ultraholomorphic classes
We start with the following definition.
Definition 4.1.Let L ∈ R N 0 >0 and S be a given sector.A function f ∈ A {L} (S) is said to be characteristic in the class A {L} (S) if, whenever f ∈ A {M} (S) ⊆ A {L} (S) for some M ∈ R N 0 >0 , we have that A {M} (S) = A {L} (S).For f ∈ A {L} (S) we consider the sequence defined by The next statement provides conditions on f which imply it is characteristic.
, S be a given sector and f ∈ A {L} (S).Then, each of the following conditions implies the next one: and the hypothesis allows us to conclude the other estimate.
(2) ⇒ (3) By assumption, there exist A, B > 0 such that , there exist C, D > 0 such that C n (f ) ≤ CD n M n for every n ∈ N 0 .The two deduced inequalities show that L n ≤ AC(BD) n M n for every n ∈ N 0 , what easily implies that A {L} (S) ⊆ A {M} (S), and we are done.

Characteristic transform
Following again the work of Rodríguez Salinas [16], we present a functional transform that modifies the derivatives at 0 of a function in a ultraholomorphic class with a precise control, what allows for the construction of characteristic functions in more general classes than the Gevrey ones, considered previously.
Definition 4.5.Let M be an (lc) sequence, L ∈ R N 0 >0 , S a sector and f ∈ A {L} (S).Then we define the T M −transform of f by This expression should be compared with the characteristic functions obtained in the ultradifferentiable setting in [27, Thm.1] and [13, Lemma 2.9].For every j ∈ N 0 let us set The following result provides estimates for this sequence in terms of the general sequence M we depart from.
Lemma 4.6.Let M ∈ R N 0 >0 , then If M is (lc), then also and so (R j ) j∈N 0 is equivalent to M.
Proof.For any j ∈ N 0 we choose n = j in the sum and get For the converse we recall that since M is (lc) we have m 0 ≤ m 1 ≤ . . .and so see [27, Thm.1] and the detailed proof in [19, (3.1.2)].Thus for all j ∈ N 0 .
Theorem 4.7.Let M be a (lc) sequence, L ∈ R N 0 >0 and for a given sector S take f ∈ A {L} (S).Then, T M (f ) ∈ A {LM} (S) with Moreover, for any A > 0, T M : A L,A (S) → A LM,A (S) is a continuous linear operator.
Proof.By definition of A {L} (S) we have that f is bounded in S by some constant C > 0. Since M is log-convex, we have that M j ≤ m j j for all j ∈ N 0 and then Consequently, the series defining T M (f ) normally converges in the whole of S, it provides a function holomorphic in S, and differentiation and limits can be interchanged with summation.For each z ∈ S and every j ∈ N 0 we observe then that and so as desired.Suppose f ∈ A L,A (S) for some A > 0, then for all j ∈ N 0 we can estimate By Lemma 4.6 we know that R j ≤ 2M j , so T M (f ) ∈ A LM,A (S), and moreover It follows that T M : A L,A (S) → A LM,A (S) is a well-defined continuous linear operator for any A > 0.
Theorem 4.8.Let M be a (lc) sequence, L ∈ R N 0 >0 and for a given sector S take f ∈ A {L} (S).
Proof.The first assertion is clear from Lemma 4.6 and (15).The second one stems from Theorem 4.2.

Construction of characteristic functions
Given a sequence M ∈ R N 0 >0 and α > 0 we construct now, under suitable assumptions, characteristic functions in A {M} (S α ).For this we are using the basic functions from Subsection 4.1 and the characteristic transform from Subsection 4.2.Theorem 4.9.Let M ∈ R N 0 >0 and α > 0.
2. If α > 1, we assume that there exists α ′ > α such that G Proof.This follows by Theorems 4.3, 4.4, 4.7 and 4.8, and from the fact that Remark 4.10.In order to guarantee that the hypotheses in the previous theorem are satisfied, one can compute the index γ(M) and check whether it is greater than α − 1.
If this is the case, the very definition of this index implies that for any β such that γ(M) > β > α − 1 the property (P β ) (see Subsection 2.3) is satisfied, and so there exists a suitable (lc) sequence L in the desired conditions.

Stability properties for ultraholomorphic classes defined by weight matrices
The aim of this section is to generalize and extend the stability result of Ider and Siddiqi [25, Thm.1], valid for Carleman-Roumieu ultraholomorphic classes in sectors not wider than a half-plane.We give the proof in the general weight matrix setting, we get rid of the restriction on the opening of the sector (thanks to the construction of characteristic functions in arbitrary sectors), and we extend the list of stability properties.
Our main result is concerned with several stability properties which will be defined next.
Definition 5.1.Let M ∈ R N 0 >0 be a sequence and U ⊆ C be an open set.Given a compact set K ⊂ U , we define We put Moreover, given a weight matrix M = {M (p) : p > 0}, we may introduce the class H {M} (U ) as (ii) inverse-closed, if for all f ∈ A {M} (S α ) such that inf z∈Sα |f (z)| > 0, we have 1/f ∈ A {M} (S α ).
(iii) closed under composition, if for all f ∈ A {M} (S α ) and for all g ∈ H {M} (U ), where U ⊆ C is an open set containing the closure of the range of f , we have g • f ∈ A {M} (S α ).
Remark 5.3.We wish to highlight that it is important to state these definitions in a clear way.We cannot relax the condition inf z∈Sα |f (z)| > 0 in the definition of inverseclosed by considering, for example, the weaker requirement: While this is enough when working with ultradifferentiable classes on compact intervals, as done in [11], our situation is different as S α is not compact.This is easily seen by considering the function z → exp(−1/z), which belongs to the class A {G 2 } (S α ) for every α ∈ (0, 1) (as a consequence of Cauchy's integral formula for the derivatives) and never vanishes in S α .However, observe that its multiplicative inverse z → exp(1/z) is not bounded, and hence it does not belong to any of the ultraholomorphic classes under consideration.
In the same vein, the open set U in (i) and (iii) has to contain the closure of the range of f , and not just the range.This is clearly seen in the forthcoming arguments involving the function z → 1/z, whose derivatives admit global analytic bounds in closed subsets of C \ {0}, but not in the whole of it.
Our first statement will consider classes in sectors S α contained in a half-plane and defined by a weight matrix M. In this case, the matrix can be changed, without altering the class, into a new matrix M α which we define now.Definition 5.4.Let M = {M (p) : p > 0} be a weight matrix (not necessarily satisfying (M sc )).Given α > 0 we assume that lim j→+∞ (j (1−α)j M (p) j ) 1/j = ∞ for all p > 0. The matrix M α := {M (p,α) : p > 0} is defined as So, every sequence in the original matrix is termwise multiplied by the Gevreylike sequence G 1−α (recall that G 1−α ≈G 1−α ), this sequence is changed into its logconvex regularization, and finally one termwise divides by G 1−α again.It is clear that = 1 (recall the convention 0 0 := 1) for all α > 0 and p > 0, and that the map p → M (p,α) j is non-decreasing for any j ∈ N 0 fixed.So, M (p,α) ≤ M (p ′ ,α) for all 0 < p < p ′ , i.e., M α is a weight matrix according to the definition given in Subsection 2.5.However, in general M α is not log-convex.
Remark 5.5.Note that if there exist some p > 0 such that lim j→+∞ (j (1−α)j M (p) j ) 1/j = ∞, then the same is valid for all p ′ > p, thanks to the fact that the M (p) ≤ M (p ′ ) .In this situation, since we also have A {M (p) } (S α ) ⊆ A {M (p ′ ) } (S α ) and the class associated to the weight matrix M is the increasing union of such classes, in order to study stability properties in it we can restrict our attention to the case described in the previous definition.
In order to prove the aforementioned equality of the classes associated with M and M α , it is convenient to recall the following result, which provides Gorny-Cartan like inequalities for holomorphic functions in sectors.
For the converse inclusion, let us consider f ∈ A {M} (S α ).There exist some C, D ∈ R >0 and p > 0 such that n , for all n ∈ N 0 .Let us fix n ∈ N 0 and distinguish two cases: ii) If not, by the construction of the log convex minorant, there exist so-called principal indices n i for i = 1, 2 (see [12, Chapitre I] and, for a detailed discussion of the regularization process and its intricacies, [23]).So, we have ln(n Therefore, with the notation of the previous theorem, we deduce from above: Now, from the previous estimate and by applying Theorem 5.6, there exist some A, q > 0 such that We conclude that f ∈ A {M α } (S α ).
We are ready to state our first main result.
Proof.(a) ⇒ (b) First recall that by the so-called Faà-di-Bruno formula for the composition we get Let now f ∈ A {M} (S α ) be given.By Theorem 5.7 we know that the classes A {M α } (S α ) and A {M} (S α ) are equal, therefore f ∈ A {M α } (S α ).In particular, f is bounded and thus any function g which is analytic in a domain containing the (compact) closure of the range of f satisfies By applying this and the fact that f ∈ A {M α } , we estimate as follows for all n ∈ N 0 and z ∈ S α : For the estimates also note that k ≤ n and w.l.o.g.(c) ⇒ (a) We follow the ideas from [25, Thm.1] and apply the constructions from the previous section.First, recall that ) lc is log-convex for any p > 0, see (16).Let p > 0 be arbitrary but from now on fixed.According to Theorem 4.9 we put By using (12) and Lemma 4.6 we estimate as follows: for all n ∈ N 0 and z ∈ S α .This estimate shows that f p ∈ A {M α } (S α ) and, in particular when being applied to n = 0, it yields sup k ) n−k and so we get and from Lemma 4.6 Take λ > 4 (note that in [24, p. 349, line 5] there is a mistake, one should write λ > C 0 (f )M α 0 ).Then, if we put f p := λ − f p , we have that ).We write g : z → 1 λ−z , then by applying again the Faà-di-Bruno-formula to the composition g • f p ∈ A {M α } (S α ) and thanks to the fact that g (k) (z) = k! (λ−z) k+1 for all k ∈ N 0 , yields: For some C, h > 0 and some index p ′ > 0 (large) we get for all n ∈ N 0 that By (18) we see and by taking into account that n i=1 (−1) ik i = (−1) n , we deduce that for every n ∈ N 0 , Each summand in this sum is strictly positive and we focus now on the one given by the choices k j = k, k i = 0 for i = j and n = jk j = jk with j, k ∈ N. Thus By involving (19) we estimate the left-hand side of (20) as follows: The last estimate is valid since (2 − α)(j + 1) + 1 ≤ 2(j + 1) + (j + 1) = 3(j + 1) ≤ 6 j for all j ∈ N, and Γ((2 − α)(j + 1) + 1) ≤ C 1 h (2−α)(j+1) 2 j! 2−α for some C 1 , h 2 ≥ 1 and all j ≥ 1 (by the properties of the Gamma function), where we have put Consequently, by (20) we get and so (21) establishes (M {rai} ) for indices p and p ′ for all choices j, k ∈ N and so for all multiplies n = jk of j ∈ N.For the remaining cases let now n ≥ 1 such that jk < n < j(k + 1) for some j, k ∈ N.Then, by using (21) (with appearing constant H), (2) and the fact that j → (j (1−α)j M (p ′ ,α) j ) 1/j is non-decreasing for each index p ′ > 0 (by log-convexity), we estimate as follows: Summarizing, property (M {rai} ) is verified for the matrix M α between the indices p and p ′ and when choosing the constant C := He2 2−α (> H). (ii) The condition that lim j→+∞ (j (1−α)j M (p) j ) 1/j = ∞ for all p > 0 can be weakened as long as the log-convex regularization of G 1−α M (p) makes sense (for example, in case M (p) = G α−1 ).In this situation, the proof of Theorem 5.7 is still valid, Theorem 4.9 can be applied and the availability of characteristic functions (needed in the previous proof of the implication (c) =⇒ (a)) is guaranteed.A similar comment can be made regarding the next corollary.
Corollary 5.10.Let M ∈ R N 0 >0 be a sequence, and 0 < α ≤ 1 be given such that lim j→+∞ (j Then the following assertions are equivalent: (e) The sequence M (α) has the property (FdB).
Remark 5.11.We may think of the situation for the ultradifferentiable class E {M} (0, +∞), consisting of those complex-valued smooth functions on the half-line (0, +∞) subject to similar growth restrictions for their derivatives as in the ultraholomorphic case, as the limiting case when taking α = 0 in the previous result, i.e. when the sector S α "collapses" to the ray (0, +∞).Then, it turns out that we (partially) recover the main result [14, Thm.1], see also [13,Thm. 3.2].
Thanks to the construction of characteristic functions in classes defined in sectors of arbitrary opening, undertaken in Subsection 4.3, we study now the stability properties for classes defined in sectors wider than a half-plane.Theorem 5.12.Let M = {M (p) : p > 0} be a weight matrix and consider α > 1.For each p > 0, we suppose that there exists some α p > α such that G (e) The matrix M satisfies the property (M {FdB} ).
Proof.The proof of (a) ⇒ (b) ⇒ (c) is similar to the one in Theorem 5.8.(c) ⇒ (a) Although the arguments are similar to those developed in the same implication in Theorem 5.8, we consider it worthy to complete the details because now we will work with the original weight matrix (instead of M α ), and the characteristic functions are different in this framework.Let p > 0 be arbitrary but from now on fixed.There exist α p > α and L (p) log-convex such that G 1−αp M (p) ≈L (p) .Then, there n for all n ∈ N 0 .According to Theorem 4.9 we put By using ( 14), Lemma 4.6 and the above inequality we have for suitable constant B p , C, D, E > 1 and for all n ∈ N 0 and z ∈ S α .This estimate shows that f p ∈ A {M} (S α ) and, in particular, it yields sup and from Lemma 4.6, Now take λ > E and put f p := λ − f p .Thus we get f p ∈ A {M} (S α ), and moreover inf z∈Sα | f p (z)| > 0. Since A {M} (S α ) is assumed to be inverse-closed, we get that z → 1 λ−fp(z) ∈ A {M} (S α ).When writing g p : z → 1 λ−z , the dependence on p is justified because λ is clearly depending on this chosen index.By applying the Faà-di-Brunoformula to the composition g p • f p we get that for some F, h > 0 and some index p ′ > 0 (large) and for all n ∈ N 0 , Using (22) and since n i=1 (−1) ik i = (−1) n , we deduce that for every n ∈ N 0 Given j, k ∈ N, we focus on the summand for k j = k, k i = 0 for i = j and n = jk j = jk, so we get that Clearly, (λ − f p (0)) k+1 ≤ h jk+1 1 for some h 1 > 0 (large) and all k ∈ N 0 .Hence, for all j, k ∈ N we have By involving (23) we estimate the left-hand side of (24) as follows: The last inequality is a consequence of the properties of the Gamma function for a suitable constant A p > 0, and we have put A p = 2/ A p .Consequently, by (24) we get and so there exists H ≥ 1 such that Equation ( 25) establishes (M {rai} ) for indices p and p ′ for all choices j, k ∈ N and so for all multiples n = jk of j ∈ N.For the remaining cases let now n ≥ 1 such that jk < n < j(k + 1) for some j, k ∈ N.Then, by using ( 25), ( 2), the equivalence and the fact that j → (L If ω has in addition (ω 2 ), then the list of equivalences can be extended by: (e) The class A {ω} (S α ) is closed under composition.
The next lemma will be necessary for stating a similar result for wide sectors.
Then there exists a weight matrix U = {U (ℓ) : ℓ > 0}, R-equivalent to M ω , and such that for each ℓ > 0, the sequence G −s U (ℓ) is equivalent to an (lc) sequence L (ℓ) depending on s.
(ii) ω s satisfies the condition (α 0 ), i.e., it is equivalent to a concave weight function.
Then the following are equivalent: If ω has in addition (ω 2 ), then the list of equivalences can be extended by: (e) The class A {ω} (S α ) is closed under composition.
(b) By [6, Thm.2.11 (v) ⇒ (ii)], we deduce that ω s is equivalent to a concave weight function, and so (α 0 ) is satisfied by ω s .Remark 6.7.In some situations it is straightforward that all the conditions on the weight function ω in the previous result are satisfied, and so all the statements (a) through (f) are equivalent.We comment on two special cases:

q-Gevrey case
In this subsection, we will work, for q > 1, with the q-Gevrey sequence, i.e M q = (q j 2 ) j≥0 .First, thanks to the fact that the sequence M q has (lc) and (dc), and moreover M q is also (lc), we can easily prove the stability properties for the class A {Mq} (S α ).For α ∈ (0, 1], the Corollary 5.10 ensures that the class A {Mq} (S α ) is stable.On the other hand, for α > 1 and for any β > α the sequence G 1−β M q is equivalent to an (lc) sequence, because the gamma index of M q is infinity.So, the Corollary 5.16 again ensures the stability.Now, we want to study the stability properties for the class A {ω Mq } (S α ).For this purpose, let us observe that we can estimate the normalized weight function ω Mq , ω Mq (t) = sup j∈N 0 ln t j q j 2 = sup j∈N 0 (j ln(t) − j 2 ln(q)), t > 1.
Obviously, ω Mq (t) is bounded above by the supremum of x ln(t) − x 2 ln(q) when x runs over (0, ∞), which is easily obtained by elementary calculus and occurs at the point ln(t) 2 ln(q) , ln 2 (t) 4 ln(q) .

Definition 5 . 2 .
Let M = {M (p) : p > 0} be a weight matrix and α > 0. The class A {M} (S α ) is said to be: (i) holomorphically closed, if for all f ∈ A {M} (S α ) and g ∈ H(U ), where U ⊆ C is an open set containing the closure of the range of f , we have g • f ∈ A {M} (S α ).

( a )
The matrix M α satisfies the property (M {rai} ).(b) The class A {M} (S α ) is holomorphically closed.(c) The class A {M} (S α ) is inverse-closed.If M has in addition (M {C ω } ) and M α has (M {dc} ), then the list of equivalences can be extended by (d) The class A {M} (S α ) is closed under composition.
(a) ⇒ (e) This follows by (ii) in Lemma 2.3.(e) ⇒ (d) This follows by repeating the arguments in the proof of (a) ⇒ (b) above (a word-by-word repetition of the proof in the ultradifferentiable setting), see [20, Thm.8.3.1].(d) ⇒ (b) For all open set U ⊆ C, the property (M {C ω } ) of M implies that the class H(U ) is contained in H {M} (U ).Since the class A {M} (S α ) is closed under composition, it is holomorphically closed too.Remark 5.9.(i) If M has (M {dc} ) then M α has it too (the converse is not clear in general).

( a )
The sequence M (α) has the property (rai).(b)The class A {M} (S α ) is holomorphically closed.(c) The class A {M} (S α ) is inverse-closed.If lim inf j→∞ ( M j ) 1/j > 0 and the sequence M (α) is (dc), then the list of equivalences can be extended by(d) The class A {M} (S α ) is closed under composition.

1 −
αp M (p) is equivalent to a (lc) sequence L (p) depending on α p .Then the following assertions are equivalent: (a) The matrix M satisfies the property (M {rai} ).(b) The class A {M} (S α ) is holomorphically closed.(c) The class A {M} (S α ) is inverse-closed.If M has in addition (M {C ω } ) and (M {dc} ), then the list of equivalences can be extended by (d) The class A {M} (S α ) is closed under composition.
to the fact that β − 1 p+1 > 1 for large p, and we can ensure that the corresponding matrix has (M {rai} ).