Conversely Convergence Theorem of Fabry Gap
Naser Abbasi, Molood Gorji*
Department of Mathematics, Faculty of science, Lorestan University, Khoramabad, Islamic Republic of Iran
To cite this article:
Naser Abbasi, Molood Gorji. Conversely Convergence Theorem of Fabry Gap. Science Journal of Applied Mathematics and Statistics. Vol. 3, No. 4, 2015, pp. 177-183. doi: 10.11648/j.sjams.20150304.12
Abstract: Our previous paper conducted to prove a variation of the converse of Fabry Gap theorem concerning the location of singularities of Taylor-Dirichlet series, on the boundary of convergence. In the present paper, we prove conversely convergence theorem of Fabry Gap. This is another proof of Fabry Gap theorem. This prove may be of interest in itself.
Keywords: Dirichlet Series, Entire Functions, Fabry Gap Theorem
The Fabry Gap theorem () states that if, is a real positive sequence such that for and as , then the Dirichlet series has at least one singularity in every interval of length exceeding on the abscissa of convergence.
We assume that the reader is familiar with the theory of Entire Functions and the theory of Dirichlet series, as used in the books [2,6,9–12].
We note that other results concerning the location of singularities of Taylor–Dirichlet series have been derived by Blambert, Parvatham, and Berland (see [3–5]).
2. Auxiliary Results and Notions
In this section, we describe the definitions and also to express and prove the lemma, we need to prove the theorem.
Definition 2.1. We denote by the class of all sequences with distinct complex terms diverging to infinity, satisfying the following conditions: (see also )
(1) There is a constant so that for all
Definition 2.2. Let the sequence and real positive numbers so that
We say that a sequence with real positive terms , not necessarily in an increasing order, belongs to the class if for all we have
and for all one of the following holds:
One observes that allows for the sequence to have coinciding terms. Also note that allows for non-coinciding terms to come very close to each other. We may now rewrite in the form of a multiplicity sequence , by grouping together all those terms that have the same modulus, and ordering them so that .We shall call this form of the reordering (see also ). given the sequences and (see also ):
where as usual
Observe that the disks in is not necessarily disjoint, since for fixed we might have for .
We state now lemmas that were proved in  regarding multiplicity sequences. They shall be used later on.
Lemma 2.3. Let be a real positive sequence and let so that is real positive too, with its reordering. Then the regions of convergence of the three series as defined in
where is a polynomial with , and
are the same. For any point inside the open convex region, the three series converge absolutely. Similarly, if instead of a real sequence we have a complex sequence .
Lemma 2.4. There exist positive constants and so that for any n one has
Lemma 2.5. For any one has
Lemma 2.6. Let be anyone of the following sequences: , , or where so that . Then
Theorem 2.7. (Phragmen-Lindelof.) Let be analytic in the region between two straight lines making an angle at the origin and on the lines themselves. Suppose that
on the lines and that as
where, uniformly in the angle. Then (5) holds throughout the region.
Theorem 2.8. Let be a complex sequence satisfying and as Let and let be its reordering. Then the entire function
satisfies the following for every as
Furthermore for every as one has:
Valiron  (p. 29) proved that if is a multiplicity-sequence that satisfies the relations
the regions of convergence of the Taylor-Dirichlet series
and its two associate series
wher are the same. For any point inside the open convex region, the three series converge absolutely.
3. Main Results
Theorem 3.1. Let be a real positive sequence for Let so that is real positive too and let be its reordering. Then any Taylor-Dirichlet series as in (3), satisfying
has at most one singularity in every interval of length exceeding on the abscissa of convergence.
Proof. We follow on the lines of the proof of Theorem XXIX in .
Let , , and as defined in (3), (11) and
From Lemma 2.3, the regions of convergence of the three series are the same. Since the are real positive numbers, we consider the non-trivial case, that is when the three series converge in identical half-planes of the form
With no loss of generality we assume that the abscissa of convergence (ordinary and absolute) is the line In other words the relation
holds. Thus, all three series converge absolutely and uniformly in any half-plane One also notes that from (12) we have
Suppose now that there exists an interval of length greater than on the line on which has two singularity ,.
Then with no loss of generality we can also assume that this interval is where . This implies the existence of some such that $ f(z) $ is analytic for , , , . We put
so that and let
The rest of the proof is broken into three steps. (to prove that ,can be obtained similarly.)
All three steps make use of the convergence of
Since is regular in the semi-strip , then for all so that , we define
where and the paths of the integrals are the segments joining the various points. Then we prove that converges as , and if we denote this limit by , one has
But now is well defined for all In fact it is analytic in
Next, we define where is the entire even function defined in Theorem 2.8 Then is an entire function in the complex plane.
Proof step 1:
Observe that the first two integrals in (21) are independent of , thus we deal with the other two. We will prove that (22) holds as
The absolute convergence of in the interval , justifies integrating it term by term to get the following:
Denote by the infinite series which depends on We will show that as
Since then. It follows from Lemma 2.6 that
On the other hand, since one has and therefore
From relation (14) one gets that , and this implies that the Dirichlet series converges absolutely for any if Thus, the series
is defined for all and is a positive decreasing function. Therefore there exists some so that for all one has . Combining this with (23) shows that as
Similarly one deduces that
Therefore for all with one has that the exists. If we denote this by then takes the form as in (22).
Next we prove that is well defined for all In fact, we prove that is analytic in
Note that the two integrals in (22) define analytic functions of in the whole complex plane. Thus, it remains to be proved that the infinite series converges uniformly on any compact subset such that
Consider such a compact Then there exists an so that for all one has for all . Let For all define
Then for all , it follows from Lemma 2.6 and (20) that
This implies uniform convergence on .
We prove that for some the relation holds, thus for
Proof step 2:
Let be as in (16). Then
Denote the infinite series by and note that Then from
Lemma 2.6 and (20), it follows that
Since is bounded on the segments and , by standard calculations one gets
Then by choosing the path of integration as the reflection in the real axis of that used in (21), we get that (25) and (26) hold for as well. Thus
From the definition of above, one deduces that
where is the entire function defined as
Note that is also written as
One observes that combining (7) and (27), gives for every
From (18) one also deduces that , and since is arbitrarily small this yields
Relation (16) implies that , thus for we have
We show that is a function of exponential type bounded in the angle . In particular, for real this implies that , thus . This eventually yields the relation which contradicts relation (15). So, two points can be obtained in two different, but the relationship is bound (15) both should be equal to zero, then there is either a single point, and this completes the proof of the theorem.
Proof step 3:
We will show that (32) holds in the angle . In order to do this, first we prove that is an entire function of exponential type. From (28) observe that it suffices to work with the function .
Consider some so that . For every so that where is the system defined in (2), we partition the sequence into two sets as follows:
Then we write
where is defined as
Similarly one defines .
Consider now . We remark that in this case the condition plays no role. Note that from Lemma 2.6, we deduce for any that
and observe that the series is bounded above by the one in (20). This implies that for some .
Next, we consider . We can also write it as
where is defined in (31). Note that for any there is a so that , thus the pseudo-multiplicity of . Since , then one gets
Fix some . Then from Lemma 2.4 we get
with the last inequality valid since . One also observes that since . Thus for all one has . This implies that there are constants and , so that for any and all one has
Next, observe that for any , we have
Combining this with (36) shows that is bounded above by
Then from Lemma 2.6 we get that
Note also that from Lemma 2.5 one gets
and combining this with (37) gives
with the series bounded above by the one in (20). This implies that for some , provided .
Since it follows that for some , provided . But according to
is the union of non-overlapping disks whose radius tends to zero. Since is an entire function, its maximum value over any such closed disk is obtained on the boundary. All these imply that for all . It then follows that is an entire function of exponential type. Combining this result with relation (32) and a Phragmen-Lindelof theorem 2.7, it yields
In particular, for real this implies that , thus
Note also that from (28) and (29), one deduces that
we can write
If we now apply (9) and (40) - (43), it yields for every
Since is arbitrary we get that
But this contradicts with relation (15), and this completes the proof of our theorem.
In this study we examine a variation of the converse of Fabry Gap theorem.
Polya's result shows that in some sense Fabry's result is the best possible. Perhaps the elementary and direct proof that mentioned above might be of some interest.
To do this, a sequence with a series of new build and reordering the call, using the convergence of three series obtain upper and lower bounds. And using the Phragmen-Lindelof theorem and we will achieve the desired result in this paper.
The author gratefully acknowledges the help of Prof. E. Zikkos to improve the original version of the paper.