Science Journal of Applied Mathematics and Statistics
Volume 3, Issue 4, August 2015, Pages: 177-183

Conversely Convergence Theorem of Fabry Gap

Naser Abbasi, Molood Gorji*

Department of Mathematics, Faculty of science, Lorestan University, Khoramabad, Islamic Republic of Iran

Email address:

(N. Abbasi)
(M. Gorji)

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


1. Introduction

The Fabry Gap theorem ([11]) 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])

(1) There is a constant  so that  for all

(2)

(3) the

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 [1]). given the sequences and  (see also [14]):

(1)

(2)

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 [1] 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

(3)

where  is a polynomial with , and

(4)

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

(5)

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

(6)

satisfies the following for every  as

(7)

and whenever

(8)

Furthermore for every  as one has:

(9)

Valiron [13] (p. 29) proved that if  is a multiplicity-sequence that satisfies the relations

(10)

the regions of convergence of the Taylor-Dirichlet series

and its two associate series

(11)

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

(12)

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 [11].

Let , ,  and  as defined in (3), (11) and

(13)

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

(14)

holds. Thus, all three series converge absolutely and uniformly in any half-plane  One also notes that from (12) we have

(15)

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

(16)

(17)

so that  and let

(18)

(19)

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

(20)

due to

Step 1:

Since is regular in the semi-strip , then for all  so that , we define

(21)

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

(22)

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

Hence

On the other hand, since one has  and therefore

(23)

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

(24)

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 .

Step 2:

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

(25)

Since  is bounded on the segments and , by standard calculations one gets

(26)

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

(27)

From the definition of  above, one deduces that

(28)

where  is the entire function defined as

(29)

Note that  is also written as

(30)

where

(31)

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

Therefore

(32)

Step 3:

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:

and

Then we write

where  is defined as

(33)

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

Thus

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

(34)

where  is defined in (31). Note that for any  there is a  so that , thus  the pseudo-multiplicity of . Since , then one gets

(35)

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

(36)

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

(37)

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

(38)

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

(39)

In particular, for real this implies that , thus

(40)

Note also that from (28) and (29), one deduces that

(41)

Then from

(42)

we can write

(43)

If we now apply (9) and (40) - (43), it yields for every

(44)

Since  is arbitrary we get that

(45)

()

But this contradicts with relation (15), and this completes the proof of our theorem.

4. Conclusion

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.

Acknowledgements

The author gratefully acknowledges the help of Prof. E. Zikkos to improve the original version of the paper.


References

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