The Scientific World Journal

Volume 2014 (2014), Article ID 571218, 4 pages

http://dx.doi.org/10.1155/2014/571218

Research Article

A Lower Bound on the Sinc Function and Its Application

Yue Hu¹ and Cristinel Mortici2,3

¹School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China

²Valahia University of Târgovişte, 18 Unirii Boulevard, 130082 Târgovişte, Romania

³Academy of Romanian Scientists, Splaiul Independenţei 54, 050094 Bucharest, Romania

Received 12 March 2014; Revised 24 June 2014; Accepted 26 June 2014; Published 8 July 2014

Academic Editor: Shanhe Wu

Copyright © 2014 Yue Hu and Cristinel Mortici. 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.

A lower bound on the sinc function is given. Application for the sequence which related to Carleman inequality is given as well.

1. Introduction

The sinc function is defined to be

This function plays a key role in many areas of mathematics and its applications [1–6].

The following result that provides a lower bound for the sinc is well known as Jordan inequality [7]:

where equality holds if and only if .

This inequality has been further refined by many authors in the past few years [8–35].

In [36], it was proposed that

We noticed that the lower bound in (3) is the fractional function. Similar result has been reported as follows [1]:

To the best of the authors’ knowledge, few results have been obtained on fractional lower bound for the sinc function. It is the first aim of the present paper to establish the following fractional lower bound for the sinc function.

Theorem 1. For any , one has

In [37], Yang proved that for any positive integer , the following Carleman type inequality holds:

whenever , , with , where

From a mathematical point of view, the sequence has very interesting properties. Yang [38] and Gyllenberg and Ping [39] have proved that, for any positive integer ,

In [40], the authors proved that

where

As an application of Theorem 1, it is the second aim of the present paper to give a better upper bound on the sequence .

Theorem 2. For any positive integer , one has

2. The Proof of Theorem 1

The proof is not based on (3). We first prove the following result.

Lemma 3. For any , one has

Proof. Set , . Then inequality (13) is equivalent to

To prove (14) by (4), it is enough to prove that

namely,

Next we prove (16). Let

We need only to prove that . Elementary calculations reveal that

Noting that, for , we have

Thus, from (19) and (18), we get

This completes the proof. Now we prove Theorem 1.

Proof. By using the power series expansions of and , we find that

where

Set , . Consider the function defined by

From (21), we get and . Lemma 3 implies

where

Elementary calculations reveal that for ,

Hence, for , we have

Therefore,

If we set

then we have

The intermediate value theorem implies that there must be at least one root with such that . Using Maple, we find that on the open interval the equation has a unique real root .

Hence, from (28) we get

By (21), (24), and (31), Theorem 1 follows.

3. The Proof of Theorem 2

First, we need an auxiliary result.

Lemma 4. For any , one has

Proof. By letting , , the requested inequality can be equivalently written as

so it suffices to show that the function

is negative on . Theorem 1 implies

Hence,

The required inequality follows. Now we prove Theorem 2.

Proof. Let

We first consider the case .

Taking the natural log gives

Taking the second derivative of both sides of (38), we have

By Lemma 4, it follows that

Thus,

and therefore for , we have

For the case , since , , and is concave up, it follows that

Using (10) from (42) and (43), we have

This proves Theorem 2.

Conflict of Interests

There is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the anonymous referees and the editor for their insightful comments and suggestions. The work of the second author was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI project no. PN-II-ID-PCE-2011-3-0087.

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