Research Article | Open Access

# A Lower Bound on the Sinc Function and Its Application

**Academic Editor:**Shanhe Wu

#### 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.

#### References

- J. Kuang,
*Applied Inequalities*, Shandong Science and Technology Press, 3rd edition, 2004. - F. Stenger,
*Numerical Methods Based on Sinc and Analytic Functions*, vol. 20 of*Springer Series in Computational Mathematics*, Springer, New York, NY, USA, 1993. View at: Publisher Site | MathSciNet - J. P. Boyd,
*Chebyshev and Fourier Spectral Methods*, Dover, New York, NY, USA, 2nd edition, 2000. - D. Borwein, J. M. Borwein, and I. E. Leonard, “Lp norms and the sinc function,”
*The American Mathematical Monthly*, vol. 117, no. 6, pp. 528–539, 2010. View at: Google Scholar - D. Borwein and J. M. Borwein, “Some remarkable properties of sinc and related integrals,”
*Ramanujan Journal*, vol. 5, no. 1, pp. 73–89, 2001. View at: Publisher Site | Google Scholar | MathSciNet - W. B. Gearhart and H. S. Schultz, “The function sin(x)/x,”
*The College Mathematics Journal*, vol. 2, no. 2, pp. 90–99, 1990. View at: Google Scholar - D. S. Mitrinovic,
*Analytic Inequalities*, Springer, New York, NY, USA, 1970. View at: MathSciNet - S.-P. Zeng and Y.-S. Wu, “Some new inequalities of Jordan type for sine,”
*The Scientific World Journal*, vol. 2013, Article ID 834159, 5 pages, 2013. View at: Publisher Site | Google Scholar - R. P. Agarwal, Y. Kim, and S. K. Sen, “A new refined jordan's inequality and its application,”
*Mathematical Inequalities and Applications*, vol. 12, no. 2, pp. 255–264, 2009. View at: Google Scholar | Zentralblatt MATH - Á. Baricz, “Some inequalities involving generalized bessel functions,”
*Mathematical Inequalities and Applications*, vol. 10, no. 4, pp. 827–842, 2007. View at: Google Scholar | Zentralblatt MATH - A. Baricz, “Jordan-type inequalities for generalized Bessel functions,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 9, no. 2, article 39, 2008. View at: Google Scholar | MathSciNet - A. Baricz, “Geometric properties of generalized Bessel functions,”
*Publicationes Mathematicae Debrecen*, vol. 73, no. 1-2, pp. 155–178, 2008. View at: Google Scholar | MathSciNet - L. Debnath and C. Zhao, “New strengthened Jordan's inequality and its applications,”
*Applied Mathematics Letters*, vol. 16, no. 4, pp. 557–560, 2003. View at: Publisher Site | Google Scholar | MathSciNet - J. Li, “An identity related to Jordan's inequality,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2006, Article ID 76782, 6 pages, 2006. View at: Publisher Site | Google Scholar | MathSciNet - J. L. Li and Y. L. Li, “On the strengthened Jordan's inequality,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 74328, 9 pages, 2007. View at: Publisher Site | Google Scholar - D. Niu, Z. Huo, J. Cao, and F. Qi, “A general refinement of Jordan's inequality and a refinement of L. Yang's inequality,”
*Integral Transforms and Special Functions*, vol. 19, no. 3-4, pp. 157–164, 2008. View at: Publisher Site | Google Scholar | MathSciNet - A. Y. Özban, “A new refined form of Jordan's inequality and its applications,”
*Applied Mathematics Letters*, vol. 19, no. 2, pp. 155–160, 2006. View at: Publisher Site | Google Scholar | MathSciNet - F. Qi and Q. D. Hao, “Refinements and sharpenings of Jordan's and Kober's inequality,”
*Mathematical Inequalities & Applications*, vol. 8, no. 3, pp. 116–120, 1998. View at: Google Scholar - F. Qi, L. Cui, and S. Xu, “Some inequalities constructed by Tchebysheff's integral inequality,”
*Mathematical Inequalities and Applications*, vol. 2, no. 4, pp. 517–528, 1999. View at: Google Scholar | Zentralblatt MATH - F. Qi, “Jordan’s inequality: refinements, generalizations, applications and related problems,”
*RGMIA Research Report Collection*, vol. 9, no. 3, article 12, 2006. View at: Google Scholar - F. Qi, D.-W. Niu, and B.-N. Guo, “Refinements, generalizations, and applications of Jordan's inequality and related problems,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 271923, 52 pages, 2009. View at: Publisher Site | Google Scholar | MathSciNet - J. Sandor, “On the concavity of sinx/x,”
*Octogon Mathematical Magazine*, vol. 13, no. 1, pp. 406–407, 2005. View at: Google Scholar - S. H. Wu, “On generalizations and refinements of Jordan type inequality,”
*RGMIA Research Report Collection*, vol. 7, article 2, 2004. View at: Google Scholar - S. H. Wu, “On generalizations and refinements of Jordan type inequality,”
*Octogon Mathematical Magazine*, vol. 12, no. 1, pp. 267–272, 2004. View at: Google Scholar - S. Wu and L. Debnath, “A new generalized and sharp version of Jordan's inequality and its applications to the improvement of the Yang Le inequality,”
*Applied Mathematics Letters*, vol. 19, no. 12, pp. 1378–1384, 2006. View at: Publisher Site | Google Scholar | MathSciNet - S. Wu and L. Debnath, “A new generalized and sharp version of Jordan's inequality and its applications to the improvement of the Yang Le inequality.II,”
*Applied Mathematics Letters*, vol. 20, no. 5, pp. 532–538, 2007. View at: Publisher Site | Google Scholar | MathSciNet - S. Wu and L. Debnath, “Jordan-type inequalities for differentiable functions and their applications,”
*Applied Mathematics Letters*, vol. 21, no. 8, pp. 803–809, 2008. View at: Publisher Site | Google Scholar | MathSciNet - S. Wu and H. M. Srivastava, “A further refinement of a Jordan type inequality and its application,”
*Applied Mathematics and Computation*, vol. 197, no. 2, pp. 914–923, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Wu, H. M. Srivastava, and L. Debnath, “Some refined families of Jordan-type inequalities and their applications,”
*Integral Transforms and Special Functions*, vol. 19, no. 3-4, pp. 183–193, 2008. View at: Publisher Site | Google Scholar | MathSciNet - L. Zhu, “Sharpening Jordan’s inequality and Yang Le inequality. II,”
*Applied Mathematics Letters*, vol. 19, no. 9, pp. 990–994, 2006. View at: Publisher Site | Google Scholar | MathSciNet - L. Zhu, “Sharpening of Jordan's inequalities and its applications,”
*Mathematical Inequalities & Applications*, vol. 9, no. 1, pp. 103–106, 2006. View at: Google Scholar - L. Zhu, “A general refinement of Jordan-type inequality,”
*Computers and Mathematics with Applications*, vol. 55, no. 11, pp. 2498–2505, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Zhu, “General forms of Jordan and Yang Le inequalities,”
*Applied Mathematics Letters*, vol. 22, no. 2, pp. 236–241, 2009. View at: Publisher Site | Google Scholar | MathSciNet - L. Zhu and J. Sun, “Six new Redheffer-type inequalities for circular and hyperbolic functions,”
*Computers & Mathematics with Applications*, vol. 56, no. 2, pp. 522–529, 2008. View at: Publisher Site | Google Scholar - Y. Qiu and L. Zhu, “The best approximation of the sinc function by a polynomial of degree $n$ with the square norm,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 307892, 12 pages, 2010. View at: Publisher Site | Google Scholar | MathSciNet - R. Redheffer, “Problem 5642,”
*The American Mathematical Monthly*, vol. 75, no. 10, pp. 1125–1126, 1968. View at: Publisher Site | Google Scholar - X. Yang, “Approximations for constant $e$ and their applications,”
*Journal of Mathematical Analysis and Applications*, vol. 262, no. 2, pp. 651–659, 2001. View at: Publisher Site | Google Scholar | MathSciNet - X. Yang, “On Carleman's inequality,”
*Journal of Mathematical Analysis and Applications*, vol. 253, no. 2, pp. 691–694, 2001. View at: Publisher Site | Google Scholar | MathSciNet - M. Gyllenberg and Y. Ping, “On a conjecture by Yang,”
*Journal of Mathematical Analysis and Applications*, vol. 264, no. 2, pp. 687–690, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Hu and C. Mortici, “On the coefficients of an expansion of ${(1+1/x)}^{x}$ related to Carleman's inequality,” http://arxiv.org/abs/1401.2236. View at: Google Scholar

#### Copyright

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.