TSWJ The Scientific World Journal 1537-744X 2356-6140 Hindawi Publishing Corporation 10.1155/2014/571218 571218 Research Article A Lower Bound on the Sinc Function and Its Application http://orcid.org/0000-0003-2776-6682 Hu Yue 1 http://orcid.org/0000-0001-5576-1921 Mortici Cristinel 2, 3 Wu Shanhe 1 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000 China hpu.edu.cn 2 Valahia University of Târgovişte, 18 Unirii Boulevard, 130082 Târgovişte Romania valahia.ro 3 Academy of Romanian Scientists, Splaiul Independenţei 54, 050094 Bucharest Romania aos.ro 2014 8 7 2014 2014 12 03 2014 24 06 2014 26 06 2014 8 7 2014 2014 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.

A lower bound on the sinc function is given. Application for the sequence bnn=1 which related to Carleman inequality is given as well.

1. Introduction

The sinc function is defined to be (1)sinc(x)={sin(x)xx0,1x=0.

This function plays a key role in many areas of mathematics and its applications .

The following result that provides a lower bound for the sinc is well known as Jordan inequality : (2)sinc(x)2π,x[0,π2], where equality holds if and only if x=π/2.

This inequality has been further refined by many authors in the past few years .

In , it was proposed that (3)sinc(x)π2-x2π2+x2,x0.

We noticed that the lower bound in (3) is the fractional function. Similar result has been reported as follows : (4)sinc(x)5353+9x2,0x13.

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 x[0,π], one has (5)sinc(x)16π4(3π2+x2)2-1.

In , Yang proved that for any positive integer m, the following Carleman type inequality holds: (6)n=1(a1a2an)1/n<en=1(1-k=1mbk(n+1)k)an, whenever an0, n=1,2,3,, with 0<n=1an<, where (7)b0=1,bn=1n(1n+1-k=0n-2bn-1-kk+1),(n=1,2,).

From a mathematical point of view, the sequence {bn}n=1 has very interesting properties. Yang  and Gyllenberg and Ping  have proved that, for any positive integer n, (8)bn>0,bn<1n(n+1).

In , the authors proved that (9)limnbn+1bn=1,(10)ebn=01xn-2h(x)dx,n2, where (11)h(x)=xx+1(1-x)1-xsinc(πx).

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

Theorem 2.

For any positive integer n2, one has (12)ebn<1n(n+1)-2-4/πn(n+1)(n+2).

2. The Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>

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

Lemma 3.

For any x(π-1/3,π], one has (13)sinc(x)16π4(3π2+x2)2-1.

Proof.

Set x=π-t, 0t<1/3. Then inequality (13) is equivalent to (14)π-t+sint16π4(π-t)(3π2+(π-t)2)2. To prove (14) by (4), it is enough to prove that (15)(π-t)+53t53+9t216π4(π-t)(3π2+(π-t)2)2; namely, (16)(π-t)(53+9t2)+53t(53+9t2)16π4(π-t)(3π2+(π-t)2)2. Next we prove (16). Let (17)g(t)=(π-t)(53+9t2)(3π2+(π-t)2)2+53t(3π2+(π-t)2)2-16π4(π-t)(53+9t2). We need only to prove that g(t)0. Elementary calculations reveal that (18)g(t)=t2(-9t5+45πt4-144π2t3+(53π+252π3)t2-4π2(36π2+53)t+636π3). Noting that, for 0t<1/3, we have (19)-9t5>-127,-144π2t3>-16π23,-4π2(36π2+53)t>-43π2(36π2+53). Thus, from (19) and (18), we get (20)g(t)t2(636π3-127-16π23-4π2(36π2+53)3)0. This completes the proof. Now we prove Theorem 1.

Proof.

By using the power series expansions of sin(x) and 16π4/(3π2+x2)2, we find that (21)1+sinc(x)-16π4(3π2+x2)2=29+n=1(-1)n-1un(x2π2)n, where (22)un=16(n+1)3n+2-π2n(2n+1)!. Set x2/π2=t, 0t1. Consider the function f(t) defined by (23)f(t)=29+n=1(-1)n-1untn. From (21), we get f(0)=2/9 and f(1)=0. Lemma 3 implies (24)f(t)0,t0<t1, where (25)t0=(1-13π)20.79. Elementary calculations reveal that for n4, (26)16(2n+1)33>(3π2)n(2n+1)!. Hence, for n4, we have (27)un>0,un-un+1=16(2n+1)3n+3-π2n(2n+1)!+π2n+2(2n+3)!>0. Therefore, (28)f(t)29+n=16(-1)n-1untn. If we set (29)g(t)=29+n=16(-1)n-1untn, then we have (30)g(0)=29>0,g(1)<f(1)=0. The intermediate value theorem implies that there must be at least one root c with (0,1) such that g(c)=0. Using Maple, we find that on the open interval (0,1) the equation g(t)=0 has a unique real root t10.89.

Hence, from (28) we get (31)f(t)0,t[0,t1]. By (21), (24), and (31), Theorem 1 follows.

3. The Proof of Theorem <xref ref-type="statement" rid="thm1.2">2</xref>

First, we need an auxiliary result.

Lemma 4.

For any x[0,1/2], one has (32)sinc2(πx)1-2x+x21-x+x2.

Proof.

By letting x=1/2-t/2π, 0tπ, the requested inequality can be equivalently written as (33)costt22+8π4t2+3π2-5π2+22, so it suffices to show that the function (34)G(t)=cost-t22-8π4t2+3π2+5π2+22 is negative on 0tπ. Theorem 1 implies (35)G(t)<0. Hence, (36)G(t)G(π)=0. The required inequality follows. Now we prove Theorem 2.

Proof.

Let (37)H(x)={xx(1-x)-xsinc(πx),0<x<11x=0,1. We first consider the case 0x1/2.

Taking the natural log gives (38)lnH(x)=(x-1)lnx-xln(1-x)+lnsinπx-lnπ. Taking the second derivative of both sides of (38), we have (39)HH′′-H2H2=x2-x+1x2(1-x)2-π2csc2(πx). By Lemma 4, it follows that (40)HH′′-H2H2>0. Thus, (41)H′′>0, and therefore for 0x1/2, we have (42)H(x)(1-2x)H(0)+2xH(12)=(2-4π)(-x)+1. For the case 1/2<x1, since H(1/2)=2/π, H(1)=1, and H is concave up, it follows that (43)H(x)2(1-x)H(12)+(2x-1)H(1)=(2-4π)(x-1)+1. Using (10) from (42) and (43), we have (44)ebn=01xn-2h(x)dx=01H(x)(xn-1-xn)dx01/2[1-(2-4π)x](xn-1-xn)dx+1/21[1+(2-4π)(x-1)](xn-1-xn)dx=1n(n+1)-(2-4π)2n+2-n-4n(n+1)(n+2)2n+1<1n(n+1)-2-4/πn(n+1)(n+2). 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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