Hindawi Publishing CorporationJournal of Inequalities and Applications Volume 2010, Article ID 838740, 4 pages doi:10.1155/2010/838740 Research Article Sharp Becker-Stark-Type Inequaliti
Trang 1Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 838740, 4 pages
doi:10.1155/2010/838740
Research Article
Sharp Becker-Stark-Type Inequalities for
Bessel Functions
Ling Zhu
Department of Mathematics, Zhejiang Gongshang University, Hangzhou, Zhejiang 310018, China
Correspondence should be addressed to Ling Zhu,zhuling0571@163.com
Received 22 January 2010; Accepted 23 March 2010
Academic Editor: Wing-Sum Cheung
Copyrightq 2010 Ling Zhu 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
We extend the Becker-Stark-type inequalities to the ratio of two normalized Bessel functions of the first kind by using Kishore formula and Rayleigh inequality
1 Introduction
In 1978, Becker and Stark1 or see Kuang 2, page 248 obtained the following two-sided rational approximation fortan x/x.
Theorem 1.1 Let 0 < x < π/2; then
8
π2− 4x2 < tanx x < π2
Furthermore, 8 and π2are the best constants in1.1.
In recent paper3, we obtained the following further result
Theorem 1.2 Let 0 < x < π/2; then
π24
8− π2
/π2
x2
π2− 4x2 < tanx x < π2
π2/3− 4x2
Furthermore, α 48 − π2/π2and β π2/3 − 4 are the best constants in 1.2.
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Moreover, the following refinement of the Becker-Stark inequality was established in
3
Theorem 1.3 Let 0 < x < π/2, and N ≥ 0 be a natural number Then
P2 N x αx2N2
π2− 4x2 < tanx x < P2 N x βx2N2
holds, where P2 Nx a0 a1x 2 · · · aN x2N , and
a n 22n2
22n2− 1π2
2n 2! |B2n2| −4· 22n
22n− 1
2n! |B2n|, n 0, 1, 2, , 1.4
where B2 n are the even-indexed Bernoulli numbers Furthermore, β a N1 and α 8 − a0 −
a1 π/22− · · · − aNπ/22N /π/22N2 are the best constants in1.3.
Our aim of this paper is to extend the tangent function to Bessel functions To achieve our goal, let us recall some basic facts about Bessel functions Suppose that ν > −1 and
consider the normalized Bessel function of the first kindJν:R → −∞, 1, defined by
Jν x 2 ν Γν 1x −ν J ν x
n≥0
−1/4 n
n!ν 1 n x2n , 1.5
where,ν 1 n Γν 1 n/Γν 1 is the well- known Pochhammer or Appell symbol,
andJ ν x defined by 4, page 40
J νx
n≥0
−1n
n!Γν 1 n
x
2
2nν
, x ∈ R. 1.6
Particularly for ν 1/2 and ν −1/2, respectively, the function J ν reduces to some elementary functions, like4, page 54 J1/2x sin x/x and J −1/2 x cos x In view of
that tanx/x J1 /2 x/J −1/2 x, inSection 3we shall extend the result ofTheorem 1.3to the ratio of two normalized Bessel functions of the first kindJν1 x and J ν x.
2 Some Lemmas
In order to prove our main result in next section, each of the following lemmas will be needed
Lemma 2.1 Kishore Formula, see 5,6 Let ν > −1, jν,n be the nth positive zero of the Bessel function of the first kind of order ν, and x ∈ R Then
x
2
J ν1x
J νx
∞
m0
where m ∈ {1, 2, 3, }, and σ ν 2m ∞n1 j −2m
ν,n is the Rayleigh function of order 2m, which showed
in [ 4 , page 502].
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Lemma 2.2 Rayleigh Inequality 5,6 Let ν > −1, and jν,n be the nth positive zero of the Bessel function of the first kind of order ν, m ∈ {1, 2, 3, }, and σ ν 2m∞n1 j −2m
ν,n is the Rayleigh function
of order 2m Then
j2
ν,1 < σ ν 2m
σ ν2∞
n1
j−2
hold.
Lemma 2.3 Let ν > −1, J νx be the normalized Bessel function of the first kind of order ν, jν,n the nth positive zero of the Bessel function of the first kind of order ν, m ∈ {1, 2, 3, ···}, σ ν 2m∞
n1 j −2m
ν,n
the Rayleigh function of order 2m, and 0 < |x| < j ν,1 Then
Ex j2
ν,1 − x2Jν1x
Jν x j ν,12 4ν 1∞
m1
A m x2m , 2.4
where A m j2
ν,1 σ ν 2m2 − σ ν 2m < 0.
Proof ByLemma 2.1and2.3 inLemma 2.2, we have
Ex j2
ν,1 − x2Jν1x
Jνx
j2
ν,1 − x22ν 1
x
J ν1x
J ν x
j2
ν,1 − x24ν 1
x2
∞
m1
σ ν 2m x2m
4ν 1j2
ν,1 − x2∞
m1
σ ν 2m x2m−2
4ν 1j2
ν,1
∞
m1
σ ν 2m x2m−2 − 4ν 1∞
m1
σ ν 2m x2m
4ν 1j2
ν,1
σ ν2∞
m2
σ ν 2m x2m−2
− 4ν 1∞
m1
σ ν 2m x2m
j2
ν,1 4ν 1∞
m1
j2
ν,1 σ ν 2m2 − σ ν 2m x2m
j2
ν,1 4ν 1∞
m1
A m x2m ,
2.5
whereA m j2
ν,1 σ ν 2m2 − σ ν 2m < 0, which follows from 2.2 inLemma 2.2
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3 Main Result and Its Proof
Theorem 3.1 Let ν > −1, J ν x be the normalized Bessel function of the first kind of order ν, j ν,n
the nth positive zero of the Bessel function of the first kind of order ν, m ∈ {1, 2, 3, }, σ ν 2m
∞
n1 j −2m
ν,n the Rayleigh function of order 2m, N ≥ 0 a natural number, and 0 < |x| < j ν,1 Let
λ 1 − j2
ν,1 /4ν 1 −N
m1 A m j2m
ν,1 /j2N2 ν,1 , and μ A N1 Then R2 N x 4ν 1λx2N2
j2
ν,1 − x2 < Jν1 x
Jνx <
R2 Nx 4ν 1μx2N2
j2
holds, where R2 Nx j2
ν,1 4ν 1N m1 A m x2m and
A n j2
ν,1 σ ν 2n2 − σ ν 2n , n ∈ {1, 2, 3, }. 3.2
Furthermore, λ and μ are the best constants in 3.1.
Proof of Theorem 3.1 Let
Hx
Ex − j2
ν,1
/4ν 1−N
m1 A m x2m
Then byLemma 2.3, we have
Hx
∞
nN1 A n x2n
x2N2 ∞
k0
SinceA n < 0 for n ∈ NbyLemma 2.3,Hx is decreasing on 0, j ν,1.
At the same time, in view of that limx → j−
ν,1 Ex 4ν 1 we have that λ
limx → j−
ν,1 Hx 1−j2
ν,1 /4ν1−N m1 A m j2m
ν,1 /j2N2 ν,1 by3.3, and μ limx → 0Hx A N1
by3.4, so λ and μ are the best constants in 3.1
Remark 3.2 Let ν −1/2 inTheorem 3.1; we obtainTheorem 1.3
References
1 M Becker and E L Stark, “On a hierarchy of quolynomial inequalities for tanx,” University of Beograd
Publikacije Elektrotehnicki Fakultet Serija Matematika i fizika, no 602–633, pp 133–138, 1978.
2 J C Kuang, Applied Inequalities, Shandong Science and Technology Press, Jinan, China, 3rd edition,
2004
3 L Zhu and J K Hua, “Sharpening the Becker-Stark inequalities,” Journal of Inequalities and Applications,
vol 2010, Article ID 931275, 4 pages, 2010
4 G N Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge
University Press, Cambridge, UK, 1995
5 N Kishore, “The Rayleigh function,” Proceedings of the American Mathematical Society, vol 14, pp 527–
533, 1963
6 ´A Baricz and S Wu, “Sharp exponential Redheffer-type inequalities for Bessel functions,” Publicationes
Mathematicae Debrecen, vol 74, no 3-4, pp 257–278, 2009.