Hindawi 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
Copyright q 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 Stark 1 or see Kuang 2, page 248 obtained the following two-sided
rational approximation for tan x/x.
Theorem 1.1. Let 0 < x < π/2; then
8
π2
tan x
<
<
.
x
π 2 − 4x2
π 2 − 4x2
1.1
Furthermore, 8 and π 2 are the best constants in 1.1.
In recent paper 3, we obtained the following further result.
Theorem 1.2. Let 0 < x < π/2; then
π 2 4 8 − π 2 /π 2 x2 tan x π 2 π 2 /3 − 4 x2
<
<
.
x
π 2 − 4x2
π 2 − 4x2
Furthermore, α 48 − π 2 /π 2 and β π 2 /3 − 4 are the best constants in 1.2.
1.2
2
Journal of Inequalities and Applications
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
P2N x αx2N2 tan x P2N x βx2N2
<
<
x
π 2 − 4x2
π 2 − 4x2
1.3
holds, where P2N x a0 a1 x2 · · · aN x2N , and
22n2 22n2 − 1 π 2
4 · 22n 22n − 1
an |B2n2 | −
|B2n |,
2n 2!
2n!
n 0, 1, 2, . . . ,
1.4
where B2n are the even-indexed Bernoulli numbers. Furthermore, β aN1 and α 8 − a0 −
a1 π/22 − · · · − aN π/22N /π/22N2 are the best constants in 1.3.
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 kind Jν : R → −∞, 1, defined by
Jν x 2ν Γν 1x−ν Jν x −1/4n
x2n ,
n!ν
1
n
n≥0
1.5
where, ν 1n Γν 1 n/Γν 1 is the well- known Pochhammer or Appell symbol,
and Jν x defined by 4, page 40
Jν x x 2nν
−1n
,
n!Γν 1 n 2
n≥0
x ∈ R.
1.6
Particularly for ν 1/2 and ν −1/2, respectively, the function Jν reduces to some
elementary functions, like 4, page 54 J1/2 x sin x/x and J−1/2 x cos x. In view of
that tan x/x J1/2 x/J−1/2 x, in Section 3 we shall extend the result of Theorem 1.3 to
the ratio of two normalized Bessel functions of the first kind Jν1 x and Jν 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 Jν1 x 2m
σν x2m ,
2 Jν x
m0
2m
where m ∈ {1, 2, 3, . . .}, and σν
in [4, page 502].
∞
−2m
n1 jν,n
2.1
is the Rayleigh function of order 2m, which showed
Journal of Inequalities and Applications
3
Lemma 2.2 Rayleigh Inequality 5, 6. Let ν > −1, and jν,n be the nth positive zero of the Bessel
2m
−2m
function of the first kind of order ν, m ∈ {1, 2, 3, . . .}, and σν
∞
n1 jν,n is the Rayleigh function
of order 2m. Then
2m
σν
2
<
jν,1
2m2
σν
∞
−2
jν,n
2
σν n1
2.2
,
1
4ν 1
2.3
hold.
Lemma 2.3. Let ν > −1, Jν x be the normalized Bessel function of the first kind of order ν, jν,n the
2m
−2m
nth positive zero of the Bessel function of the first kind of order ν, m ∈ {1, 2, 3, · · ·}, σν
∞
n1 jν,n
the Rayleigh function of order 2m, and 0 < |x| < jν,1 . Then
∞
J x
ν1
2
2
Ex jν,1
− x2
4ν 1 Am x2m ,
jν,1
Jν x
m1
2m2
2
where Am jν,1
σν
2m
− σν
2.4
< 0.
Proof. By Lemma 2.1 and 2.3 in Lemma 2.2, we have
J x
ν1
2
− x2
Ex jν,1
Jν x
2ν 1 J x
ν1
2
jν,1
− x2
x
Jν x
∞
4ν 1 2m
2
− x2
σν x2m
jν,1
x2 m1
∞
2m
2
4ν 1 jν,1
− x2
σν x2m−2
m1
2
4ν 1jν,1
∞
2m 2m−2
σν
x
m1
2
2
σν 4ν 1jν,1
2
jν,1
4ν 1
∞ m1
2
jν,1
4ν 1
2m2
2
where Am jν,1
σν
2m
− σν
− 4ν 1
∞
∞
2m 2m−2
x
2m2
x
− 4ν 1
m2
2
jν,1
σν
2.5
2m 2m
σν
m1
σν
∞
∞
2m 2m
σν
m1
2m
− σν
x2m
Am x2m ,
m1
< 0, which follows from 2.2 in Lemma 2.2.
x
4
Journal of Inequalities and Applications
3. Main Result and Its Proof
Theorem 3.1. Let ν > −1, Jν x be the normalized Bessel function of the first kind of order ν, jν,n
2m
the nth positive zero of the Bessel function of the first kind of order ν, m ∈ {1, 2, 3, . . .}, σν
∞ −2m
j
the
Rayleigh
function
of
order
2m,
N
≥
0
a
natural
number,
and
0
<
|x|
<
j
.
Let
ν,1
n1 ν,n
2m
2N2
2
λ 1 − jν,1
/4ν 1 − N
m1 Am jν,1 /jν,1 , and μ AN1 . Then
R2N x 4ν 1λx2N2 Jν1 x R2N x 4ν 1μx2N2
<
<
2
2
Jν x
jν,1
− x2
jν,1
− x2
2
holds, where R2N x jν,1
4ν 1
N
m1
2n2
2
An jν,1
σν
3.1
Am x2m and
2n
− σν
,
3.2
n ∈ {1, 2, 3, . . .}.
Furthermore, λ and μ are the best constants in 3.1.
Proof of Theorem 3.1. Let
Hx 2
2m
Ex − jν,1
/4ν 1 − N
m1 Am x
x2N2
.
3.3
Then by Lemma 2.3, we have
Hx ∞
nN1 An x
x2N2
2n
∞
AN1k x2k .
3.4
k0
Since An < 0 for n ∈ N by Lemma 2.3, Hx is decreasing on 0, jν,1 .
− Ex
At the same time, in view of that limx → jν,1
4ν 1 we have that λ N
2m
2N2
2
− Hx 1−j
limx → jν,1
by 3.3, and μ limx → 0 Hx AN1
ν,1 /4ν1− m1 Am jν,1 /jν,1
by 3.4, so λ and μ are the best constants in 3.1.
Remark 3.2. Let ν −1/2 in Theorem 3.1; we obtain Theorem 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 Á. Baricz and S. Wu, “Sharp exponential Redheffer-type inequalities for Bessel functions,” Publicationes
Mathematicae Debrecen, vol. 74, no. 3-4, pp. 257–278, 2009.