Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES INVOLVING BESSEL FUNCTIONS OF THE
FIRST KIND
volume 5, issue 4, article 94,
2004.
EDWARD NEUMAN
SDepartment of Mathematics
Southern Illinois University
Carbondale, IL 62901-4408, USA.
Received 25 September, 2004;
accepted 13 October, 2004.
Communicated by: A. Lupaş
EMail: edneuman@math.siu.edu
URL: http://www.math.siu.edu/neuman/personal.html
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
171-04
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Abstract
An inequality involving a function fα (x) = Γ(α + 1)(2/x)α Jα (x) (α > − 12 ) is
obtained. The lower and upper bounds for this function are also derived.
2000 Mathematics Subject Classification: 33C10, 26D20.
Key words: Bessel functions of the first kind, Inequalities, Gegenbauer polynomials.
Inequalities Involving Bessel
Functions of the First Kind
Contents
1
Introduction and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
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1.
Introduction and Definitions
In this note we deal with the function
(1.1)
α
2
fα (x) = Γ(α + 1)
Jα (x),
x
x ∈ R, α > − 12 and Jα stands for the Bessel function of the first kind of order
α. It is known (see, e.g., [1, (9.1.69)]) that
X
2 n
∞
1
x
x2
,
fα (x) = 0 F1 −; α + 1; −
=
−
4
n!(α
+
1)
4
n
n=0
where (a)k = Γ(a + k)/Γ(a) (k = 0, 1, . . .). It is obvious from the above
representation that fα (−x) = fα (x) and also that fα (0) = 1. The function
under discussion admits the integral representation
Z 1
(1.2)
fα (x) =
cos(xt)dµ(t)
−1
(see, e.g., [1, (9.1.20)]) where dµ(t) = µ(t)dt with
1
1
2 α− 12
2α
(1.3)
µ(t) = (1 − t )
2 B α+ , α+
2
2
being the Dirichlet measure on the interval [−1, 1] and B(·, ·) stands for the beta
function. Clearly
Z 1
(1.4)
dµ(t) = 1.
−1
Inequalities Involving Bessel
Functions of the First Kind
Edward Neuman
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Thus µ(t) is the probability measure on the interval [−1, 1].
In [2], R. Askey has shown that the following inequality
(1.5)
fα (x) + fα (y) ≤ 1 + fα (z)
holds true for all α ≥ 0 and z 2 = x2 + y 2 . This provides a generalization of
Grünbaum’s inequality ([4]) who has established (1.5) for α = 0.
In this note we give a different upper bound for the sum fα (x) + fα (y) (see
(2.1)). Also, lower and upper bounds for the function in question are derived.
Inequalities Involving Bessel
Functions of the First Kind
Edward Neuman
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2.
Main Results
Our first result reads as follows.
Theorem 2.1. Let x, y ∈ R. If α > − 12 , then
(2.1)
[fα (x) + fα (y)]2 ≤ [1 + fα (x + y)][1 + fα (x − y)].
Proof. Using (1.2), some elementary trigonometric identities, Cauchy-Schwarz
inequality for integrals, and (1.4) we obtain
Z 1
|fα (x) + fα (y)| ≤
| cos(xt) + cos(yt)|dµ(t)
−1
Z 1
(x
+
y)t
(x
−
y)t
cos
dµ(t)
=2
cos
2
2
−1
Z 1
21 Z 1
12
(x
+
y)t
(x
−
y)t
≤2
cos2
dµ(t)
cos2
dµ(t)
2
2
−1
−1
12
Z 1
1
=2
(1 + cos(x + y)t)dµ(t)
2 −1
12
Z 1
1
(1 + cos(x − y)t)dµ(t)
×
2 −1
1
2
1
2
= [1 + fα (x + y)] [1 + fα (x − y)] .
Hence, the assertion follows.
Inequalities Involving Bessel
Functions of the First Kind
Edward Neuman
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When x = y, inequality (2.1) simplifies to 2fα2 (x) ≤ 1 + fα (2x) which bears
resemblance of the double-angle formula for the cosine function 2 cos2 x =
1 + cos 2x.
Our next goal is to establish computable lower and upper bounds for the
function fα . We recall some well-known facts about Gegenbauer polynomials
Ckα (α > − 12 , k ∈ N) and the Gauss-Gegenbauer quadrature formulas. They are
1
orthogonal on the interval [−1, 1] with the weight function w(t) = (1 − t2 )α− 2 .
The explicit formula for Ckα is
[k/2]
Ckα (t) =
X
(−1)m
m=0
Γ(α + k − m)
(2t)k−2m
Γ(α)m!(k − 2m)!
(see, e.g., [1, (22.3.4)]). In particular,
2
(2.2)
= 2α(α + 1)t − α,
= α(α + 1)[2(α + 2)t3 − 3t].
3
The classical Gauss-Gegenbauer quadrature formula with the remainder is [3]
Z 1
k
X
1
(2.3)
(1 − t2 )α− 2 g(t)dt =
wi g(ti ) + γk g (2k) (η),
C2α (t)
2
−1
C3α (t)
i=1
where g ∈ C 2k ([−1, 1]), γk is a positive number and does not depend on g,
and η is an intermediate point in the interval (−1, 1). Recall that the nodes ti
(1 ≤ i ≤ n) are the roots of Ckα and the weights wi are given explicitly by [5,
(15.3.2)]
(2.4)
wi = π 22−2α
Γ(2α + k)
1
·
2
2
k![Γ(α)]
(1 − ti )[(Ckα )0 (ti )]2
Inequalities Involving Bessel
Functions of the First Kind
Edward Neuman
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(1 ≤ i ≤ k).
We are in a position to prove the following.
Theorem 2.2. Let α > − 12 . If |x| ≤
!
x
(2.5)
cos p
2(α + 1)
π
2
, then
≤ fα (x)
"
≤
1
2α + 1 + (α + 2) cos
3(α + 1)
s
3
x
2(α + 2)
!#
.
Inequalities Involving Bessel
Functions of the First Kind
Edward Neuman
Equalities hold in (2.5) if x = 0.
Proof. In order to establish the lower bound in (2.5) we use the Gauss-Gegenbauer
quadrature formula (2.3) with g(t) = cos(xt) and k = 2. Since g (4) (t) =
x4 cos(xt) ≥ 0 for t ∈ [−1, 1] and |x| ≤ π2 ,
Z 1
1
(2.6)
w1 g(t1 ) + w2 g(t2 ) ≤
(1 − t2 )α− 2 cos(xt)dt.
−1
1
−t1 = t2 = p
2(α + 1)
2α
1
,
2
1
).
2
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Making use of (2.2) and (2.4) we obtain
1
2
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and w1 = w2 = 2 B(α + α +
This in conjunction with (2.6) gives
! Z
1
1
1
x
1
2α
2 B α+ , α+
cos p
≤
(1 − t2 )α− 2 cos(xt)dt.
2
2
2(α + 1)
−1
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Application of (1.3) together with the use of (1.2) gives the asserted result. In
order to derive the upper bound in (2.5) we use again (2.3). Letting g(t) =
cos(xt) and k = 3 one has g (6) (t) = −x6 cos(xt) ≤ 0 for |t| ≤ 1 and |x| ≤ π2 .
Hence
Z 1
1
(2.7)
(1 − t2 )α− 2 cos(xt)dt ≤ w1 g(t1 ) + w2 g(t2 ) + w3 g(t3 ).
−1
It follows from (2.2) and (2.4) that
s
−t1 = t3 =
Inequalities Involving Bessel
Functions of the First Kind
3
,
2(α + 2)
t2 = 0
and
Edward Neuman
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α+2
1
1
w1 = w3 = 22α B α + , α +
,
2
2 6(α + 1)
1
1 2α + 1
2α
w2 = 2 B α + , α +
.
2
2 3(α + 1)
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This in conjunction with (2.7), (1.3), and (1.2) gives the desired result. The
proof is complete.
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Sharper lower and upper bounds for fα can be obtained using higher order
quadrature formulas (2.3) with even and odd numbers of knots, respectively.
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References
[1] M. ABRAMOWITZ AND I.A. STEGUN (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover
Publications, Inc., New York, 1965.
[2] R. ASKEY, Grünbaum’s inequality for Bessel functions, J. Math. Anal.
Appl., 41 (1973), 122–124.
[3] K.E. ATKINSON, An Introduction to Numerical Analysis, 2nd ed., Wiley,
New York, 1989.
[4] F. GRÜNBAUM, A property of Legendre polynomials, Proc. Nat. Acad.
Sci., USA, 67 (1970), 959–960.
[5] G. SZEGÖ, Orthogonal polynomials, in Colloquium Publications, Vol. 23,
4th ed., American Mathematical Society, Providence, RI, 1975.
Inequalities Involving Bessel
Functions of the First Kind
Edward Neuman
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