Volume 8 (2007), Issue 1, Article 11, 6 pp.
REDHEFFER TYPE INEQUALITY FOR BESSEL FUNCTIONS
ÁRPÁD BARICZ
FACULTY OF M ATHEMATICS AND C OMPUTER S CIENCE
"BABE Ş -B OLYAI " U NIVERSITY,
S TR . M. KOG ĂLNICEANU NR . 1
RO-400084 C LUJ -NAPOCA , ROMANIA
bariczocsi@yahoo.com
Received 23 August, 2006; accepted 09 February, 2007
Communicated by F. Qi
A BSTRACT. In this short note, by using mathematical induction and infinite product representations of the functions Jp : R → (−∞, 1] and Ip : R → [1, ∞), defined by
Jp (x) = 2p Γ(p + 1)x−p Jp (x) and Ip (x) = 2p Γ(p + 1)x−p Ip (x),
an extension of Redheffer’s inequality for the function Jp and a Redheffer-type inequality for
the function Ip are established. Here Jp and Ip , denotes the Bessel function and modified Bessel
function, while Γ stands for the Euler gamma function. At the end of this work a lower bound for
the Γ function is deduced, using Euler’s infinite product formula. Our main motivation to write
this note is the publication of C.P. Chen, J.W. Zhao and F. Qi [2], which we wish to complement.
Key words and phrases: Bessel functions, Modified Bessel functions, Redheffer’s inequality.
2000 Mathematics Subject Classification. 33C10, 26D05.
1. I NTRODUCTION AND P RELIMINARIES
The following inequality
sin x
π 2 − x2
≥ 2
, for all x ∈ R
x
π + x2
(1.1)
is known in literature as Redheffer’s inequality [4, 5]. Motivated by this inequality recently C.P.
Chen, J.W. Zhao and F. Qi [2] (see also the survey article of F.Qi [3]) using mathematical induction and infinite product representation of cos x, sinh x and cosh x established the following
Redheffer-type inequalities:
(1.2)
cos x ≥
π 2 − 4x2
π 2 + 4x2
π
,
and
cosh
x
≤
, for all |x| ≤ .
2
2
2
2
π + 4x
π − 4x
2
Research partially supported by the Institute of Mathematics, University of Debrecen, Hungary. The author is grateful to Professor Szilárd
András and to the referee for their instructive comments which were helpful.
220-06
2
Á RPÁD BARICZ
Moreover, the authors found the hyperbolic analogue of inequality (1.1), by showing that
sinh x
π 2 + x2
≤ 2
for all |x| < π.
x
π − x2
As we mentioned above, the proofs of inequalities (1.2) and (1.3) by C.P. Chen, J. W. Zhao and
F. Qi are based on the following representations [1, p. 75 and 85] of cos x, sinh x and cosh x:
Y
Y
4x2
4x2
1−
(1.4)
cos x =
,
cosh x =
1+
,
2π2
2π2
(2n
−
1)
(2n
−
1)
n≥1
n≥1
(1.3)
and
(1.5)
x2
sinh x Y
=
1+ 2 2
x
nπ
n≥1
respectively. In this paper our aim is to show that the idea of using mathematical induction and
infinite product representation is also fruitful for Bessel functions as well as for the Γ function.
2. A N E XTENSION OF R EDHEFFER ’ S I NEQUALITY AND ITS H YPERBOLIC A NALOGUE
Our first main result reads as follows.
Theorem 2.1. Let us consider the functions Jp : R → (−∞, 1] and Ip : R → [1, ∞), defined
by the relations
Jp (x) = 2p Γ(p + 1)x−p Jp (x) and Ip (x) = 2p Γ(p + 1)x−p Ip (x),
where Jp and Ip are the well-known Bessel function, and modified Bessel function respectively.
Furthermore suppose that p > −1 and let jp,n be the n-th positive zero of the Bessel function
2
Jp . If ∆p (n) := jp,n+1
− jp,1 jp,n − jp,n jp,n+1 ≥ 0, where n = 1, 2, . . ., then the following
inequalities hold
q
2
jp,1
− x2
, for all |x| ≤ αp := min
jp,1 , ∆p (n) ,
(2.1)
Jp (x) ≥ 2
n≥1,p>−1
jp,1 + x2
(2.2)
Ip (x) ≤
2
jp,1
+ x2
, for all |x| < jp,1 .
2
jp,1
− x2
Remark 2.2. For later use it is worth mentioning that in particular for p = −1/2 and p = 1/2
respectively the functions Jp and Ip reduce to some elementary functions [1, p. 438 and 443],
such as
p
(2.3)
J−1/2 (x) = π/2 · x1/2 J−1/2 (x) = cos x,
p
sin x
J1/2 (x) = π/2 · x−1/2 J1/2 (x) =
,
x
with their hyperbolic analogs
p
(2.4)
I−1/2 (x) = π/2 · x1/2 I−1/2 (x) = cosh x,
p
sinh x
.
I1/2 (x) = π/2 · x−1/2 I1/2 (x) =
x
Recall that J−1/2 has the infinite product representation (1.4) and [1, p. 75]
sin x Y
x2
J1/2 (x) =
=
1− 2 2 .
x
nπ
n≥1
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 11, 6 pp.
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R EDHEFFER T YPE I NEQUALITY FOR B ESSEL F UNCTIONS
3
Thus using the relations (2.6) and (2.3) it is clear that j−1/2,n =p(2n − 1)π/2 and j1/2,n = nπ
for all n = 1, 2, . . .. Consequently for all n = 1, 2, . . . we have ∆1/2 (n) = j1/2,1 = π and
q
π√
π√
π
∆−1/2 (n) =
2n + 3 ≥
5 > = j−1/2,1 ,
2
2
2
which imply that α−1/2 = π/2 and α1/2 = π. So in view of (2.3), if we take in (2.1) p = −1/2
and p = 1/2 respectively, then we reobtain the first inequality from (1.2) and Redheffer’s
inequality (1.1) respectively, with the intervals of validity [−π/2, π/2] and [−π, π], respectively.
The situation is similar to inequality (2.2), namely if we choose in (2.2) p = −1/2 and p = 1/2
respectively, then by using (2.4), we reobtain the second inequality from (1.2) and inequality
(1.3), with the same intervals of validity, i.e. [−π/2, π/2] and [−π, π], respectively.
Proof of Theorem 2.1. First observe that to prove (2.1) it is enough to show that
1 − x2
1 + x2
holds for all |x| ≤ αp /jp,1 . It is known that for the Bessel function of the first kind Jp the
following infinite product formula [6, p. 498]
Y
x2
p
−p
(2.6)
Jp (x) = 2 Γ(p + 1)x Jp (x) =
1− 2
jp,n
n≥1
Jp (xjp,1 ) ≥
(2.5)
is valid for arbitrary x and p 6= −1, −2, . . .. From this we deduce
!
n
2 2
i
Y
x
j
1 − x2 h
p,1
(1 + x2 ) lim Qp,n , where Qp,n :=
1− 2
.
(2.7)
Jp (xjp,1 ) =
2
n→∞
1+x
j
p,k
k=2
In what follows we want to prove by mathematical induction that
(1 + x2 )Qp,n ≥ 1 +
(2.8)
x2 jp,1
jp,n
holds for all p > −1, n ≥ 2 and |x| ≤ αp /jp,1 . For n = 2 clearly by assumptions we have
x2 x2 jp,1
2
2
(1 + x )Qp,2 − 1 +
= 2 ∆p (1) − jp,1
x2 ≥ 0.
jp,2
jp,2
Now suppose that (2.8) holds for some m ≥ 2. From the definition of Qp,m , we easily get
2 x2 jp,1
Qp,m+1 = Qp,m · 1 − 2
, for all m = 2, 3, 4, . . . ,
jp,m+1
thus
x2 jp,1
(1 + x )Qp,m+1 − 1 +
jp,m+1
2
2 x2 jp,1
x2 jp,1
=(1 + x )Qp,m 1 − 2
− 1+
jp,m+1
jp,m+1
2
x2 jp,1
x2 jp,1
x2 jp,1
≥ 1+
1− 2
− 1+
jp,m
jp,m+1
jp,m+1
x2 jp,1 2
=
∆p (m) − jp,1
x2 ≥ 0,
2
jp,m jp,m+1
2
and hence
p by induction (2.8) follows. Here we used the fact that from the hypothesis we obtain
|x| ≤ ∆p (m)/jp,1 ≤ jp,m+1 /jp,1 . On the other hand from the MacMahon expansion [6, p.
506],
jp,n = (n + p/2 − 1/4)π + O(n−1 ), n → ∞,
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 11, 6 pp.
http://jipam.vu.edu.au/
4
Á RPÁD BARICZ
we have jp,n → ∞, as n tends to infinity. Finally from (2.8) we obtain
x2 jp,1
2
lim (1 + x )Qp,n ≥ lim 1 +
= 1,
n→∞
n→∞
jp,n
which in view of (2.7) implies (2.5). This completes the proof of (2.1).
Proceeding similarly as in the proof of (2.1) now we prove (2.2). It suffices to show that
Ip (xjp,1 ) ≤
(2.9)
1 + x2
1 − x2
holds for all |x| < 1. Analogously, using the factorisation (2.6), it is known that for the modified
Bessel function of the first kind Ip the following infinite product formula
Y
x2
p
−p
Ip (x) = 2 Γ(p + 1)x Ip (x) =
1+ 2
jp,n
n≥1
is also valid for arbitrary x and p 6= −1, −2, . . .. From this we get that
n
2
i
Y
x2 jp,1
1+x h
2
Ip (xjp,1 ) =
(1 − x ) lim Rp,n , where Rp,n :=
1+ 2
n→∞
1 − x2
jp,k
k=2
2
(2.10)
!
.
In what follows we want to show by mathematical induction that
(1 − x2 )Rp,n ≤ 1 −
(2.11)
x2 jp,1
jp,n
holds for all p > −1, n ≥ 2 and |x| < 1. For n = 2, clearly we have
x2 jp,1
x2 2
2
(1 − x )Rp,2 − 1 −
= 2 −∆p (1) − jp,1
x2 ≤ 0.
jp,2
jp,2
Now suppose that (2.11) holds for some m ≥ 2. From the definition of Rp,m , we easily get
2 x2 jp,1
Rp,m+1 = Rp,m · 1 + 2
, for all m = 2, 3, 4, . . . ,
jp,m+1
thus
x2 jp,1
(1 − x )Rp,m+1 − 1 −
jp,m+1
2
2 x2 jp,1
x2 jp,1
=(1 − x )Rp,m 1 + 2
− 1−
jp,m+1
jp,m+1
2
x2 jp,1
x2 jp,1
x2 jp,1
≤ 1−
1+ 2
− 1−
jp,m
jp,m+1
jp,m+1
x2 jp,1 2
=
−∆p (m) − jp,1
x2 ≤ 0,
2
jp,m jp,m+1
2
and hence by induction, (2.11) follows. Finally using again the fact that jp,n → ∞, as n tends
to infinity, from (2.11) we obtain
x2 jp,1
2
lim (1 − x )Rp,n ≤ lim 1 −
= 1,
n→∞
n→∞
jp,n
which in view of (2.10) implies (2.9). Thus the proof is complete.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 11, 6 pp.
http://jipam.vu.edu.au/
R EDHEFFER T YPE I NEQUALITY FOR B ESSEL F UNCTIONS
3. A L OWER B OUND FOR
THE
5
Γ F UNCTION
In an effort to popularize the method that C.P. Chen, J. W. Zhao and F. Qi used in the previous
section, we illustrate it below by giving a lower bound for the Γ function.
Theorem 3.1. If x ∈ (0, 1], then
(3.1)
1 − x e(1−γ)x
·
,
1+x
x
− log n = 0.5772156649 . . . is the Euler constant.
Γ(x) ≥
where γ = limn→∞ 1 + 21 + 13 + · · · +
1
n
Proof. From the well-known Euler infinite product formula [1, p. 255] for the Γ function,
Y
1
x −x
=
1
+
e n,
xeγx Γ(x) n≥1
n
we have
(3.2)
n i
Y
e(1−γ)x
1+xh
x −x
=
(1 − x) lim Sn , where Sn =
1+
e k , n = 2, 3, . . ..
n→∞
xΓ(x)
1−x
k
k=2
Observe that to prove (3.1), it is enough to show that for all n = 2, 3, . . .
x
(3.3)
(1 − x)Sn < 1 −
n
holds, hence from this we get lim (1 − x)Sn ≤ 1, and consequently from (3.2) the inequality
n→∞
(3.1) follows. To prove (3.3) we use mathematical induction again. For n = 2 we easily get for
x ∈ (0, 1) that
(1 − x) 1 + x2
x
x/2
(1 − x)S2 < 1 −
⇐⇒ e >
,
2
1 − x2
which clearly holds because
X
(1 − x) 1 + x2
1 xk
x/2
e −
=
x+
> 0.
k
1 − x2
k!
2
k≥1
Now suppose that (3.3) holds for some m ≥ 2. Then from (3.2) and (3.3) we obtain that
x
x
x
x
− m+1
(1 − x)Sm+1 − 1 −
=(1 − x)Sm 1 +
e
− 1−
m+1
m+1
m+1
x
x
x
x
− m+1
< 1−
1+
e
− 1−
m
m+1
m+1
and this is negative if and only if
x
x
X x
1− m
1 + m+1
x
1
1
xk
m+1
−
e
=
+
−
1
+
> 0.
x
m m
k! (m + 1)k
1 − m+1
k≥1
Remark 3.2. Numerical experiments in Maple6 show that the lower bound from (3.1)√is far
from being the best possible one. For example for x = 0.5 we have that Γ(0.5) = π =
1.772453851 . . . , while the right hand side of (3.1) is just 0.8235978287. . .. Similarly for x =
0.25 we have Γ(0.25) = 3.6256099082 . . . , while the right hand side of (3.1) is just 2.667561665. . ..
In fact graphics in Maple6 suggest that the function f : (−1, ∞) → R defined by
f (x) = Γ(x) −
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 11, 6 pp.
1 − x e(1−γ)x
·
1+x
x
http://jipam.vu.edu.au/
6
Á RPÁD BARICZ
is convex and satisfies the inequality f (x) ≥ 1, for all x ∈ (−1, 0] or x ∈ [1, ∞). Moreover
f (x) ∈ (0.94, 1], for all x ∈ [0, 1].
R EFERENCES
[1] M. ABRAMOVITZ AND I.A. STEGUN (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1965.
[2] Ch.-P. CHEN, J.-W. ZHAO AND F. QI, Three inequalities involving hyperbolically trigonometric
functions, Octogon Math. Mag., 12(2) (2004), 592–596. RGMIA Res. Rep. Coll., 6(3) (2003), Art.
4, 437-443. [ONLINE: http://rgmia.vu.edu.au/v6n3.html].
[3] F. QI, Jordan’s inequality: Refinements, generalizations, applications and related problems, RGMIA
Res. Rep. Coll., 9(3) (2006), Art. 12. [ONLINE: http://rgmia.vu.edu.au/v9n3.html].
Bùděngshì Yānjiū Tōngxùn (Communications in Studies on Inequalities) 13(3) (2006) 243–259.
[4] R. REDHEFFER, Correction, Amer. Math. Monthly, 76(4) (1969), 422.
[5] R. REDHEFFER, Problem 5642, Amer. Math. Monthly, 76 (1969), 422.
[6] G. N. WATSON, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1962.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 11, 6 pp.
http://jipam.vu.edu.au/