Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 2, Article 45, 2004
SOME PROBLEMS AND SOLUTIONS INVOLVING MATHIEU’S SERIES AND ITS
GENERALIZATIONS
H.M. SRIVASTAVA AND ŽIVORAD TOMOVSKI
D EPARTMENT OF M ATHEMATICS AND S TATISTICS
U NIVERSITY OF V ICTORIA
V ICTORIA , B RITISH C OLUMBIA V8W 3P4
C ANADA
harimsri@math.uvic.ca
I NSTITUTE OF M ATHEMATICS
S T. C YRIL AND M ETHODIUS U NIVERSITY
MK-1000 S KOPJE
M ACEDONIA
tomovski@iunona.pmf.ukim.edu.mk
Received 14 October, 2003; accepted 1 May, 2004
Communicated by P. Cerone
Dedicated to Professor Blagoj Sazdo Popov on the Occasion of his Eightieth Birthday
A BSTRACT. The authors investigate several recently posed problems involving the familiar
Mathieu series and its various generalizations. For certain families of generalized Mathieu series,
they derive a number of integral representations and investigate several one-sided inequalities
which are obtainable from some of these general integral representations or from sundry other
considerations. Relevant connections of the results and open problems (which are presented or
considered in this paper) with those in earlier works are also indicated. Finally, a conjectured
generalization of one of the Mathieu series inequalities proven here is posed as an open problem.
Key words and phrases: Mathieu’s series, Integral representations, Bessel functions, Hypergeometric functions, One-sided
inequalities, Fourier transforms, Riemann and Hurwitz Zeta functions, Eulerian integral, Polygamma
functions, Laplace integral representation, Euler-Maclaurin summation formula, Riemann-Liouville
fractional integral, Lommel function of the first kind.
2000 Mathematics Subject Classification. Primary 26D15, 33C10, 33C20, 33C60; Secondary 33E20, 40A30.
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant
OGP0007353.
146-03
2
H.M. S RIVASTAVA AND Ž IVORAD T OMOVSKI
1. I NTRODUCTION , D EFINITIONS ,
AND
P RELIMINARIES
The following familiar infinite series:
S (r) :=
(1.1)
∞
X
n=1
2n
2
(n + r2 )2
r ∈ R+
is named after Émile Leonard Mathieu (1835-1890), who investigated it in his 1890 work [13]
on elasticity of solid bodies.
For the Mathieu series S (r) defined by (1.1), Alzer et al. [2] showed that the best constants
κ1 and κ2 in the following two-sided inequality:
1
1
<
S
(r)
<
κ1 + r2
κ2 + r2
(1.2)
(r 6= 0)
are given by
1
1
and
κ2 = ,
2ζ (3)
6
where ζ (s) denotes the Riemann Zeta function defined by (see, for details, [20, Chapter 2])
 ∞
∞
X
X 1
1
1


=
(R (s) > 1)


s
−s

1 − 2 n=1 (2n − 1)s
 n=1 n
(1.3)
ζ (s) :=

∞

X

(−1)n−1

1−s −1

(R (s) > 0; s 6= 1) .
 (1 − 2 )
ns
n=1
κ1 =
A remarkably useful integral representation for S (r) in the elegant form:
Z
1 ∞ x sin (rx)
(1.4)
S (r) =
dx
r 0
ex − 1
was given by Emersleben [6]. In fact, by applying (1.4) in conjunction with the generating
function:
∞
X zn
z
=
Bn
ez − 1 n=0
n!
(1.5)
(|z| < 2π)
for the Bernoulli numbers
Bn
(n ∈ N0 := {0, 1, 2, . . .}) ,
Elbert [5] derived the following asymptotic expansion for S (r):
(1.6)
S (r) ∼
∞
X
k=0
(−1)k
B2k
1
1
1
= 2− 4−
− ···
2k+2
r
r
6r
30r6
(r → ∞) .
More recently, Guo [10] made use of the integral representation (1.4) in order to obtain a number of interesting results including (for example) bounds for S (r). For various subsequent
developments using (1.4), the interested reader may be referred to the works by (among others)
Qi et al. ([16] to [19]). (See also an independent derivation of the asymptotic expansion (1.6)
by Wang and Wang [24]).
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M ATHIEU ’ S S ERIES AND I TS G ENERALIZATIONS
3
Several interesting problems and solutions dealing with integral representations and bounds
for the following mild generalization of the Mathieu series (1.1):
Sµ (r) :=
(1.7)
∞
X
n=1
2n
2
(n + r2 )µ
r ∈ R+ ; µ > 1
can be found in the recent works by Diananda [4], Guo [10], Tomovski and Trenčevski [23], and
Cerone and Lenard [3]. Motivated essentially by the works of Cerone and Lenard [3] (and Qi
[17]), we propose to investigate the corresponding problems involving a family of generalized
Mathieu series, which is defined here by
Sµ(α,β)
(1.8)
(r; a) =
Sµ(α,β)
(r; {ak }∞
k=1 )
:=
∞
X
n=1
r, α, β, µ ∈ R
+
2aβn
(aαn + r2 )µ
,
where (and throughout this paper) it is tacitly assumed that the positive sequence
∞
a := {ak }k=1 = {a1 , a2 , a3 , . . . , ak , . . .}
lim ak = ∞
k→∞
is so chosen (and then the positive parameters α, β, and µ are so constrained) that the infinite
series in the definition (1.8) converges, that is, that the following auxiliary series:
∞
X
1
µα−β
n=1 an
is convergent. We remark in passing that, in a very recent research report (which appeared after
the submission of this paper to JIPAM), Pogány [14] considered a substantially more general
form of the definition (1.8). As a matter of fact, Pogány’s investigation [14] was based largely
upon such main mathematical tools as the Laplace integral representation of general Dirichlet
series and the familiar Euler-Maclaurin summation formula (cf., e.g., [20, p. 36 et seq.]).
Clearly, by comparing the definitions (1.1), (1.7), and (1.8), we obtain
S2 (r) = S (r)
(1.9)
and
Sµ (r) = Sµ(2,1) (r; {k}∞
k=1 ) .
Furthermore, the special cases
(2,1)
S2
(r; {ak }∞
k=1 ) ,
Sµ(2,1) (r; {k γ }∞
k=1 ) ,
and
Sµ(α,α/2) (r; {k}∞
k=1 )
were investigated by Qi [17], Tomovski [22], and Cerone and Lenard [3].
2. A C LASS OF I NTEGRAL R EPRESENTATIONS
First of all, we find from the definition (1.8) that
∞ ∞
X
X
µ+m−1
1
∞
(α,β)
2 m
Sµ (r; {ak }k=1 ) = 2
−r
,
(µ+m)α−β
m
m=0
n=1 an
J. Inequal. Pure and Appl. Math., 5(2) Art. 45, 2004
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4
H.M. S RIVASTAVA AND Ž IVORAD T OMOVSKI
so that
(2.1)
Sµ(α,β)
(r; {k γ }∞
k=1 )
=2
∞ X
µ+m−1
m
m=0
−r2
m
ζ (γ [(µ + m) α − β])
r, α, β, γ ∈ R+ ; γ (µα − β) > 1
in terms of the Riemann Zeta function defined by (1.3).
Now, by making use of the familiar integral representation (cf., e.g., [20, p. 96, Equation 2.3
(4)]):
1
ζ (s) =
Γ (s)
(2.2)
∞
Z
0
xs−1
dx
ex − 1
(R (s) > 1)
in (2.1), we obtain
(2.3)
Sµ(α,β)
(r; {k γ }∞
k=1 )
2
=
Γ (µ)
∞
xγ(µα−β)−1
ex − 1
0
· 1 Ψ1 (µ, 1) ; (γ (µα − β) , γα) ; −r2 xγα dx,
Z
r, α, β, γ ∈ R+ ; γ (µα − β) > 1 ,
where p Ψq denotes the Fox-Wright generalization of the hypergeometric p Fq function with p
numerator and q denominator parameters, defined by [21, p. 50, Equation 1.5 (21)]
(2.4)
p Ψq
[(α1 , A1 ) , . . . , (αp , Ap ) ; (β1 , B1 ) , . . . , (βq , Bq ) ; z]
∞ Qp
m
X
j=1 Γ (αj + Aj m) z
Q
:=
·
,
q
j=1 Γ (βj + Bj m) m!
m=0
Aj ∈ R+ (j = 1, . . . , p) ; Bj ∈ R+ (j = 1, . . . , q) ; 1 +
q
X
j=1
Bj −
p
X
!
Aj > 0 ,
j=1
so that, obviously,
(2.5)
p Ψq
[(α1 , 1) , . . . , (αp , 1) ; (β1 , 1) , . . . , (βq , 1) ; z]
=
Γ (α1 ) · · · Γ (αp )
p Fq (α1 , . . . , αp ; β1 , . . . , βq ; z) .
Γ (β1 ) · · · Γ (βq )
In its special case when
γα = q
(q ∈ N := {1, 2, 3, . . .}) ,
we can apply the Gauss-Legendre multiplication formula [21, p. 23, Equation 1.1 (27)]:
m
Y
1
1
j
−
1
(1−m)
mz−
2
(2.6)
Γ (mz) = (2π) 2
m
Γ z+
m
j=1
1
2
z ∈ C \ 0, − , − , . . . ; m ∈ N
m m
J. Inequal. Pure and Appl. Math., 5(2) Art. 45, 2004
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M ATHIEU ’ S S ERIES AND I TS G ENERALIZATIONS
5
on the right-hand side of our integral representation (2.3). We thus find that
β
xq[µ− α ]−1
(2.7)
ex − 1
0
q β
x
2
· 1 Fq µ; ∆ q; q µ −
; −r
dx
α
q
β
−1
+
r, α, β ∈ R ; µ − > q ; q ∈ N ,
α
where, for convenience, ∆ (q; λ) abbreviates the array of q parameters
Sµ(α,β)
q/α ∞
=
r; k
k=1
2
Γ q µ − αβ
Z
∞
λ+q−1
λ λ+1
,
,...,
(q ∈ N) .
q
q
q
For q = 2, (2.7) can easily be simplified to the form:
Z ∞ 2[µ−(β/α)]−1
x
2
(α,β)
2/α ∞
(2.8) Sµ
r; k
=
k=1
Γ (2 [µ − (β/α)]) 0
ex − 1
β
β 1 r 2 x2
· 1 F2 µ; µ − , µ − + ; −
dx
α
α 2
4
1
β
+
.
r, α, β ∈ R ; µ − >
α
2
A further special case of (2.8) can be deduced in terms of the Bessel function Jν (z) of order
ν:
(2.9)
ν+2m
∞
X
(−1)m 12 z
Jν (z) : =
m! Γ (ν + m + 1)
m=0
ν
1
z
z2
2
=
; ν + 1; −
.
0 F1
Γ (ν + 1)
4
Thus, by setting β = 21 α and µ 7→ µ + 1 in (2.8), and applying (2.9) as well as (2.6) with m = 2,
we obtain the following known result [3, p. 3, Theorem 2.1]:
∞ (α,α/2)
(2,1)
(2.10)
Sµ+1
r; k 2/α k=1 = Sµ+1 (r; {k}∞
k=1 ) = Sµ+1 (r)
√
Z ∞ µ+ 1
x 2
π
Jµ− 1 (rx) dx
=
1
x
2
(2r)µ− 2 Γ (µ + 1) 0 e − 1
r, µ ∈ R+ .
In a similar manner, a limit case of (2.8) when β → 0 would formally yield the formula:
∞
X
2
(α,0)
2/α ∞
(2.11)
Sµ
r; k
=
k=1
2
(n + r2 )µ
n=1
√
Z ∞ µ− 1
x 2
2 π
Jµ− 1 (rx) dx
=
1
x
2
(2r)µ− 2 Γ (µ) 0 e − 1
1
+
r∈R ; µ>
,
2
J. Inequal. Pure and Appl. Math., 5(2) Art. 45, 2004
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6
H.M. S RIVASTAVA AND Ž IVORAD T OMOVSKI
which is, in fact, equivalent to the following 1906 result of Willem Kapteyn (1849-1927) [25,
p. 386, Equation 13.2 (9)]:
Z
∞
(2.12)
0
∞
xν
1 X
1
(2λ)ν
Jν (λx) dt = √ Γ ν +
πx
2 n=1 (n2 π 2 + λ2 )ν+ 12
e −1
π
(R (ν) > 0; |J (λ)| < π) .
Furthermore, a rather simple consequence of (2.11) or (2.12) in the form:
(2.13)
√
Z ∞ s− 1
x 2
1
2 π
−2s
Js− 1 (cx) dx
=
c
+
s
1
x
2
(n2 + c2 )
(2c)s− 2 Γ (s) 0 e − 1
n=−∞
1
R (s) > ; |c| < 1 .
2
∞
X
appears erroneously in the works by (for example) Hansen [11, p. 122, Entry (6.3.59)] and
Prudnikov et al. [15, p. 685, Entry 5.1.25.1]. And, by making use of the Trigamma function
ψ 0 (z) defined, in general, by [20, p. 22, Equation 1.2 (52)]
dm+1
dm
{log
Γ
(z)}
=
{ψ (z)}
dz m+1
dz m
−
m ∈ N0 := N ∪ {0} ; z ∈ C \ Z−
;
Z
:=
{0,
−1,
−2,
.
.
.}
0
0
ψ (m) (z) :=
(2.14)
or, equivalently, by
(2.15)
ψ
(m)
m+1
(z) := (−1)
∞
X
1
m+1
m! ζ (m + 1, z)
m+1 =: (−1)
(k + z)
k=0
m ∈ N; z ∈ C \ Z−
0
m!
in terms of the Hurwitz (or generalized) Zeta function ζ (s, a) [20, p. 88, Equation 2.2 (1)
et seq.], both Hansen [11, p. 111, Entry (6.1.137)] and Prudnikov et al. [15, p. 687, Entry
5.1.25.28] have recorded the following explicit evaluation of the classical Mathieu series:
(2.16)
S (r) :=
∞
X
k=1
ψ 0 (−ir) − ψ 0 (ir)
2k
=
2ir
(k 2 + r2 )2
i :=
√
−1 .
We remark in passing that, in light of one of the familiar relationships:

r
cos z
2 
(2.17)
J∓ 1 (z) =
·
,
2
πz  sin z
a special case of (2.10) when µ = 1 would immediately yield the well-exploited integral representation (1.4).
J. Inequal. Pure and Appl. Math., 5(2) Art. 45, 2004
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M ATHIEU ’ S S ERIES AND I TS G ENERALIZATIONS
7
Next, in the theory of Bessel functions, it is fairly well known that (cf., e.g., [7, p. 49,
Equation 7.7.3 (16)])
Z ∞
(2.18)
e−st tλ−1 Jν (ρt) dt
0

ρ ν
Γ (ν + λ)

=
s−λ
2 F1
2s
Γ (ν + 1)
1
2
(ν + λ) , 12 (ν + λ + 1) ;
2
−
ν + 1;

ρ 
s2
(R (s) > |J (ρ)| ; R (ν + λ) > 0) .
Since
(2.19)
1 F0
; z) = (1 − z)−λ
(λ;
(|z| < 1; λ ∈ C) ,
the integral formula (2.18) would simplify considerably when λ = ν + 1 and when λ = ν + 2,
giving us [see also Equations (2.10) and (2.11) above]
Z ∞
Γ ν + 12
(2ρ)ν
−st ν
e t Jν (ρt) dt = √ ·
(2.20)
π (s2 + ρ2 )ν+ 12
0
1
R (s) > |J (ρ)| ; R (ν) > −
2
and
Z
∞
e
(2.21)
0
Γ ν + 32
2s (2ρ)ν
t
Jν (ρt) dt = √
·
3
π
(s2 + ρ2 )ν+ 2
(R (s) > |J (ρ)| ; R (ν) > −1) ,
−st ν+1
respectively. While each of the special cases (2.20) and (2.21), too, together with the parent
formula (2.18), are readily accessible in many different places in various mathematical books
and tables (cf., e.g., [26, p. 72]), (2.20) appears slightly erroneously in [7, p. 49, Equation 7.7.3
(17)]. The integral formula (2.21) would follow also when we differentiate both sides of (2.20)
partially with respect to the parameter s.
Now we turn once again to our definition (1.8) which, for α = 2, yields
∞
X
2aβn
∞
(2,β)
+
(2.22)
Sµ (r; {ak }k=1 ) =
r,
β,
µ
∈
R
.
(a2n + r2 )µ
n=1
Making use of the integral formulas (2.20) and (2.21), we find from (2.16) that
!
√
Z ∞ X
∞
1
π
2
∞
(2.23)
aβn e−an x xµ− 2 Jµ− 1 (rx) dx
Sµ(2,β) (r; {ak }k=1 ) =
µ− 12
2
(2r)
Γ (µ) 0
n=1
r, β, µ ∈ R+
and
(2.24)
√
Sµ(2,β) (r; {ak }∞
k=1 ) =
µ− 32
(2r)
Z
π
Γ (µ)
∞
0
r, β, µ ∈ R
J. Inequal. Pure and Appl. Math., 5(2) Art. 45, 2004
∞
X
!
aβ−1
e−an x
n
1
xµ− 2 Jµ− 3 (rx) dx
2
n=1
+
,
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8
H.M. S RIVASTAVA AND Ž IVORAD T OMOVSKI
respectively.
A special case of the integral representation (2.24) when
β=1
and
µ 7−→ µ + 1
was given by Cerone and Lenard [3, p. 9, Equation (4.5)].
Finally, in view of the Eulerian integral formula:
Z ∞
Γ (λ)
(2.25)
e−st tλ−1 dt = λ
(R (s) > 0; R (λ) > 0) ,
s
0
we find from the definition (1.8) that
Sµ(α,β)
(2.26)
Z ∞
2
2
=
xµ−1 e−r x ϕ (x) dx
Γ (µ) 0
r, α, β, µ ∈ R+ ,
(r; {ak }∞
k=1 )
where, for convenience,
ϕ (x) :=
(2.27)
∞
X
aβn exp (−aαn x) .
n=1
In terms of the generalized Mathieu series Sµ (r) defined by (1.7), a special case of the integral representation (2.26) when
α = 2,
β = 1,
and
ak = k
(k ∈ N) ,
was given by Tomovski and Trenčevski [23, p. 6, Equation (2.3)].
3. B OUNDS D ERIVABLE FROM
THE I NTEGRAL
R EPRESENTATION (2.8)
For the generalized hypergeometric p Fq function of p numerator and q denominator parameters, which is defiend by (2.4) and (2.5), we first recall here the following equivalent form of a
familiar Riemann-Liouville fractional integral formula (cf., e.g., [8, p. 200, Entry 13.1 (95)]:
(3.1)
p+1 Fq+1
(ρ, α1 , . . . , αp ; ρ + σ, β1 , . . . , βq ; z)
Z 1
Γ (ρ + σ)
tρ−1 (1 − t)σ−1 p Fq (α1 , . . . , αp ; β1 , . . . , βq ; zt) dt
=
Γ (ρ) Γ (σ) 0
(p 5 q + 1; min {R (ρ) , R (σ)} > 0; |z| < 1 when p = q + 1) ,
which, for
p=q−1=0
β
β1 = µ −
α
J. Inequal. Pure and Appl. Math., 5(2) Art. 45, 2004
,
ρ = µ,
σ=
1 β
− ,
2 α
and z = −
r 2 x2
,
4
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M ATHIEU ’ S S ERIES AND I TS G ENERALIZATIONS
9
immediately yields
(3.2)
1 F2
β 1 r 2 x2
β
µ; µ − , µ − + ; −
α
α 2
4
µ−(β/α)−1
Γ µ − αβ Γ µ − αβ + 12
2
=
β
1
rx
Γ (µ) Γ 2 − α
Z 1 √ µ+(β/α)−1
√
1
·
t
(1 − t)−(β/α)− 2 Jµ−(β/α)−1 rx t dt
0
β
1
r, x, µ ∈ R ;
<
α
2
+
.
In terms of the Lommel function sµ,ν (z) of the first kind, defined by [7, p. 40, Equation 7.5.5
(69)]
1
1
3 1
1
3 z2
z µ+1
,
(3.3)
sµ,ν (z) =
1 F2 1; µ − ν + ; µ + ν + ; −
(µ − ν + 1) (µ + ν + 1)
2
2
2 2
2
2
4
the special case µ = 1 of (3.2) can be found recorded as a Riemann-Liouville fractional integral
formula by Erdélyi et al. [8, p. 194, Entry 13.1 (64)] (see also [8, p. 195, Entry 13.1 (65)]).
Now we turn to a recent investigation by Landau [12] in which several best possible uniform
bounds for the Bessel functions were obtained by using monotonicity arguments. Following
also the work of Cerone and Lenard [3, Section 3], we choose to recall here two of Landau’s
inequalities given below. The first inequality:
|Jν (x)| 5
(3.4)
bL
ν 1/3
holds true uniformly in the argument x and is the best possible in the exponent 13 , with the
constant bL given by
(3.5)
bL = 21/3 sup {Ai (x)} ∼
= 0.674885 . . . ,
x
where Ai (z) denotes the Airy function satisfying the differential equation:
(3.6)
d2 w
− zw = 0
dz 2
w = Ai (z) .
The second inequality:
|Jν (x)| 5
(3.7)
cL
x1/3
holds true uniformly in the order ν ∈ R+ and is the best possible in the exponent 31 , with the
constant cL given by
(3.8)
cL = sup x1/3 J0 (x) ∼
= 0.78574687 . . . .
x
J. Inequal. Pure and Appl. Math., 5(2) Art. 45, 2004
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10
H.M. S RIVASTAVA
AND
Ž IVORAD T OMOVSKI
By appealing appropriately to the bounds in (3.4) and (3.7), we find from (3.2) that
β
β 1 r2 x2 (3.9) 1 F2 µ; µ − , µ − + ; −
α
α 2
4 µ−(β/α)−1 − 13
β
Γ µ − αβ Γ µ − αβ + 12 Γ µ2 + 2α
+ 21
2
β
·
5 bL
µ− −1
β
rx
α
Γ (µ) Γ µ2 − 2α
+1
β
β
β
1
+
r, x ∈ R ; µ − > 1; µ + > −1;
<
α
α
α
2
and
2 2 β
β
1
r
x
(3.10) 1 F2 µ; µ − , µ − + ; −
α
α 2
4 µ−(β/α)−1
β
Γ µ − αβ Γ µ − αβ + 12 Γ µ2 + 2α
+ 13
cL
2
5
·
β
Γ (µ) Γ µ2 − 2α
+ 56
(rx)1/3 rx
β
β
2 β
1
+
r, x ∈ R ; µ − > 1; µ + > − ;
<
,
α
α
3 α
2
where bL and cL are given by (3.5) and (3.8), respectively.
Finally, we apply the inequalities (3.9) and (3.10) in our integral representation (2.8). We
thus obtain the following bounds for the generalized Mathieu series occurring in (2.8):
(3.11)
√
bL π
− 13
β
µ− −1
α
(2r)µ−(β/α)−1
β
Γ µ − αβ + 1 Γ µ2 + 2α
+ 21
β
·
ζ µ− +1
β
α
Γ (µ) Γ µ2 − 2α
+1
1
β
+ β
r, x, α, β ∈ R ;
< ; µ− >1
α
2
α
Sµ(α,β)
2/α ∞
r; k
5
k=1
Sµ(α,β)
2/α ∞
r; k
5
k=1
and
(3.12)
√
cL π
2
2µ−(β/α)−1 rµ−(β/α)− 3 Γ µ − αβ + 23 Γ
·
Γ (µ) Γ µ2 −
µ
β
+ 2α
+ 21
2
β
+1
2α
β 2
ζ µ− +
α 3
β
1
β
r, x, α, β ∈ R ;
< ; µ− >1 ,
α
2
α
where we have employed the integral representation (2.2) for the Riemann Zeta function ζ (s),
+
bL and cL being given (as before) by (3.5) and (3.8), respectively.
In their special case when
1
β −→ α
and
µ 7−→ µ + 1,
2
the bounds in (3.11) and (3.12) would correspond naturally to those given earlier by Cerone
and Lenard [3, p. 7, Theorem 3.1]. The second bound asserted by Cerone and Lenard [3, p.
J. Inequal. Pure and Appl. Math., 5(2) Art. 45, 2004
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M ATHIEU ’ S S ERIES AND I TS G ENERALIZATIONS
11
7, Equation (3.12)] should, in fact, be corrected to include Γ (µ + 1) in the denominator on the
right-hand side.
4. I NEQUALITIES A SSOCIATED
WITH
G ENERALIZED M ATHIEU S ERIES
We first prove the following inequality which was recently posed as an open problem by Qi
[17, p. 7, Open Problem 2]:
Z ∞
2
Z ∞
x sin (rx)
2
2
(4.1)
dx > 2r
x2 e−r x f (x) dx
x
e −1
0
0
!
∞
X
2
r ∈ R+ ; f (x) :=
ne−n x ,
n=1
which, in view of the integral representation (1.4), is equivalent to the inequality:
Z ∞
2
2
(4.2)
[S (r)] > 2
x2 e−r x f (x) dx,
0
where f (x) is defined as in (4.1).
Proof. Since the infinite series:
∞
X
ne−(n
2 +r 2
)x
n=1
is uniformly convergent when x ∈ R+ , for the right-hand side of the inequality (4.2), we have
!
Z ∞
Z ∞
∞
X
2 +r 2 x
2
−
n
) dx
2
x2 e−r x f (x) dx = 2
x2
ne (
0
0
=2
=4
∞
X
n=1
∞
X
n=1
n=1
Z
∞
n
x2 e−(n
2 +r 2
)x dx
0
n
=: 2S3 (r) ,
(n2 + r2 )3
where we have used the Eulerian integral formula (2.25). Hence it is sufficient to prove the
following inequality:
(4.3)
[S (r)]2 > 2S3 (r) ,
which was, in fact, conjectured by Alzer and Brenner [2] and proven by Wilkins [27] by remarkably applying series and integral representations for the Trigamma function ψ 0 (z) defined
by (2.14) for m = 1.
We conclude our present investigation by remarking that it seems to be very likely that the
inequality (4.1) can be generalized to the following form:
J. Inequal. Pure and Appl. Math., 5(2) Art. 45, 2004
http://jipam.vu.edu.au/
12
H.M. S RIVASTAVA
AND
Ž IVORAD T OMOVSKI
Open Problem. Prove or disprove that
Z ∞
µ
Z ∞
x sin (rx)
2
µ
(4.4)
dx > r Γ (µ + 1)
xµ e−r x f (x) dx
x
e −1
0
0
!
∞
X
2
r, µ ∈ R+ ; f (x) :=
ne−n x
n=1
or, equivalently, that
{Γ (µ + 1)}2
Sµ+1 (r)
[S (r)] >
2 r, µ ∈ R+ ,
µ
(4.5)
since
Z
(4.6)
∞
xµ e−r
2x
f (x) dx = Γ (µ + 1)
0
∞
X
n=1
n
Γ (µ + 1)
=:
Sµ+1 (r) ,
µ+1
2
(n2 + r2 )
by virtue of the Eulerian integral formula (2.25) once again.
The open problem (4.1), which we have completely solved here, corresponds to the special
case µ = 2 of the Open Problem (4.4) posed in this paper.
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