M athematical
I nequalities
& A pplications
Volume 20, Number 4 (2017), 973–986
doi:10.7153/mia-20-61
SOME FAMILIES OF GENERALIZED MATHIEU–TYPE
POWER SERIES, ASSOCIATED PROBABILITY
DISTRIBUTIONS AND RELATED INEQUALITIES INVOLVING
COMPLETE MONOTONICITY AND LOG–CONVEXITY
Ž IVORAD T OMOVSKI AND K HALED M EHREZ
Dedicated to Professor Neven Elezović
on the occasion of his sixtieth birthday
(Communicated by M. Praljak)
Abstract. By making use of the familiar Mathieu series and its generalizations, the authors derive
a number of new integral representations and present a systematic study of probability density
functions and probability distributions associated with some generalizations of the Mathieu series. In particular, the mathematical expectation, variance and the characteristic functions, related
to the probability density functions of the considered probability distributions are derived. As a
consequence, some interesting inequalities involving complete monotonicity and log-convexity
are derived.
1. Introduction
The following familiar infinite series
∞
S (r) =
2n
∑ (n2 + r2)2
n=1
r ∈ R+
(1)
is named after Emile Leonard Mathieu (1835–1890), who investigated it in his 1890
work [13] on elasticity of solid bodies. Let C, R+ , N and Z−
0 be the sets of complex
numbers, positive real numbers, positive integers, and non-positive integers, respectively. Integral representations of (1) is given by (see [9])
S (r) =
1
r
Z∞
0
t sin (rt)
dt.
et − 1
(2)
Mathematics subject classification (2010): 3B15, 33E20, 60E10, 11M35.
Keywords and phrases: Mathieu series, integral representations, probability distributions, inequality,
Hurwitz-Lerch zeta function, Mittag-Leffler function, characteristic function, completely monotonic function,; log-convex function, Turán type inequality.
c D l , Zagreb
Paper MIA-20-61
973
974
Ž. T OMOVSKI AND K. M EHREZ
Several interesting problems and solutions dealing with integral representations and
bounds for the following slight generalization of the Mathieu series with a fractional
power
∞
2n
r ∈ R+ ; µ > 1
Sµ (r) = ∑ 2
(3)
µ
2
n=1 (n + r )
can be found in the works by Cerone and Lenard [4], Diananda [7], and Tomovski
and Trencevski [21]. Motivated essentially by the works of Cerone and Lenard [4],
Srivastava and Tomovski in [18] defined a family of generalized Mathieu series
(α ,β )
Sµ
(α ,β )
(r;a) = Sµ
(r; {ak }∞
k=1 ) =
β
∞
2an
r, α , β , µ ∈ R+
∑ (aα + r2)µ
n=1
n
(4)
where it is tacitly assumed that the positive sequence
=
∞
=
{a
,
a
,
a
,
.
.
.}
lim
a
a = {an }∞
n
1 2 3
n=1
n→∞
is so chosen that the infinite series in definition (4) converges, that is, that the following
auxiliary series
∞
1
∑
µα −β
n=1 an
is convergent. Comparing the definitions (1), (3) and (4), we see that S2 (r) = S (r) and
(2,1)
(2,1)
Sµ (r) = Sµ (r, {n}∞
(r; {an }∞
n=1 ) , S µ (r) =
n=1 ). Furthermore, the special cases S2
(2,1)
(α ,α /2)
(2,1)
(r; {nγ }∞
(r; {n}∞
Sµ (r; {n}∞
n=1 ) , S µ
n=1 ) and S µ
n=1 ) were investigated by Cerone-Lenard [4], Diananda [7]; and Tomovski [22]. For more details the interested
reader is referred to the papers [8, 15, 18, 5, 20, 21, 22, 23, 24, 25].
In this paper we consider a power series
(α ,β )
(α ,β )
Sµ ,ν (r,a; z) = Sµ ,ν (r, {ak }∞
k=1 ; z) =
∞
β
2an (ν ) zn
∑ (aα + r2)nµ n!
n=1
(5)
n
r, α , β , µ ∈ R+ ; |z| 6 1 .
We denote
(α ,β )
(α ,β )
Sµ ,ν (r,a; 1) ≡ Sµ ,ν (r,a) ,
(α ,β )
(α ,β )
Sµ ,ν (r,a; −1) ≡ Seµ ,ν (r,a) ,
(α ,β )
(α ,β )
Sµ ,1 (r,a; 1) ≡ Sµ
(r,a)
(α ,β )
(α ,β )
S µ ,1 (r,a; −1) ≡ Seµ
(r,a)
(6)
(7)
For an = n , α = 2 , β = 1 , ν = 1 and µ with µ + 1 the series (5) was introduced
and considered by Tomovski and Pogany in [25].
This paper is organized as follows: in Sections 2 and 3, we present new integral
and series representations for generalised Mathieu series. In particular, we present a
new type integral for the Mathieu series for a special case. In section 4, we introduce,
develop and investigate probability distribution functions (PDF) associated with the
975
S OME FAMILIES OF GENERALIZED M ATHIEU - TYPE POWER SERIES
Mathieu series and their generalizations. As consequences some inequalities are derived. In Section 5, we show the complete monotonicity and log-convexity properties
for generalised Mathieu series. Moreover, as consequences of these results, we presented some functional inequalities as well as lower and upper bounds for generalised
Mathieu series.
2. Integral expression of Mathieu series
Our first main results is the next theorem.
T HEOREM 1. Let r, α , β , ν , µ > 0 and γ (µα − β ) > 1 . Then the Mathieu type
(α ,β )
power series Sµ ,ν (r, {kγ }∞
k=1 ; z) has the integral representation
(α ,β )
Sµ ,ν (r, {kγ }∞
k=1 ; z) =
2ν z
Γ (µ )
Z ∞ γ [µα −β ]−1 −t
e
t
1Ψ1
(1−ze−t )ν +1
0
(µ , 1) ; (γ ( µα −β ) , γα ) ; −r2t γα dt,
(8)
where p Ψq denotes the Fox-Wright generalization of the hypergeometric p Fq function
with p numerator and q denominator parameters, defined by [18, p. 50, Eq. 1.5 (21)]
p
∞
p Ψq [(α1 , A1 ) , . . . , (α p , A p ) ; (β1 , B1 ) , . . . , (βq , Bq ) ; z] =
∏ j=1 Γ(α j +nA j ) zn
∑ ∏q Γ(β j +nB j ) . n! , (9)
j=1
n=0
with
p
q
+
+
!
A j ∈ R , j = 1, . . . , p; B j ∈ R , j = 1, . . . , q; 1 + ∑ B j − ∑ A j > 0 .
j=1
j=1
Proof. First of all, we find from the definition (5) that
(α ,β )
Sµ ,ν (r, {ak }∞
k=1 ; z) = 2
∞
∑
m=0
−r2
µ +m−1
m
So,
(α ,β )
Sµ ,ν (r, {kγ }∞
k=1 ; z) = 2z
∞
∑
m=0
∞
= 2ν z
= 2ν z
∑
m=0
∞ ∑
m=0
−r2
µ +m−1
m
µ +m−1
m
µ +m−1
m
m
−r
−r
m
∞
(ν )n zn
(µ +m)α −β
n!
n=1 an
∞
∑
(ν )
(10)
zn
∑ (n + 1)[(µ +m)n+1
α −β ]γ
(n + 1)!
n=0
∞
2 m
2 m
(ν + 1) zn
∑ (n + 1)[(µ +m)αn−β ]γ +1 n!
n=0
Φν∗ +1 (z, [(µ + m) α − β ] γ + 1, 1).
Here Φ∗ν +1 denotes a Hurwitz-Lerch Zeta function, defined by Lin and Srivastava [12]
Φν∗ (z, s, a) =
∞
(ν )n zn
s
n=0 n! (n + a)
∑
976
Ž. T OMOVSKI AND K. M EHREZ
ν ∈ C; a ∈ C \ Z−
0 , s ∈ C, |z| < 1; ℜ (s − ν ) > 1 when |z| = 1 .
A special case of Φν∗ with ν = 1 give us the well-known Hurwitz-Lerch Zeta function
∞
Φ (z, s, a) =
zn
∑ (n + a)s
n=0
a∈
C \ Z−
0,
s ∈ C, |z| < 1; ℜ (s) > 1 when |z| = 1 .
Now, by making use of the familiar integral representation (see [12])
Φν∗ (z, s, a) =
1
Γ (s)
Z ∞ s−1 −at
t e
(1 − ze−t )ν
0
dt
(ℜ (a) > 0; ℜ (s) > 0 when |z| 6 1, z 6= 1; ℜ (s) > 1 when z = 1)
we get
(α ,β )
Sµ ,ν (r, {kγ }∞
k=1 ; z)
µ +m−1
m
∞
= 2ν z
∑
−r2
m Z
∞ t γ [( µ +m)α −β ]−1 e−t
dt
Γ (γ [(µ + m) α − β ])
0
(1 − ze−t )ν +1
"
#
Z ∞ γ [µα −β ]−1 −t
µ +m−1
∞
e
t
2 γα m
m
−r t
dt
= 2ν z
ν +1 ∑ Γ (γ [(µ + m) α − β ])
0 (1 − ze−t )
m=0
m=0
=
2ν z
Γ (µ )
Z ∞ γ [µα −β ]−1 −t
e
t
0
This ends the proof.
1 Ψ1
(1 − ze−t )ν +1
(µ , 1) ; (γ (µα − β ), γα ) ; −r2t γα dt.
3. Concluding Remarks
(α ,β )
1. As the convergence interval of Sµ ,ν (r, {kγ }∞
k=1 ; z) is [−1, 1], we easily conclude the following representations
(α ,β )
Sµ ,ν (r, {kγ }) ≡
2ν
Γ (µ )
Z ∞ γ [µα −β ]−1 −t
e
t
(µ , 1) ; (γ (µα − β ) , γα ) ; −r2t γα dt,
Z ∞ γ [µα −β ]−1 −t
t
e
(µ , 1) ; (γ (µα − β ) , γα ) ; −r2t γα dt.
0
1 Ψ1
(1 − e−t )ν +1
(11)
and
(α ,β )
Seµ ,ν (r, {kγ }) ≡
2ν
Γ (µ )
0
1 Ψ1
(1 + e−t )ν +1
(12)
2. Specially, for z = e−x < 1, we find that
(α ,β )
Sµ ,ν
2ν e−x Z ∞ t γ (µα −β )−1 e−t
−x
=
r, {kγ }∞
k=1 ; e
Γ ( µ ) 0 1 − e−(x+t) ν +1
× 1 Ψ1 ( µ , 1) ; (γ ( µα − β ), γα ) ; −r2t γα dt,
(13)
S OME FAMILIES OF GENERALIZED M ATHIEU - TYPE POWER SERIES
977
and
2ν e−x Z ∞ t γ (µα −β )−1 e−t
(α ,β )
−x
;
e
=
Seµ ,ν r, {kγ }∞
k=1
Γ ( µ ) 0 1 + e−(x+t) ν +1
× 1 Ψ1 ( µ , 1) ; (γ ( µα − β ), γα ) ; −r2t γα dt.
(14)
3. In a similar manner, we get
n
o∞
(α ,β )
Sµ ,ν r, kq/α
;z
k=1
∞ β
2 m ∗
µ
+m−1
γ,1
= 2ν ∑
Φν +1 z, q (µ + m) −
−r
m
α
m=0


i
h
µ +m−1
Z ∞ q µ − αβ −1 −t
∞
e 
t
m
m
i −r2t q 
h
= 2ν
 dt
ν +1  ∑
β
0 (1 − ze−t )
m=0 Γ q (µ + m) − α
=
2ν
h
i
Γ q µ − αβ
h
i
Z ∞ q µ − αβ −1 −t
e
t
0
(1 − ze−t )ν +1
q β
2 t
µ
;
∆
q;
q
µ
−
dt.
F
;
−r
1 q
α
q
β
+
−1
r, α , β , ν ∈ R ; µ − > q ; q ∈ N ,
α
where for convenience, ∆ (q; λ ) abbreviates the array of q parameters
λ λ +1
λ +q−1
,
,...,
(q ∈ N) .
q
q
q
4. For q = 2 , this integral representation, can easily be simplified to the form:
n
o∞
(α ,β )
;z =
Sµ ,ν r, k2/α
2ν
h
i
Z ∞ 2 µ − αβ −1 −t
t
e
i
h
ν +1
0 (1 − ze−t )
Γ 2 µ − αβ
β
β 1 r2 t 2
dt,
× 1 F2 µ ; µ − , µ − + ; −
α
α 2
4
β
1
+
r, α , β , ν ∈ R ; µ − >
.
α
2
k=1
5. In a similar manner, a limit case, when β → 0 would formally yield the formula:
Z ∞
n
o∞
2ν
1 r2 t 2
t 2µ −1 e−t
(α ,0)
2/α
;z =
dt
Sµ ,ν r, k
0 F1 −; , µ + ; −
k=1
Γ (2µ ) 0 (1 − ze−t )ν +1
2
4
1
r, α , ν ∈ R+ ; µ >
.
2
978
Ž. T OMOVSKI AND K. M EHREZ
6. On the other hand, we have
(2,1)
S2,ν (r, {k}∞
k=1 ; z) =
=
∞
∑
[(n + ir) + (n − ir)] (ν )n zn
[(n + ir)(n − ir)]2 n!
n=1
1
[Φ∗ (z, 2, ir) − Φν∗ (z, 2, −ir)]
2ir ν
(15)
As a matter of fact, in terms of the Riemann-Liouville fractional derivative operator
Dνz , defined by
Dzµ { f
(z)} =
(
R
z
1
(z − t)−µ −1 f (t) dt (ℜ ( µ ) < 0)
nΓ(−µ ) 0 o
m
µ −m
d
f (z) (m − 1 6 ℜ (µ ) < m) (m ∈ N)
dzm Dz
it is easily seen from the series definitions in (5) and (15) that
Φ∗ν (z, s, a) =
we obtain
(2,1)
S2,ν (r; {n}∞
n=1 ; z) =
1
Dνz −1 zν −1 Φ (z, s, a)
Γ (ν )
ν −1 ν −1
1
Dz
z Φ (z, 2, ir) − Dνz −1 zν −1 Φ (z, 2, −ir) ,
2irΓ (ν )
(16)
where 0 6 Re(ν ) < 1.
7. In [10], the authors defined generalized β -Mittag-Leffler functions
∞
Eβ ,ν ,γ (x) =
xk
∑ [Γ(ν k + γ )]β .
(17)
k=0
For ν = γ = 1 and β ∈ N, we get the hyper-Bessel function. We define a new family
of β -Mittag-Leffler functions
∞
(τ )
Eβ ,ν ,γ (x) =
(τ )k xk
∑ k![Γ(ν k + γ )]β .
(18)
k=0
(1)
Specially, for τ = 1, we obtain Eβ ,ν ,γ (x) = Eβ ,ν ,γ (x).
(α ,β )
The Mathieu series Sµ ,ν (r, {Γ(γ n + δ )}∞
n=1 , z) admits the following series representation:
(α ,β )
S µ ,ν (r, {Γ(γ n + δ )}∞
n=1 , z)
∞ ∞
(ν )n zn
m
= ∑ µ +m−1 −r2 ∑
( µ +m)α −β
m
n!
m=0
n=1 [Γ(γ n + δ )]
"
#
∞ m (ν )
1
2
E(µ +m)α −β ,γ ,δ (z) −
= ∑ µ +m−1 −r
.
m
[Γ (δ )](µ +m)α −β
m=0
979
S OME FAMILIES OF GENERALIZED M ATHIEU - TYPE POWER SERIES
(α ,β )
8. The Mathieu series Sµ ,1 (r, {Γ(γ n + δ )}∞
n=1 , z) admits the following series
representation:
∞ ∞
zn
(α ,β )
2 m
µ +m−1
,
z)
=
Sµ ,1 (r, {Γ(γ n + δ )}∞
−r
∑ m
∑
n=1
( µ +m)α −β
m=0
n=1 [Γ(γ n + δ )]
∞ m ∞
zn
= z ∑ µ +m−1 −r2 ∑
( µ +m)α −β
m
n=0 [Γ(γ n + γ + δ )]
m=0
∞ m
= z ∑ µ +m−1 −r2 E(µ +m)α −β ,γ ,γ +δ (z).
m
m=0
(α ,β )
In the next Theorem we give an integral expression of Mathieu series S3/2,1 (r; a; 1),
by using the Fourier transform.
T HEOREM 2. Let r, α , β > 0 and a = (ak )k>1 be a sequences such that the function
(α ,β )
fr
∞
(t) =
2(aαn t 2 − r2 )
∑
α −β
2
(19)
(aαn t 2 + r2 )2
n=1 an
converges for all t > 0. Then we have
(α ,β )
S3/2,1 (r; a, 1)
2
=
π
Z ∞ p
t
1
(α ,β )
t 2 − 1 fr
(t)dt.
Proof. In [26, Eq. 6.6], the author give the Fourier transform of the function
1
ϕc,µ (x) = (c2 +x
2 )µ with c > 0 and µ > 1/2 :
F ϕc,µ (ξ ) =
21−µ
|ξ |µ −1/2 Kµ −1/2 (c |ξ |),
Γ(µ )cµ −1/2
(20)
where Kα is the modified Bessel function of the second kind. On the other hand, using
the representation integral (see for example [2, Theorem 4.17]),
x α Z +∞
α − 12
1
1
dt, α > − , x > 0. (21)
e−xt t 2 − 1
Kα (x) = √
1
2
π Γ(α + 2 ) 2
1
and Fubini’s Theorem that the even function xα Kα (x) belongs to L1 [0, ∞). So, by using
the inversion formula and (20) we deduce that
21−µ
ϕc,µ (x) = √
F |ξ |µ −1/2 Kµ −1/2 (c |ξ | (x),
2π Γ(µ ) cµ −1/2
(22)
Z ∞
22−µ
µ −1/2
cos(xξ )ξ
=√
Kµ −1/2 (cξ )d ξ
2π Γ(µ ) cµ −1/2 0
In view of the representation integral (see [26, Def. 5.10])
Kµ (z) =
Z ∞
0
e−z cosh(t) cosh(µ t)dt,
(23)
980
Ž. T OMOVSKI AND K. M EHREZ
and (22) we obtain
22−µ
ϕc,µ (x) = √
2π Γ(µ )cµ −1/2
Z ∞
0
cosh((µ −1/2)t)
Z
∞
0
cos(xξ )ξ µ −1/2 e−cξ cosh(t) d ξ dt.
(24)
Now, let µ = 3/2. Combining the following formula (see [6, Eq. 2. p. 391])
Z ∞
0
cos(xξ )ξ e−c cosh(t)ξ d ξ =
c2 cosh(t)2 − x2
(c2 cosh(t)2 + x2 )2
(25)
and (24) we obtain
Z
∞
1
(c2 cosh(t)2 − x2 )
ϕc,3/2 (x) = √
dt
cosh(t) 2
(c cosh(t)2 + x2 )2
π Γ(3/2) c 0
Z ∞ p
(c2t 2 − x2 )
1
=√
dt.
t t2 − 1 2 2
(c t + x2 )2
π Γ(3/2) c 1
(26)
α
Letting c = an2 and x = r in the (26), we get
β
2an
2
=
π
(aαn + r2 )3/2
Z ∞ p
β
t2 − 1
t
1
2an (aαn t 2 − r2 )
α /2
an (aαn t 2 + r2 )2
dt.
The interchanging between integral and summation gives the desired result.
4. Mathieu probability distribution
The main objective of this section is to introduce, develop and investigate probability distribution functions (PDF) associated with the Mathieu series and their generalizations. We define a discrete random variable X defined on some fixed standard
probability space (Ω, ̥, P) possesing a Mathieu distribution with parameter r > 0,
(α ,α )
Pµ ,ν
(α ,α )
2nα (ν )n
1
(nα + r2 )µ n! S(µα,ν,α ) (r)
r, α , µ , ν ∈ R+ , n ∈ N,
(n, r) = P (X = n) =
(α ,α )
(α ,α )
where Sµ ,ν (r) = Sµ ,ν (r, {n}∞
n=1 ) . Pµ ,ν
∞
(α ,α )
∑ Pµ ,ν
n=1
(n, r) is normalized, since
(α ,α )
Sµ ,ν (r)
2nα (ν )
1
∑ (nα + r2)µn n! (α ,α ) = (α ,α ) = 1.
Sµ ,ν (r) Sµ ,ν (r)
n=1
∞
(n, r) =
(27)
(α ,α )
T HEOREM 3. The expected value EX of a Mathieu distribution Pµ ,ν
(α ,α +1)
(r)
S µ ,ν
(α ,α )
Sµ ,ν (r)
and variance Var (X) is given by
(α ,α +2)
Sµ ,ν
h
i2
(α ,α )
(α ,α +1)
(r) Sµ ,ν (r) − Sµ ,ν
(r)
.
2
(α ,α )
Sµ ,ν (r)
(n, r) is
981
S OME FAMILIES OF GENERALIZED M ATHIEU - TYPE POWER SERIES
Moreover the following Turán type inequality holds true
2
(α ,α +2)
(α ,α )
(α ,α +1)
Sµ ,ν
(r) Sµ ,ν (r) > Sµ ,ν
(r) .
(28)
Proof. By computation we have
(α ,α +1)
∞
∞
EX =
∑ nP (X = n) =
n=1
Sµ ,ν
(r)
2nα +1 (ν )n
1
= (α ,α )
,
∑ (nα + r2)µ n!
(α ,α )
Sµ ,ν (r)
Sµ ,ν (r)
n=1
and
EX 2 =
∞
∑ n2 P (X = n) =
n=1
(α ,α +2)
∞
(r)
Sµ ,ν
2nα +2 (ν )n
1
∑ (nα + r2)µ n! (α ,α ) = (α ,α ) .
Sµ ,ν (r)
Sµ ,ν (r)
n=1
Thus
(α ,α +2)
Var (X) = EX 2 − (EX)2 =
S µ ,ν
i
h
(α ,α )
(α ,α +1) 2
(r) S µ ,ν (r) − Sµ ,ν
.
2
(α ,α )
Sµ ,ν (r)
Using the fact that Var(x) is nonnegative, we easy get that the Turán type inequality
(28) holds true. (2,1)
T HEOREM 4. The characteristic funtion of the Mathieu distribution Pµ +1,ν (n, r),
r > 0 is given by
√
Z∞ uν µ +1/2
π
e u
1
,
Re (µ ) > −
fx (t) =
ν Jµ −1/2 (ru) du
2
(2r)µ −1/2 Γ (µ + 1)S µ ,ν (r) (eu − eit )
0
(29)
where Jµ −1/2 (.) is the Bessel function.
Proof. Using the formula:
2n
(n2 + r2 )
µ +1
=
√
π
Z∞
e−nt t µ +1/2 Jµ −1/2 (rt) dt
(2r)µ −1/2
0
1
ℜ (µ ) > −
2
we obtain
∞
fx (t) =
2n (ν )
1
∑ eitn (n2 + r2)µn+1 n! Sµ ,ν (r)
n=1
=
√
π
Z∞
µ +1/2
#
(ν )n −u it n
du
e e
Jµ −1/2 (ru) ∑
n=1 n!
"
∞
u
(2r)µ −1/2 Γ ( µ + 1)S µ ,ν (r)
0
√
Z∞ uν µ +1/2
π
e u
=
ν Jµ −1/2 (ru) du.
µ −1/2
Γ ( µ + 1)S µ ,ν (r) (eu − eit )
(2r)
0
982
Ž. T OMOVSKI AND K. M EHREZ
So, the proof of Theorem 4 is complete.
C OROLLARY 1. Let ν > 0 and µ > 1. Then the following inequality holds
In particular,
h
i2
(2,3)
(2,2)
(2,1)
Sµ ,ν (r, {n}) S µ ,ν (r, {n}) > Sµ ,ν (r, {n}) .
(30)
h
i2
(2,1)
(2,3)
(2,2)
S2,ν (r, {n}) S2,ν (r, {n}) > S2,ν (r, {n}) .
Proof. We consider a special case
(2,1)
Pµ ,ν (n, r) = P (Y = n) =
2n (ν )n
2
(n + r2 )µ
1
n!
(2,1)
Sµ ,ν (r)
(31)
r, µ , ν ∈ R+ , n ∈ N.
Then
(2,2)
EY =
Sµ ,ν (r, {n})
EY 2 =
Sµ ,ν (r, {n})
(2,1)
Sµ ,ν (r, {n})
(2,3)
(2,1)
Sµ ,ν (r, {n})
Applying the elementary inequality EY 2 > (EY )2 ,we obtain
h
i2
(2,1)
(2,3)
(2,2)
S µ ,ν (r, {n}) S µ ,ν (r, {n}) > Sµ ,ν (r, {n}) .
T HEOREM 5. Let ν > 1. Then the following inequality
holds true.
h
i2
(2,1)
(2,2)
(2,1)
(2,1)
S1,ν (r, {n}) S2,ν (r, {n}) > 2r2 S3,1 (r, {n}) + S2,1 (r, {n})
(32)
Proof. From Corollary 1, we have
h
i
h
i2
(2,1)
(2,1)
(2,1)
(2,2)
S1,ν (r, {n}) − r2 S2,ν (r, {n}) S2,ν (r, {n}) > S2,ν (r, {n}) .
(33)
i2 h
i2
h
(2,2)
(2,2)
S2,ν (r, {n}) > S2,1 (r, {n}) ,
(35)
On the other hand, using the fact that the function ν 7→ (ν )n is increasing on [1, ∞), we
deduce that
h
i2 h
i2
(2,1)
(2,1)
S2,ν (r, {n}) > S2,1 (r, {n}) ,
(34)
and
S OME FAMILIES OF GENERALIZED M ATHIEU - TYPE POWER SERIES
In [28], Wilkins proved the following inequality
"
#2
∞
∞
n
n
∑ (n2 + r2)2 > ∑ (n2 + r2)3 .
n=1
n=1
Combining (34) and (36) we get
h
i2
(2,1)
(2,1)
S2,ν (r, {n}) > 2S3,1 (r, {n}) .
In view of (37) and (35) we get the inequality (32).
983
(36)
(37)
5. Functional inequalities for Mathieu series
Before we present our main results in this section, we recall some standard definitions and basic facts. A non-negative function f defined on (0, ∞) is called completely
monotonic if it has derivatives of all orders and
(−1)n f (n) (x) > 0, n > 1
and x > 0 [16, 3, 14]. This inequality is known to be strict unless f is a constant. By
the celebrated Bernstein theorem, a function is completely monotonic if and only if it
is the Laplace transform of a non-negative measure [16, Theorem 1.4].
(α ,β ) √
T HEOREM 6. Let α , β , ν , µ > 0 and 06z61 . Then the function r7→S µ ,ν ( r, a, z)
is completely monotonic and log-convex on (0, ∞). In particular, for r1 , r2 , ν > 0, the
following chain of inequalities:
"
!#2
r
r1 + r2
(α ,β )
(α ,β ) √
(α ,β ) √
Sµ ,ν
, a, z
6 Sµ ,ν ( r1 , a, z)S µ ,ν ( r2 , a, z)
2
(38)
√
(α ,β )
6 2ζµ ,ν (α , β , z)S µ ,ν ( r1 + r2 , a, z)
holds true if the following auxiliary series
∞
ζν ,µ (α , β , z) =
∑
n=1
(ν )n zn
α µ −β
n!an
is convergent.
β
n
2an (ν )n z
Proof. As the function r 7→ n!(a
α +r)µ is completely monotonic on (0, ∞) for all
n
α , β , µ , ν > 0. Using the fact that sums of completely monotonic functions are com(α ,β ) √
pletely monotonic too, we deduce that the function r 7→ Sµ ,ν ( r, a, z) is completely
monotonic and log-convex on (0, ∞), since every completely monotonic function is
log-convex (see [27, p. 167]). Thus for all r1 , r2 > 0, and t ∈ [0, 1] we get
!
r
tr1 + (1 − t)r2
(α ,β ) √
(α ,β )
(α ,β ) √
, a, z 6 [Sµ ,ν ( r1 , a, z)]t [Sµ ,ν ( r2 , a, z)]1−t .
Sµ ,ν
(39)
2
984
Ž. T OMOVSKI AND K. M EHREZ
Choosing t = 1/2 , the above inequality reduce to the first inequality in (38). For the
(α ,β ) √
second inequality in (38), we observe that the function r 7→ Sµ ,ν ( r, a, z) is decreasing on (0, ∞), and thus
∞
(ν )n zn
(α ,β ) √
(α ,β )
Sµ ,ν ( r, a, z) 6 S µ ,ν (0, a, z) = 2 ∑
= 2ζµ ,ν (α , β , z).
α µ −β
n=1 n!an
S
(α ,β )
√
( r,a,z)
So, the function r 7→ 2µζ,νµ ,ν (α ,β ,z) maps (0, ∞) into (0, 1) and its completely monotonic
on (0, ∞). On the other hand, according to Kimberling [11], if a function f , defined on
(0, ∞), is continuous and completely monotonic on (0, ∞) into (0, 1), then the log f is
super-additive, that is for all x, y > 0 , we have
log f (x + y) > log f (x) + log f (y) or f (x + y) > f (x) f (y).
Therefore we conclude the second inequality in (38).
L EMMA 1. [1] Let f , g : [a, b] → R, be two continuous functions which are differentiable on (a, b). Further, let g′(x) 6= 0 on (a, b). If f ′ /g′ is increasing (or decreasing) on (a, b), then the functions
f (x) − f (a)
f (x) − f (b)
and
,
g(x) − g(a)
g(x) − g(b)
are also increasing (or decreasing) on (a, b).
C OROLLARY 2. Let α , β , µ > 0 and 0 6 z 6 1. Then the following inequality
2ζµ ,ν (α , β , z)e
ζ
(α ,β ,z)
− µ 2µζ+1,ν(α ,β ,z) r2
µ ,ν
(α ,β )
6 Sµ ,ν (r, a, z)
(40)
holds true if the following auxiliary series
∞
ζµ ,ν (α , β , z) =
∑
(ν )n zn
n=1
α µ −β
n!an
is convergent.
Proof. Since the function r 7→
0, we obtain that r 7→
(α ,β ) √
Sµ ,ν ( r,a,z)
2ζµ ,ν (α ,β ,z)
(α ,β ) √
(Sµ ,ν ( r,a,z))′
(α ,β ) √
Sµ ,ν ( r,a,z)
F(r) = log
is log-convex on (0, ∞) for al α , β , ν >
is increasing on (0, ∞). Let
!
(α ,β ) √
Sµ ,ν ( r, a, z)
, and G(r) = r.
2ζµ ,ν (α , β , z)
Then the function
H(r) =
F(r)
F(r) − F(0)
=
,
G(r) G(r) − G(0)
S OME FAMILIES OF GENERALIZED M ATHIEU - TYPE POWER SERIES
985
is increasing on (0, ∞), by means of Lemma 1. So, by using the l’Hospital’s rule we
get
!
(α ,β ) √
Sµ ,ν ( r, a, z)
ζµ +1,ν (α , β , z)
log
> rF ′ (0) = −r µ
.
(41)
2ζµ ,ν (α , β , z)
2ζµ ,ν (α , β , z)
Therefore we conclude the asserted inequality (40).
R EMARKS .
1. Taking z = β = ν = 1, a = (n)n>1 and α = 2 in (40), we obtain the following
inequality
ζ (2µ + 1) 2
2ζ (2µ − 1) exp −µ
(42)
r 6 Sµ (r), r > 0,
2µ − 1
where ζ (.) denotes the Riemann zeta function defined by
∞
ζ (p) =
1
∑ np .
n=1
2. We note that if we choose µ = 2 and µ = 3/2 in (42) we obtain the following
inequalities
ζ (5) 2
2ζ (3) exp −2
(43)
r 6 S(r),
ζ (3)
and
π2
π 2 r2
6 S3/2 (r),
exp −
3
10
(44)
holds true for all r > 0.
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(Received October 6, 2016)
Živorad Tomovski
University “St. Cyril and Methodius”
Faculty of Natural Sciences and Mathematics, Institute of Mathematics
Republic of Macedonia
and
Department of Mathematics
University of Rijeka
Radmile Matejcic 2, 51000 Rijeka, Croatia
e-mail: [email protected]
Khaled Mehrez
Département de Mathématiques ISSAT Kasserine
Université de Kairouan
Tunisia
and
Département de Mathématiques Faculté de Science de Tunis
Université Tunis El Manar
Tunisia
e-mail: [email protected]
Mathematical Inequalities & Applications
www.ele-math.com
[email protected]