ON SOME TURÁN-TYPE INEQUALITIES
A. LAFORGIA AND P. NATALINI
Received 14 September 2005; Accepted 20 September 2005
We prove Tur
´
an-type inequalities for some special functions by using a generalization of
the Schwarz inequality.
Copyright © 2006 A. Laforgia and P. Natalini. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, dis-
tribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The importance, in many fields of mathematics, of the inequalities of the type
f
n
(x) f
n+2
(x) − f
2
n+1
(x) ≤ 0, (1.1)
where n
= 0,1,2, ,iswellknown.Theyarenamed,byKarlinandSzeg
¨
o, Tur
´
an-type
inequalities because the first of this type of inequalities was proved by Tur
´
an [12]. More
precisely, by using the classical recurrence relation [10, page 81]
(n +1)P


P
n
(x) P
n+1
(x)
P
n+1
(x) P
n+2
(x)





≤
0, −1 ≤ x ≤ 1, (1.4)
where P
n
(x) is the Legendre polynomial of degree n.In(1.4) equality occurs only if
x
=±1. This classical result has been extended in several directions: ultraspherical poly-
nomials, Laguerre and Hermite polynomials, Bessel functions of first kind, modified
Bessel functions, and so forth.
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006, Article ID 29828, Pages 1–6
DOI 10.1155/JIA/2006/29828
2OnsomeTur

(x) and for the zeros of ultraspherical, Laguerre, and
Hermite polynomials have been established in [2, 3, 6], respectively.
Recently, in [7], we have proved Tur
´
an-type inequalities for some special functions, as
well as the polygamma and the Riemann zeta functions, by using the following general-
ization of the Schwarz inequality:

b
a
g(t)

f (t)

m
dt ·

b
a
g(t)

f (t)

n
dt ≥


b
a
g(t)

1
2
s(s
− 1)π
−s/2
Γ

s
2

ζ(s), (1.8)
where ζ is the Riemann ζ-function. This function has the following representation (see
[5]):
ξ

s +
1
2

=
∞

k=0
b
k
s
2k
, (1.9)
where the coefficients b
k

e
−πn
2
e
4t
. (1.11)
A. Laforgia and P. Natalini 3
In [1] the fol l ow ing Tur
´
an-type inequalities were proved:
b
2
k
−
k +1
k
b
k+1
b
k−1
≥ 0, k = 0,1, , (1.12)
which are very important in the theory of the Riemann ξ-function (see [5]).
In the third one, we will use the modified Bessel functions of the third kind K
ν
(x),
x>0, defined as follows:
K
ν
(x) =
π

Theorem 2.1. For n
= 1,2, ,denotebyh
n
=

n
k
=1
(1/k) the partial sum of the harmonic
series. Let
a
n
= h
n
− logn, (2.1)
then

a
n
− γ

a
n+2
− γ

≥

a
n+1
− γ

− 1

dt,Rez>0. (2.5)
By putting z
= n in (2.5), for n = 1,2, ,weobtainfrom(2.4)and(2.5),
n

k=1
1
k
− γ =

∞
0

e
−t
t
−
e
−nt
e
−t
− 1

dt =

∞
0
e

−nt
t
dt
= logn, (2.7)
we have
n

k=1
1
k
− logn − γ =

∞
0
e
t
− 1 − t
t

e
t
− 1

e
−nt
dt. (2.8)
By (1.6)withg(t)
= (e
t
− 1 − t)/t(e


e
−(n+2)t
dt ≥


∞
0
e
t
− 1 − t
t

e
t
− 1

e
−(n+1)t
dt

2
(2.9)
that is the inequality (2.2).

Theorem 2.2. For k = 1,2, ,letb
k
(k = 1,2, ) be the coefficients in (1.9), then
b
2

2k
dt

2
. (2.11)
Dividing (2.11)by(2k)! this inequality becomes
(2k +2)!
(2k)!
b
k+1
(2k − 2)!
(2k)!
b
k−1
≤ b
2
k
, k = 1,2, , (2.12)
from which, since ((2k +2)!/(2k)!)((2k
− 2)!/(2k)!) = ((2k +1)(k +1))/k(2k − 1), we ob-
tain the conclusion of Theorem 2.2.

Remark 2.3. It is important to note that inequalities (1.12)and(2.10)togethergive
k +1
k
b
k+1
b
k−1
≤ b

and a = 0, b = +∞,weget

∞
0
t
m−1
e
−β/t−γt
dt ·

∞
0
t
n−1
e
−β/t−γt
dt ≥


∞
0
t
(m+n)/2−1
e
−β/t−γt
dt

2
. (2.15)
A. Laforgia and P. Natalini 5


βγ

·
K
μ

2

βγ

≥
K
2
(ν+μ)/2

2

βγ

(2.17)
which, putting x
= 2

βγ, is equivalent to the conclusion of Theorem 2.4.
In the particular case μ
= ν +2,wefind
K
ν
(x) · K

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