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
n+1(x) ≤ 0,
(1.1) fn(x) fn+2(x) − f 2
where n = 0,1,2,..., is well known. They are named, by Karlin and Szeg¨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 = 0,1,... (1.2) (n + 1)Pn+1(x) = (2n + 1)xPn(x) − nPn−1(x), P−1(x) = 0, P0(x) = 1
and the differential relation [10, page 83]
(cid:2) 1 − x2
(cid:3) P(cid:3) n(x) = nPn−1(x) − nxPn(x),
(1.3)
he proved the following inequality:
≤ 0, −1 ≤ x ≤ 1,
(cid:4) (cid:4) (cid:4) (cid:4) (cid:4)
(cid:4) (cid:4) Pn(x) Pn+1(x) (cid:4) (cid:4) (cid:4) Pn+1(x) Pn+2(x)
(1.4)
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 29828, Pages 1–6 DOI 10.1155/JIA/2006/29828
where Pn(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.
2 On some Tur´an-type inequalities
For example, Lorch [8] established Tur´an-type inequalities for the positive zeros cνk, k = 1,2,..., of the general Bessel function
(1.5) 0 ≤ α < π, Cν(x) = Jν(x)cosα − Yν(x)sinα,
where Jν(x) and Yν(x) denote the Bessel functions of the first and the second kind, re- spectively, while the corresponding results for the positive zeros c(cid:3) νk, ν ≥ 0, k = 1,2,..., of the derivative C(cid:3) ν(x) = (d/dx)Cν(x) and for the zeros of ultraspherical, Laguerre, and Hermite polynomials have been established in [2, 3, 6], respectively.
(cid:5)
(cid:8) (cid:5)
(cid:5)
(cid:9)2
b
b
b
(cid:6)
(cid:7)
(cid:6)
(cid:7)
(cid:6)
(cid:7)(m+n)/2
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:
mdt ·
ndt ≥
a
a
a
, (1.6) g(t) f (t) g(t) f (t) g(t) f (t) dt
where f and g are two nonnegative functions of a real variable and m and n belong to a set S of real numbers, such that the integrals in (1.6) exist.
As mentioned in [7] this approach represents an alternative method with respect to the classical ones used by the above-cited authors and based, prevalently, on the Sturm theory.
In this paper, we continue, in this direction, to investigate about Tur´an-type inequal- ities satisfied by some special functions. In the next section, we will give three results. In the first one, we will use the well-known psi function defined as follows:
, (1.7) x > 0, ψ(x) = Γ(cid:3)(x) Γ(x)
(cid:11)
with the usual notation for the gamma function. In the second one, we will use the so-called Riemann ξ-function which can be defined (see [11, page 16], cf. [9, page 285]) by
(cid:10) s(s − 1)π−s/2Γ
(1.8) ζ(s), ξ(s) = 1 2 s 2
(cid:10)
(cid:11)
∞(cid:12)
=
where ζ is the Riemann ζ-function. This function has the following representation (see [5]):
k=0 where the coefficients bk are given by the formula
(cid:5) ∞
(1.9) s + ξ bks2k, 1 2
0
∞(cid:12)
(cid:3)
(1.10) t2kΦ(t)dt, k = 0,1,..., bk = 8 22k (2k)!
(cid:2) 2π2n4e9t − 3πn2e5t
n=1
(1.11) e−πn2e4t . Φ(t) =
A. Laforgia and P. Natalini 3
In [1] the following Tur´an-type inequalities were proved:
− k + 1 k
(1.12) k = 0,1,..., bk+1bk−1 ≥ 0, b2 k
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 Kν(x), x > 0, defined as follows:
, ν (cid:6)= 0, ±1, ±2,... , (1.13) I−ν(x) − Iν(x) sinνπ Kν(x), n = 0, ±1, ±2,..., Kν(x) = π 2 Kn(x) = limν→n
∞(cid:12)
where
k=0
(1.14) Iν(x) = (x/2)ν+2k k!Γ(ν + k + 1)
are the modified Bessel functions of the first kind.
(cid:13) n k=1(1/k) the partial sum of the harmonic
2. The results
Theorem 2.1. For n = 1,2,..., denote by hn = series. Let
(2.1) an = hn − logn,
(cid:3)(cid:2)
(cid:3)
≥
(cid:3)2,
then
(cid:2) an − γ
(cid:2) an+1 − γ
(2.2) an+2 − γ
where γ is the Euler-Mascheroni constant defined by
(2.3) γ = −ψ(1) = 0,5772156649 ....
n(cid:12)
Proof. For the psi function, we use the following expression:
− γ,
k=1
(2.4) ψ(n + 1) = n = 1,2,..., 1 k
(cid:10)
(cid:11)
(cid:5) ∞
and the following integral representation:
0
− e−zt e−t − 1
(2.5) dt, Re z > 0. ψ(z + 1) = e−t t
(cid:10)
(cid:5) ∞
(cid:5) ∞
(cid:5) ∞
n(cid:12)
By putting z = n in (2.5), for n = 1,2,..., we obtain from (2.4) and (2.5), (cid:11)
− γ =
(cid:3) dt.
0
0
0
− e−nt e−t − 1
k=1
(2.6) dt = dt + 1 k e−t t e−t − e−nt t e−nt et − 1 − t (cid:2) et − 1 t
4 On some Tur´an-type inequalities
(cid:5) ∞
Since
0
(2.7) dt = logn, e−t − e−nt t
(cid:5) ∞
n(cid:12)
we have
− logn − γ =
(cid:3) e−ntdt.
0
k=1
(2.8) 1 k et − 1 − t (cid:2) et − 1 t
(cid:5) ∞
(cid:5) ∞
(cid:14) (cid:5) ∞
(cid:15)2
By (1.6) with g(t) = (et − 1 − t)/t(et − 1), f (t) = e−t and a = 0, b = +∞, we get
(cid:3) e−ntdt ·
(cid:3) e−(n+2)tdt ≥
(cid:3) e−(n+1)tdt
0
0
0
(2.9) et − 1 − t (cid:2) et − 1 t et − 1 − t (cid:2) et − 1 t et − 1 − t (cid:2) et − 1 t
(cid:2)
that is the inequality (2.2). Theorem 2.2. For k = 1,2,..., let bk (k = 1,2,...) be the coefficients in (1.9), then
− (2k + 1)(k + 1) k(2k − 1)
(2.10) k = 1,2,.... bk+1bk−1 ≤ 0, b2 k
(cid:5) ∞
(cid:5) ∞
(cid:14) (cid:5) ∞
(cid:15)2
Proof. By (1.6) and (1.10), with g(t) = 8Φ(t), f (t) = (2t)2 and a = 0, b = +∞, we get
0
0
0
(2.11) 8Φ(t)(2t)2k+2dt · 8Φ(t)(2t)2k−2dt ≥ 8Φ(t)(2t)2kdt .
Dividing (2.11) by (2k)! this inequality becomes
(2.12) k = 1,2,..., bk+1 bk−1 ≤ b2 k, (2k + 2)! (2k)! (2k − 2)! (2k)!
from which, since ((2k + 2)!/(2k)!)((2k − 2)!/(2k)!) = ((2k + 1)(k + 1))/k(2k − 1), we ob- (cid:2) tain the conclusion of Theorem 2.2.
Remark 2.3. It is important to note that inequalities (1.12) and (2.10) together give
≤ k + 1 k
(2.13) k = 1,2,.... bk+1bk−1, bk+1bk−1 ≤ b2 k k + 1 k 2k + 1 2k − 1
Theorem 2.4. Let Kν(x), x > 0, be the modified Bessel function of the third kind. Then, for ν > −1/2 and μ > −1/2,
(ν+μ)/2(x).
(cid:14) (cid:5) ∞
(cid:5) ∞
(cid:5) ∞
(2.14) Kν(x) · Kμ(x) ≥ K 2
0
0
0
Proof. By (1.6) with g(t) = e−β/t−γt, f (t) = t−1 and a = 0, b = +∞, we get (cid:15)2 (2.15) tm−1e−β/t−γtdt · tn−1e−β/t−γtdt ≥ t(m+n)/2−1e−β/t−γtdt .
A. Laforgia and P. Natalini 5
(cid:16)
(cid:17)ν/2
(cid:19)
(cid:5) ∞
(cid:20)
(cid:18) 2
Using the following formula (see [4, Integral 3.471(9)]):
0
, (2.16) , tν−1e−β/t−γtdt = 2 Kν βγ ν > − 1 2 β γ
(cid:19)
(cid:19)
(cid:19)
(cid:18)
(cid:20)
(cid:20)
(cid:18)
(cid:20)
from (2.15) we have
(cid:18) 2
≥ K 2
· Kμ
(ν+μ)/2
(cid:19)
(2.17) 2 2 βγ βγ βγ Kν
βγ, is equivalent to the conclusion of Theorem 2.4. which, putting x = 2 In the particular case μ = ν + 2, we find
ν+1(x),
. Kν(x) · Kν+2(x) ≥ K 2 ν > − 1 2 (2.18) (cid:2)
(cid:21) π 0 (log sinx)ndx (n = 0,1,...) for which we have
Concluding Remark 2.5. By means of (1.6) Tur´an-type inequalities for many complicated integrals as well as, for example, sn =
n+1(x),
(2.19) sn(x)sn+2(x) ≥ s2
can be obtained.
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A. Laforgia: Department of Mathematics, Roma Tre University, Largo San Leonardo Murialdo, 100146 Rome, Italy E-mail address: laforgia@mat.uniroma3.it
P. Natalini: Department of Mathematics, Roma Tre University, Largo San Leonardo Murialdo, 100146 Rome, Italy E-mail address: natalini@mat.uniroma3.it
6 On some Tur´an-type inequalities
