Turán type inequalities for Krätzel functions

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by  Árpád Baricz

File: BJP_Kratzel.tex, printed: 2011-01-14, 1.09
TURA´N TYPE INEQUALITIES FOR KRA¨TZEL FUNCTIONS
A´RPA´DBARICZ†, DRAGANAJANKOV,AND TIBOR K. POGA´NY
Abstract. Complete monotonicity, Laguerre and Tur´an type inequalities are established for
theso-called Kr¨atzel function Zν, defined by
ρ
∞
Zρν(u)=Z tν−1e−tρ−utdt,
0
1 where u > 0 and ρ,ν ∈ R. Moreover, we prove the complete monotonicity of a determinant
1
function of which entries involvetheKr¨atzel function.
0
2

n
1. Introduction
a
J
In1941 whilestudyingthezeros ofLegendrepolynomials, theHungarian mathematician Paul

3
Tura´n discovered the following inequality
1

[P (x)]2 > P (x)P (x),
] n n−1 n+1
A
where x < 1, n 1,2,... and P stands for the classical Legendre polynomial. This
C n
| | ∈ { }
inequality was publishedbyP.Tura´nonlyin1950in[31]. However, sincethepublicationin1948
.
h
by G. Szego˝ [30] of the above famous Tura´n inequality for Legendre polynomials, many authors
t
a have deduced analogous results for classical (orthogonal) polynomials and special functions. In
m
the last 62 years it has been shown by several researchers that the most important (orthogonal)
[ polynomials (e.g. Laguerre, Hermite, Appell, Bernoulli, Jacobi, Jensen, Pollaczek, Lommel,

1 Askey-Wilson, ultraspherical polynomials) and special functions (e.g. Bessel, modified Bessel,
v gamma, polygamma, Riemann zeta functions) satisfy a Tura´n inequality. In 1981 one of the
3
PhD students of P. Tura´n, L. Alp´ar [2] in Tura´n’s biography mentioned that the above Tura´n
2
5 inequality had a wide-ranging effect, this inequality was dealt with in more than 60 papers. The
2 Tura´n type inequalities now have a more extensive literature and recently some of the results
.
1 have been applied successfully in problems that arise in information theory, economic theory
0 and biophysics. Motivated by these applications, the Tura´n type inequalities have recently
1
come under the spotlight once again and it has been shown that, for example, the Gauss and
1
: Kummer hypergeometric functions, as well as the generalized hypergeometric functions, satisfy
v
naturally some Tura´n type inequalities. For the most recent papers on this subject we refer to
i
X [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [17], [21], [19], [29]. For more details see also the
r references therein.
a
Motivated by the above immense research on Tura´n type inequalities, in this paper our aim
is to deduce complete monotonicity, lower bounds and Tura´n type inequalities for the so-called
Kra¨tzel function, defined below. The Kra¨tzel function is defined for u > 0, ρ R and ν C,
∈ ∈
being such that Re(ν) < 0 for ρ 0, by the integral
≤
∞
(1.1) Zρν(u) = tν−1e−tρ−utdt.
Z0
2000 Mathematics Subject Classification. Primary 33C15, Secondary 26D07.
Keywordsandphrases. Kr¨atzelfunctions,ModifiedBesselfunctions,Tur´antypeinequalities,Completemono-
tonicity, Vandermondedeterminant, Tur´an determinant, H¨older-Rogers and Chebyshev integral inequality.
†TheresearchofA´.BariczwassupportedbytheJ´anosBolyaiResearchScholarshipoftheHungarianAcademy
ofSciencesandbytheRomanianNationalAuthorityforScientificResearchCNCSIS-UEFISCSU,projectnumber
PN-II-RU-PD388/2010.
1
2 A´.BARICZ,D.JANKOV,T.K.POGA´NY
For ρ 1 the function (1.1) was introduced by E. Kra¨tzel [24] as a kernel of the integral
≥
transform
∞
(Kρf)(u) = Zν(ut)f(t)dt,
ν ρ
Z0
which was applied to the solution of some ordinary differential equations. The study of the
Kra¨tzel function (1.1) and the above integral transform were continued by several authors. For
example, in [23] the authors deduced explicit forms of Zν in terms of the generalized Wright
ρ
function, while in [22] the authors obtained the asymptotic behavior of this function at zero
and infinity and gave applications to evaluation of integrals involving Zν. Such investigations
ρ
now are of a great interest in connection with applications, see [22] and [23] and the references
therein for more details. We note that the Kra¨tzel function occurs in the study of astrophysical
thermonuclear functions, which are derived on the basis of Boltzmann-Gibbs statistical mechan-
ics, see [28]. It is also important to note that the Kra¨tzel function Zν is related to the modified
1
Bessel function of the second kind K . More precisely, in view of the formula [4, p. 237]
ν
(1.2) Kν(u) = 2uν+ν1 ∞t−ν−1e−t−u42tdt
Z0
we have that for all u > 0 and ν C
∈
u2 u ν u ν
Zν = 2 K (u) =2 K (u)
1 4 2 −ν 2 ν
(cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
and consequently
(1.3) Zν(u) = 2uν/2K (2√u).
1 ν
Note that this function is useful in chemical physics. More precisely, the function
u 2ν−1Zν(uq2/4) = (q√u)νK (q√u)
7→ 1 ν
is related to theHartree-Frock energy andis usedas a basis function for thehelium isoelectronic
series. See [15] and [16] for more details. Moreover, the function
u 2/π2ν−1Zν(u2/4) = 2/πuνK (u)
7→ 1 ν
p p
is called in the literature as reduced Bessel function and plays an important role in theoretical
chemistry. See [33] and the references therein for more details.
This paper is organized as follows: in the next section we present some monotonicity, log-
convexity properties, complete monotonicity and lower bounds for the Kra¨tzel function, while
in the third section we prove the complete monotonicity of a Tura´n determinant which entries
involve the Kra¨tzel function Zν. The main result of the third section is actually a generalization
ρ
of a Tura´n-type inequality, of which counterpart is a conjecture at the end of this paper.
We close these preliminaries with the following definitions, which will be used in the sequel.
A function f : (0, ) R is said to be completely monotonic if f has derivatives of all orders
∞ →
and satisfies
( 1)mf(m)(u) 0
− ≥
for all u > 0 and m 0,1,... . A function g : (0, ) (0, ) is said to be logarithmically
∈ { } ∞ → ∞
convex, or simply log-convex, if its natural logarithm lng is convex, that is, for all u ,u > 0
1 2
and α [0,1] we have
∈
g(αu +(1 α)u ) [g(u )]α[g(u )]1−α.
1 2 1 2
− ≤
Note that every completely monotonic function is log-convex, see [34, p. 167].
TURA´N TYPE INEQUALITIES FOR KRA¨TZEL FUNCTIONS 3
2. Monotonicity and convexity properties of the Kr¨atzel functions
Our first main result reads as follows:
Theorem 1. If ν,ρ R and u > 0, then the following assertions are true:
∈
a. The Kr¨atzel function Zν satisfies the recurrence relation
ρ
(2.1) νZν(u) = ρZν+ρ(u) uZν−1(u).
ρ ρ − ρ
b. The function u Zν(u) is completely monotonic on (0, ).
7→ ρ ∞
c. The function ν Zν(u) is log-convex on R.
7→ ρ
d. The function u Zν(u) is log-convex on (0, ).
7→ ρ ∞
e. For n 1,2,... the following Laguerre type inequality holds
∈ { }
(2.2) Zν(u) (n) 2 Zν(u) (n−1) Zν(u) (n+1).
ρ ≤ ρ ρ
h i
f. Suppose that ρ > 0.(cid:2)Then t(cid:3)he follow(cid:2)ing ine(cid:3)quality(cid:2) holds(cid:3)
(2.3) Zρ−ν(u) ≥ 2u21−νΓ(ν)K1(2√u)
provided that ν 1. Moreover, if 0 < ν 1, then the above inequality is reversed. In
≥ ≤
particular, the inequality
(2.4) uν−1K (u) 2ν−1Γ(ν)K (u)
ν 1
≥
is valid for all ν 1. If 0 < ν 1, then the above inequality is reversed. In the above
≥ ≤
inequality equality holds if u tends to zero or ν = 1.
We note here that similar result to that of (2.4) was proved by M.E.H. Ismail [18]. More
precisely, M.E.H. Ismail proved among others that for all u > 0 and ν > 1/2 the inequality
(2.5) uνK (u)eu > 2ν−1Γ(ν)
ν
is valid and it is sharp as u 0. It should be mentioned here that for ν 1 the inequality (2.4)
→ ≥
is better than (2.5) since for all ν 1 and u > 0 we have
≥
uνK (u) 2ν−1Γ(ν)uK (u) > 2ν−1Γ(ν)e−u.
ν 1
≥
Observethatinfact(2.5)andthelater inequalityfollow fromthefact(see[25])thatthefunction
u uνeuK (u)
ν
7→
isstrictly increasingon(0, )forallν > 1/2andconsequently forallu > 0wehaveueuK (u) >
1
∞
1. Here we used tacitly that when ν > 0 is fixed and u tends to zero, the asymptotic relation
uνK (u) 2ν−1Γ(ν)
ν
∼
holds.
Proof of Theorem 1. a. By using (1.3) and the recurrence relation [32, p. 79]
2ν
K (u) K (u) = K (u)
ν−1 ν+1 ν
− − u
we obtain
uZν−1(u) Zν+1(u) = νZν(u).
1 − 1 − 1
We note that the above recurrence relation can be verified also by using integration by parts as
follows
∞
νZ1ν(u) = e−t−ut νtν−1 dt
Z0∞(cid:16) u(cid:17)(cid:0) (cid:1)
= 1 e−t−ut tνdt
− t2
Z0 (cid:16) (cid:17)(cid:16) (cid:17)
= Zν+1(u) uZν−1(u).
1 − 1
4 A´.BARICZ,D.JANKOV,T.K.POGA´NY
Moreover, by using the same idea we immediately have
∞
νZρν(u) = e−tρ−ut νtν−1 dt
Z0∞(cid:16) (cid:17)u(cid:0) (cid:1)
= ρtρ−1 e−tρ−ut tνdt
− t2
Z0 (cid:16) (cid:17)(cid:16) (cid:17)
= ρZν+ρ(u) uZν−1(u).
ρ − ρ
b. The change of variable 1/t = s in (1.1) yields
∞
(2.6) Zν(u) = s−ν−1e−s−ρ e−usds,
ρ
Z0 (cid:16) (cid:17)
i.e. the Kra¨tzel function Zν is the Laplace transform of the function s s−ν−1e−s−ρ. This
ρ 7→
in view of the Bernstein-Widder theorem (see [34]) implies that the function u Zν(u) is
7→ ρ
completely monotonic, i.e. for all n 0,1,... , ν,ρ R and u > 0 we have
∈{ } ∈
( 1)n Zν(u) (n) > 0.
− ρ
We note that this can be verified also direc(cid:2)tly by u(cid:3)sing that for all n 0,1,... , ν,ρ R and
∈ { } ∈
u > 0
(2.7) Zν(u) (n) = ( 1)nZν−n(u),
ρ − ρ
which follows via mathematical in(cid:2)duction(cid:3) easily from (1.1) or (2.6).
c. Recall the H¨older-Rogers inequality [27, p. 54], that is,
b b 1/p b 1/q
(2.8) f(t)g(t)dt f(t)pdt g(t)qdt ,
| | ≤ | | | |
Za (cid:20)Za (cid:21) (cid:20)Za (cid:21)
wherep > 1,1/p+1/q = 1,f andg arerealfunctionsdefinedon[a,b]and f p, g q areintegrable
| | | |
functions on [a,b]. Using (1.1) and (2.8) we obtain that
∞
Zραν1+(1−α)ν2(u) = tαν1+(1−α)ν2−1e−tρ−utdt
Z0
∞
= tα(ν1−1)+(1−α)(ν2−1)e−tρ−utdt
Z0
= ∞ tν1−1e−tρ−ut α tν2−1e−tρ−ut 1−αdt
Z0 (cid:16) (cid:17) (cid:16) (cid:17)
∞ α ∞ 1−α
tν1−1e−tρ−utdt tν2−1e−tρ−utdt
≤
(cid:20)Z0 (cid:21) (cid:20)Z0 (cid:21)
= Zν1(u) α Zν2(u) 1−α
ρ ρ
holds for all α ∈ [0,1], ν1,ν2,ρ ∈ R(cid:2) and u(cid:3)>(cid:2)0, i.e. t(cid:3)he function ν 7→ Zρν(u) is log-convex on R.
d. This follows from the fact that the integrand in (1.1) or (2.6) is a log-linear function of u
and by using the H¨older-Rogers inequality (2.8) we have that
∞
Zρν(αu1 +(1−α)u2)= tν−1e−tρ−1t(αu1+(1−α)u2)dt
Z0
= ∞ tν−1e−tρ−ut1 α tν−1e−tρ−ut2 1−αdt
Z0 (cid:16) (cid:17) (cid:16) (cid:17)
∞ α ∞ 1−α
tν−1e−tρ−ut1dt tν−1e−tρ−ut2dt
≤
(cid:20)Z0 (cid:21) (cid:20)Z0 (cid:21)
= Zν(u ) α Zν(u ) 1−α
ρ 1 ρ 2
holds for all α ∈ [0,1], ν,ρ ∈ R and(cid:2) u1,u2(cid:3)>(cid:2)0, i.e. t(cid:3)he function u 7→ Zρν(u) is log-convex on
(0, ).
∞
TURA´N TYPE INEQUALITIES FOR KRA¨TZEL FUNCTIONS 5
Alternatively, we may use part b of this theorem. More precisely, it is known that every
completely monotonic function is log-convex (see [34, p. 167]), and then in view of part b the
Kra¨tzel function Zν is log-convex on (0, ). Moreover, as a third proof we may use part c of
ρ ∞
this theorem. Namely, since the function ν Zν(u) is log-convex, it follows that the following
Tura´n type inequality holds for all ν ,ν ,ρ 7→R aρnd u > 0
1 2
∈
ν1+ν2 2
(2.9) Z 2 (u) Zν1(u)Zν2(u).
ρ ≤ ρ ρ
(cid:20) (cid:21)
Now, let choose ν = ν 2 and ν =ν, then we obtain the Tura´n type inequality
1 2
−
fν(u) = Zν−1(u) 2 Zν−2(u)Zν(u) 0,
ρ ρ − ρ ρ ≤
which is valid for all ν,ρ R and u(cid:2)> 0. This(cid:3) in turn together with (2.7) implies that
∈
Zν(u) ′ ′ Zν−1(u) ′ fν(u)
ρ ρ ρ
= = 0,
"(cid:2)Zρν(u)(cid:3) # −" Zρν(u) # − Zρν(u) 2 ≥
i.e. the function u Zν(u) ′/Zν(u) is increasing on (0, (cid:2) ) for(cid:3)all ν,ρ R.
7→ ρ ρ ∞ ∈
e. This follows also from part c of this theorem. More precisely, in view of (2.7) the Laguerre
(cid:2) (cid:3)
type inequality (2.2) is equivalent to the Tura´n type inequality
Zν−n(u) 2 Zν−n−1(u)Zν−n+1(u),
ρ ≤ ρ ρ
which clearly follows from (2.9(cid:2)) by cho(cid:3)osing ν1 = ν n 1 and ν2 = ν n+1.
f. Let us recall the Chebyshev integral inequality−[27,−p. 40]: If f,g : [−a,b] R are integrable
functions, both increasing or both decreasing and p : [a,b] R is a positive i→ntegrable function,
→
then
b b b b
(2.10) p(t)f(t)dt p(t)g(t)dt p(t)dt p(t)f(t)g(t)dt.
≤
Za Za Za Za
Note that if one of the functions f or g is decreasing and the other is increasing, then (2.10) is
reversed. We shall use this inequality. For this by using (2.6) let us write Z−ν(u) as
ρ
∞
Z−ν(u) = e−uttν−1e−t−ρdt
ρ
Z0
and let p(t)= e−ut, f(t)= tν−1 and g(t) = e−t−ρ. Clearly f is increasing (decreasing) on (0, )
if and only if ν 1 (ν 1). Since g′(t)/g(t) = ρt−ρ−1, it follows that g is increasing if and o∞nly
≥ ≤
if ρ > 0. Moreover,
∞ ∞ 1
p(t)dt = e−utdt =
u
Z0 Z0
and
∞ ∞ ∞
p(t)f(t)dt= e−uttν−1dt = u−ν e−ssν−1ds= u−νΓ(ν).
Z0 Z0 Z0
Similarly, integration by parts and (2.6) imply
∞p(t)g(t)dt = ∞e−ute−t−ρdt= ρ ∞t−ρ−1e−t−ρe−utdt= ρZρ(u).
u u ρ
Z0 Z0 Z0
Now, by choosing ν = 0 in (2.1), and using (1.3) we obtain that
ρZρ(u) = uZ−1(u) = 2√uK (2√u)
ρ 1 1
and appealing to the Chebyshev integral inequality (2.10) the proof of the inequality (2.3) is
done.
Finally, observe that by using the relation [32, p. 79] K (u) = K (u) we obtain easily
ν −ν
(2.11) Z−ν(u) = u−νZν(u),
1 1
and if we let ρ= 1 in (2.3), then by using (2.11) we immediately obtain (2.4), and with this the
proof is complete. (cid:3)
6 A´.BARICZ,D.JANKOV,T.K.POGA´NY
3. Tur´an type inequalities for Kr¨atzel functions
Let us consider now the Tura´n type inequality
(3.1) Zν(u) 2 Zν−ρ(u)Zν+ρ(u) < 0,
ρ − ρ ρ
which holds for all ν,ρ R and(cid:2)u > 0.(cid:3)This inequality is actually a particular case of part c of
∈
Theorem 1. More precisely, by choosing ν = ν ρ and ν = ν +ρ in (2.9) the proof of (3.1) is
1 2
−
done. However, we give here an alternative proof. Just observe that
1 ∞ ∞
Zρν(u) 2−Zρν−ρ(u)Zρν+ρ(u) = 2 (ts)ν−1e−tρ−sρ−u(1t+s1)[2−(t/s)ρ−(s/t)ρ]dtds
Z0 Z0
(cid:2) (cid:3)
and by using the elementary inequality (t/s)ρ + (s/t)ρ 2, the integrand becomes negative,
≥
which proves (3.1). Moreover the above integral representation yields the following complete
monotonicity result: the function
u7→ ZZρν−ν(ρu(u)) ZZνρ+ν(ρu(u)) = 12 ∞ ∞(ts)ν−1e−tρ−sρ−u(1t+1s)[(t/s)ρ+(s/t)ρ−2]dtds
(cid:12) ρ ρ (cid:12) Z0 Z0
(cid:12) (cid:12)
is not on(cid:12)(cid:12)ly positive, but even c(cid:12)(cid:12)ompletely monotonic on (0, ) for all ν,ρ R. The next result is
∞ ∈
an analogue of [19, Theorem 2.1] for the Tura´n determinant of Kra¨tzel functions and provides
a generalization of the above result and of part b of Theorem 1. Note that in view of (1.3)
the following result in particular for ρ= 1 gives better Tura´n type inequalities for the modified
Bessel function of the second kind K than [19, Theorem 2.5]. For more details, compare the
ν
first Tura´n type inequality in [19, Remark 2.6] with the right-hand side of (3.3) below.
Theorem 2. If ν,ρ R and n 1,2,... , then the function
∈ ∈ { }
Zν−ρ(u) Zν(u) ... Zν+(n−1)ρ(u)
ρ ρ ρ
Zν(u) Zν+ρ(u) ... Zν+nρ(u)
u Aν (u) = (cid:12)(cid:12) ρ. ρ . ρ . (cid:12)(cid:12)
7→ ρ,n (cid:12) .. .. .. (cid:12)
(cid:12) (cid:12)
(cid:12) ν+(n−1)ρ ν+nρ ν+(2n−1)ρ (cid:12)
(cid:12) Zρ (u) Zρ (u) ... Zρ (u) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
is completely monotonic on (0,(cid:12) ). (cid:12)
(cid:12) (cid:12)
∞
Proof. By using (1.1) we have
Zν−ρ(u) Zν(u) ... Zν+(n−1)ρ(u)
ρ ρ ρ
Zν(u) Zν+ρ(u) ... Zν+nρ(u)
Aνρ,n(u) =(cid:12)(cid:12)(cid:12) ρ... ρ ... ρ ... (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) ν+(n−1)ρ ν+nρ ν+(2n−1)ρ (cid:12)
(cid:12) Zρ (u) Zρ (u) ... Zρ (u) (cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) tν−ρ−1 tν−1 ... tν+(n−1)ρ(cid:12)(cid:12)−1
(cid:12) 0 0 0 (cid:12)
=Z[0,∞)n+1(cid:12)(cid:12)(cid:12)(cid:12) t1ν...−1 tν1+...ρ−1 ... t1ν+n...ρ−1 (cid:12)(cid:12)(cid:12)(cid:12)j=n0e−tρj−tujdt0dt1...dtn
(cid:12) ν+(n−1)ρ−1 ν+nρ−1 ν+(2n−1)ρ−1 (cid:12)Y
(cid:12) tn tn ... tn (cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) tν−ρ−1 tν−1 ... tν+(n−1)ρ−1 (cid:12)(cid:12)
(cid:12) σ(0) σ(0) σ(0) (cid:12)
= (cid:12)(cid:12)(cid:12) tσν.−(11) tνσ+(1.ρ)−1 ... tσν+(1n.)ρ−1 (cid:12)(cid:12)(cid:12) n e−tρj−tujdt0dt1...dtn,
. . .
Z[0,∞)n+1(cid:12)(cid:12) . . . (cid:12)(cid:12)jY=0
(cid:12) ν+(n−1)ρ−1 ν+nρ−1 ν+(2n−1)ρ−1 (cid:12)
t t ... t
(cid:12) σ(n) σ(n) σ(n) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
TURA´N TYPE INEQUALITIES FOR KRA¨TZEL FUNCTIONS 7
where σ is a permutation on 0,1,...,n . Now, let sgn(σ) be denote the sign of σ and S be
n
{ }
the symmetric group on n symbols. Then we obtain
t0 tρ ... tnρ
0 0 0
Aνρ,n(u) = (cid:12)(cid:12) t..ρ1 t21..ρ ... t(1n+..1)ρ (cid:12)(cid:12)sgn(σ) n tνj−ρ−1e−tρj−tujdt0dt1...dtn
Z[0,∞)n+1(cid:12)(cid:12)(cid:12) tn.ρ t(n+.1)ρ ... t2.nρ (cid:12)(cid:12)(cid:12) jY=0
(cid:12) n n n (cid:12)
(cid:12) (cid:12)
ρ nρ
(cid:12) 1 t ... t (cid:12)
= (cid:12)(cid:12)(cid:12) 1.. t..ρ10 ... tn10..ρ (cid:12)(cid:12)sgn(σ) t00(cid:12)tρ1...tnnρ n tνj−ρ−1e−tρj−tuj dt0dt1...dtn
Z[0,∞)n+1(cid:12)(cid:12)(cid:12) 1. t.ρn ... tnn.ρ (cid:12)(cid:12)(cid:12) (cid:0) (cid:1)jY=0(cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) ρ (cid:12) nρ
(cid:12) 1 t0 .(cid:12).. t0
= (n+1 1)! Z(cid:12)[0,∞)n+1(cid:12)(cid:12)(cid:12)(cid:12) 1... t...ρρ1 .(cid:12).. tnn1...ρρ (cid:12)(cid:12)(cid:12)(cid:12)jY=n0(cid:18)tνj−ρ−1e−tρj−tuj(cid:19)
(cid:12) 1 tn ... tn (cid:12)
(cid:12) (cid:12)
sgn(σ)tσ(0(cid:12)(cid:12))(tρ)σ(1)...(tρ)σ(n)(cid:12)(cid:12)dt dt ...dt
× 0 (cid:12) 1 n (cid:12) 0 1 n
σ∈XSn+1
n
= 1 tρ tρ tν−ρ−1e−tρj−tuj
(n+1)! j − i j
Z[0,∞)n+11≤iY<j≤n(cid:16) (cid:17)jY=0(cid:18) (cid:19)
sgn(σ)tσ(0)(tρ)σ(1)...(tρ)σ(n)dt dt ...dt
× 0 1 n 0 1 n
σ∈XSn+1
n
= (n+1 1)! tρj −tρi 2 tνj−ρ−1e−tρj−tuj dt0dt1...dtn,
Z[0,∞)n+11≤iY<j≤n(cid:16) (cid:17) jY=0(cid:18) (cid:19)
where we used that by Leibniz’s formula the Vandermonde determinant can be written as
ρ nρ
1 t ... t
0 0
ρ nρ
1 t ... t
(cid:12)(cid:12)(cid:12) ... ...1 1... (cid:12)(cid:12)(cid:12)= sgn(σ)tσ0(0)(tρ1)σ(1)...(tρn)σ(n) = tρj −tρi .
(cid:12) ρ nρ (cid:12) σ∈XSn+1 1≤iY<j≤n(cid:16) (cid:17)
(cid:12) 1 tn ... tn (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Summar(cid:12)izing (cid:12)
(cid:12) (cid:12)
n
Aνρ,n(u) = (n+11)! e−jP=0(cid:18)tρj+tuj(cid:19) tρj −tρi 2 n tνj−ρ−1dt0dt1...dtn
Z[0,∞)n+1 1≤iY<j≤n(cid:16) (cid:17) jY=0
and then for all ν,ρ R, u > 0 and m 0,1,... we obtain
∈ ∈ { }
m n
0< (−1)m Aνρ,n(u) (m) = (n+11)!  n t1 e−jP=0(cid:18)tρj+tuj(cid:19)
Z[0,∞)n+1 j=0 j
(cid:2) (cid:3) X
  n
2
ρ ρ ν−ρ−1
t t t dt dt ...dt ,
× j − i j 0 1 n
1≤iY<j≤n(cid:16) (cid:17) jY=0
which completes the proof. (cid:3)
Now, let us consider the function Φν :(0, ) R, defined by
ρ ∞ →
ν−ρ ν+ρ
Z (u)Z (u)
Φν(u) = 1 ρ ρ .
ρ − Zν(u) 2
ρ
(cid:2) (cid:3)
8 A´.BARICZ,D.JANKOV,T.K.POGA´NY
Recall the asymptotic expansion (see [22, 24])
Zν(u) ανu22(νρ−+1ρ)e−βρuρ+ρ1,
ρ ∼ ρ
where
ανρ = ρ2+π1ρ1−22(νρ++11) and βρ = (1+1/ρ)ρρ+11,
r
which holds for large values of u and fixed ρ > 0, ν R. By using the above asymptotic relation
∈
we obtain that lim Φν(u) = 0, which shows that in (3.1) the constant 0 is the best possible.
u→∞ ρ
Moreover, based on numerical experiments we believe, but are unable to prove the following
conjecture.
Conjecture. Ifν > ρ > 0,thenthefunctionΦν isstrictlyincreasing on(0, ),andconsequently
ρ ∞
the following Tura´n type inequality holds
(3.2) ρ/(ρ ν) Zν(u) 2 < Zν(u) 2 Zν−ρ(u)Zν+ρ(u).
− ρ ρ − ρ ρ
Note that for ν,ρ > 0 fixed if(cid:2)u tends(cid:3) to z(cid:2)ero, th(cid:3)en the asymptotic relation (see [22, 24])
ρZν(u) Γ(ν/ρ)
ρ ∼
is valid. Using this relation we obtain
Γ(ν/ρ 1)Γ(ν/ρ+1)
lim Φν(u) = 1 − = ρ/(ρ ν)
u→0 ρ − Γ2(ν/ρ) −
for all ν > ρ > 0, which shows that in (3.2) the constant ρ/(ρ ν) is the best possible.
−
It is worthwhile to note here that in fact the inequality (3.2) is motivated by the follow-
ing result. If the above conjecture were be true then (3.2) together with (3.1) would yield a
generalization of (3.3), since for ρ= 1 the inequalities (3.2) and (3.1) reduce to (3.3).
Theorem 3. Let K be the modified Bessel function of the second kind. Then the following
ν
Tura´n type inequalities hold for all ν > 1 and u > 0
(3.3) 1/(1 ν)[K (u)]2 < [K (u)]2 K (u)K (u) < 0.
ν ν ν−1 ν+1
− −
Moreover, the right-hand side of (3.3) holds true for all ν R. These inequalities are sharp in
∈
the sense that the constants 1/(1 ν) and 0 are the best possible.
−
For the sake of completeness it should be mentioned that the right-hand side of (3.3) was
first proved by M.E.H. Ismail and M.E. Muldoon [20], and later by A. Laforgia and P. Natalini
[26] and recently was deduced also by A´. Baricz [11, 12] and J. Segura [29], by using different
approaches. The left-hand side of (3.3) was deduced very recently by using completely different
methods by A´. Baricz [12] and J. Segura [29]. See also [13] for more details on (3.3). Note that
the left-hand side of (3.3) provides actually an upper bound for the effective variance of the
generalized Gaussian distribution. More precisely, in [1] the authors used (without proof) the
inequality 0< v < 1/(µ 1) for µ = ν +4, where
eff
−
K (u)K (u)
µ−1 µ+1
v = 1
eff [K (u)]2 −
µ
is the effective variance of the generalized Gaussian distribution.
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Department of Economics, Babe¸s-Bolyai University, Cluj-Napoca 400591, Romania
E-mail address: [email protected]
Dragana Jankov : Department of Mathematics, University of Osijek, 31000 Osijek, Croatia
E-mail address: [email protected]
Faculty of Maritime Studies, University of Rijeka, Rijeka 51000, Croatia
E-mail address: [email protected]