Kragujevac Journal of Mathematics
Volume 36 Number 1 (2012), Pages 73–76.
SOME REMARKS ON RESULTS OF MORTICI
TSERENDORJ BATBOLD
Abstract. In this paper, we establish new generalizations of Turán-type inequality
involving the Gamma and Polygamma functions.
1. Introduction
P. Turán ([1]) proved that the Legendre polynomials Pn (x) satisfy the determinantal
inequality
(1.1)
¯
¯ P (x)
Pn+1 (x)
¯
n
¯
¯ Pn+1 (x) Pn+2 (x)
¯
¯
¯
¯ ≤ 0,
¯
−1 ≤ x ≤ 1
where n = 0, 1, 2, . . . and equality occurs only if x = ±1. This classical result has
been extended in several directions: ultraspherical polynomials, Lagguerre and Hermite polynomials, Bessel functions of the first kind, modified Bessel functions, etc.
Today there is a huge literature on Turán inequalities, since they have important applications in complex analysis, number theory, combinatorics, theory of mean-values,
or statistics and control theory.
Recently, Mortici ([2], [3]) proved some Turán-type inequalities for some special
functions as well as the polygamma functions.
The aim of this paper is to prove new generalizations of Turán-type inequalities for
the Gamma and Polygamma functions in ([2],[3]).
2. Main Results
P
= 1 and
Lemma 2.1 (Hölder’s inequality). If p1 , . . . , pn > 0 are such that ni=1 p−1
i
f1 , . . . , fn are non-negative functions such that these integrals exist, then the following
Key words and phrases. Turán-type inequalities, Gamma function, Polygamma functions,
Hölder’s inequality.
2010 Mathematics Subject Classification. 26D07, 33B15.
Received: October 16, 2011 .
73
74
TSERENDORJ BATBOLD
inequality holds:
n µZ ∞
Y
pi
(2.1)
pi
fi (t)dt
0
i=1
Z ∞ ÃY
n
¶1
≥
0
!
fi (t) dt.
i=1
In what follows, we use the integral representations, for x > 0 and n = 1, 2, . . .
(n)
(2.2)
Γ
(2.3)
(n)
ψ
(x) =
0
e−t tx−1 logn tdt,
n+1
(x) = (−1)
Z ∞ n −tx
t e
1 − e−t
0
1 Z ∞ tx−1
dt,
ζ(x) =
Γ(x) 0 et − 1
(2.4)
and
(2.5)
Z ∞
θ(x) = (−1)
n
Z ∞µ
0
dt,
x > 1,
¶
1
1 1 n−1 −tx
− +
t e dt,
t
e −1 t 2
where Γ is the gamma function, ψ (n) is the n-th polygamma function and ζ is the
Reimann-zeta function.
We first give a generalization of Mortici ([2, Theorem 2.1], [3, Theorem 2.2]).
Theorem 2.1. Let p1 , . . . , pn be conjugate parameters such that pi > 1, i = 1, . . . , n,
P
P
and ni=1 p1i = 1, and let m1 , . . . , mn ≥ 1 be integers such that ni=1 mpii is an integer.
Then the following inequality holds for xi > 0, i = 1, . . . , n:
¯ ³P
n ¯
n
¯
¯1
Y
¯ (mi )
¯
¯ pi
¯ψ
(xi )¯ ≥ ¯ψ i=1
¯
(2.6)
i=1
mi
pi
´Ã
!¯
n
X
xi ¯¯
¯.
pi ¯
i=1
Proof. By Hölder’s inequality (2.1) and (2.3), we have
n ¯
¯1
Y
¯ (mi )
¯ψ
(xi )¯¯ pi
=
i=1
n
Y
ÃZ
∞ tmi e−txi
i=1
≥
Z ∞
1−e
0


0
n
Y
i=1
Pn
Z ∞
t
t
mi
pi
mi
i=1 pi
dt
−t
e
(1 −
!1

−txi
pi
1
e−t ) pi
−t
pi
³P
 dt
n
xi
i=1 pi
´
e
dt
−t
1
−
e
0
´Ã
¯ ³P
!¯
n
n
mi
¯
X
xi ¯¯
¯
i=1 pi
= ¯¯ψ
¯.
¯
i=1 pi
=
Hence (2.6) is valid.
¤
The next result is a generalization of Mortici ([2, Theorem 2.2], [3, Theorem 2.1,
2.3]).
SOME REMARKS ON RESULTS OF MORTICI
75
Theorem 2.2. Let p1 , . . . , pn be conjugate parameters such that pi > 1, i = 1, . . . , n,
P
P
and ni=1 p1i = 1, and let m1 , . . . , mn ≥ 1 be integers such that ni=1 mpii is an integer.
Then the following inequalities hold:
n ³
Y
(2.7)
Γ
(mi )
³P
´1
pi
(xi )
≥Γ
n
mi
i=1 pi
´Ã
xi > 0, i = 1, . . . , n,
(2.8)
,
xi > 1, i = 1, . . . , n,
!¯
n
X
xi ¯¯
¯,
pi ¯
xi > 0, i = 1, . . . , n.
i=1
(ζ(xi ))
1
pi
Γ
³P
n
xi
i=1 pi
≥Q
n
1
pi
i=1 (Γ(xi ))
i=1
¯ ³P
n ¯
n
¯1
¯
Y
¯ (mi )
¯ pi
¯
¯θ
(xi )¯ ≥ ¯θ i=1
¯
(2.9)
´
mi
pi
ζ
!
,
i=1
n
Y
n
X
xi
pi
à n
!
X xi
i=1
´Ã
pi
i=1
i=1
Proof. We just use Hölder’s inequality (2.1) and the proofs are similar to the proof
of Theorem 2.1. We omit the details.
¤
The next result is a generalization of Mortici ([2, Theorem 3.1]).
Theorem 2.3. Let p1 , . . . , pn be conjugate parameters such that pi > 1, i = 1, . . . , n,
P
P
and ni=1 p1i = 1, and let m1 , . . . , mn ≥ 0 be even integers such that nj=1 mpjj is even
integers. Then the following inequality holds for xi > 0, i = 1, . . . , n:
³P
(2.10)
n
mi
i=1 pi
exp Γ
´Ã
!
n
X
xi
pi
i=1
n ³
Y
´1
exp Γ(mi ) (xi )
≤
pi
.
i=1
Proof. Using (2.2) and Weighted AM-GM inequality
n
X
Γ(mi ) (xi )
pi
i=1
=
n
X
i=1
=
≥
³P
−Γ
n
mi
i=1 pi
i=1
pi
0
i=1 pi
Z ∞ ÃY
n
0
!
n
X
xi
pi
³P
Z ∞
n
1 Z ∞ −t xi −1 mi
−t
e t
log tdt −
e t i=1
³P
Z ∞ ÃX
n
1 xi − nj=1
0
´Ã
t
t
xi
− p1
pi
i
³P
´
xj
pj
xj
n
j=1 pj
0
³P
log
´
log
mi −
mi
− p1
pi
i
mj
n
j=1 pj
xi
pi
´
´
−1
³P
log
!
t − 1 e−t t
³P
mj
n
j=1 pj
´
n
mi
i=1 pi
³P
n
xi
i=1 pi
³P
!
t − 1 e−t t
´
tdt
´
−1
n
xi
i=1 pi
³P
log
´
−1
n
mi
i=1 pi
³P
log
´
n
mi
i=1 pi
tdt
´
tdt
i=1
= 0.
The conclusion follows by exponentiating the inequality
n
X
Γ(mi ) (xi )
i=1
pi
³P
n
≥Γ
mi
i=1 pi
´Ã
n
X
xi
i=1
pi
!
.
¤
76
TSERENDORJ BATBOLD
Acknowledgement: The author wish to thank the anonymous referee for through
review and insightful suggestions.
References
[1] P. Turán, On the zeros of the polynomials of Legendre, Casopis pro Pestovani Mat. a Fys. 75
(1950), 113–122.
[2] C. Mortici, Turn-type inequalities for the gamma and polygamma functions, Acta Univ. Apulensis. 23 (2010), 117–121.
[3] C. Mortici, New inequalities for some special functions via the Cauchy-Buniakovsky-Schwarz
inequality, Tamkang J. Math. 42 (2011), no. 1, 53–57.
Institute of Mathematics
National University of Mongolia
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