Review Article
A Class of Logarithmically Completely Monotonic Functions and
Their Applications
Senlin Guo
Department of Mathematics, Zhongyuan University of Technology, Zhengzhou, Henan 450007, China
Correspondence should be addressed to Senlin Guo; sguo@hotmail.com
Received 7 April 2014; Accepted 10 May 2014; Published 13 July 2014
Academic Editor: Qiu-Ming Luo
Copyright © 2014 Senlin Guo. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the recent investigations on a class of functions which are logarithmically completely monotonic. Two open problems are
also presented.
1. Introduction
Recall [1] that a positive function is said to be logarithmi-
cally completely monotonic (LCM) on an open interval if
has derivatives of all orders on and for all ∈ N :=
{1, 2, 3, . . .},
(−1)
[ln ()]
()
≥ 0.
(1)
LCM functions are related to completely monotonic
(CM) functions [2], strongly logarithmically completely
monotonic (SLCM) functions [3], almost strongly completely
monotonic (ASCM) functions [3], almost completely mono-
tonic (ACM) functions [4], Laplace transforms, and Stieltjes
transforms and have wide applications. It is evident that the
set of SLCM functions is a nontrivial subset of the set of
LCM functions, which is a nontrivial subset of the set of CM
functions, and that the set of CM functions is a nontrivial
subset of the set of ACM functions. It was established [3] that
the set of SLCM functions is a nontrivial subset of the set of
ASCM functions and that the set of SLCM functions on the
interval (0, ∞) is disjoint with the set of strongly completely
monotonic (SCM) functions (see [5] for its definition) on the
interval (0, ∞).
It is well known that the classical Euler gamma function
is defined for >0 by
Γ () = ∫
∞
0
−1
−
d. (2)
e logarithmic derivative of Γ(), denoted by
() =
Γ
()
Γ ()
, (3)
is called psi function, and
()
for ∈ N are called polygamma
functions.
For , ∈ R and ≥0, define
,,
() := [
Γ ( + )
+−
]
, ∈ (0, ∞) , (4)
which is encountered in probability and statistics.
Since
,,
() ( > 0) is logarithmically completely
monotonic if and only if
,,1
() is logarithmically com-
pletely monotonic and
,,
() ( < 0) is logarithmically
completely monotonic if and only if
,,−1
() is logarith-
mically completely monotonic, we only need to study the
logarithmically complete monotonicity of the function
,,±1
() = [
Γ ( + )
+−
]
±1
, ∈ (0, ∞) . (5)
In [6, eorem 3.2], it was proved that the function
1/2,0,1
() is decreasing and logarithmically convex from
(0, ∞) onto (
√
2, ∞) and that the function
1,0,1
() is
increasing and logarithmically concave from (0, ∞) onto
(1, ∞).
Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2014, Article ID 757462, 5 pages
http://dx.doi.org/10.1155/2014/757462