COMPLETE MONOTONICITY OF A DIFFERENCE BETWEEN THE EXPONENTIAL AND TRIGAMMA FUNCTIONS

The Pure and Applied Mathematics.
2014.
May,
21(2):
141-145

- Received : April 06, 2014
- Accepted : May 12, 2014
- Published : May 31, 2014

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In the paper, by directly verifying an inequality which gives a lower bound for the first order modified Bessel function of the first kind, the authors supply a new proof for the complete monotonicity of a difference between the exponential function
e
^{1/t}
and the trigamma function 𝜓′(
t
) on (0, ∞).
on (0, ∞) was discovered and employed, where 𝜓(
t
) denotes the digamma function
and Γ is the classical Euler gamma function which may be defined for ℜ(
z
) > 0 by
The functions 𝜓′(
z
) and 𝜓′′(
z
) are respectively called the trigamma function and the tetragamma function. As a whole, the derivatives 𝜓
^{(k)}
(
z
) for
k
∈ {0} ∪
are called polygamma functions.
An infinitely differentiable function
f
defined on an interval
I
is said to be a completely monotonic function on
I
if it satisfies
for all
k
∈ {0} ∪
on
I
. Some properties of the completely monotonic functions can be found in, for example,
[2
,
8]
.
In
[5, Theorem 3.1]
and
[6, Theorem 1.1]
, the following theorem was proved by three methods totally.
Theorem 1.1.
The function
is completely monotonic on
(0, ∞)
and
The second main result of the paper
[6]
is
[6, Theorem 1.2]
which has been referenced in
[4, Section 1.2]
and
[5, Lemma 2.1]
as follows.
Theorem 1.2.
For k
∈ {0} ∪
and z
≠ 0,
let
For
ℜ(
z
) > 0,
the function H_{k}
(
z
)
has the integral representations
and
where the hypergeometric series
for b_{i}
∉ {0, −1, −2, ... },
the shifted factorial
(
a
)
_{0}
= 1
and
for n
> 0
and any real or complex number a, and the modified Bessel function of the first kind
for ν
∈
and z
∈
.
When
k
= 0, the integral representations (1.6) and (1.7) may be written as
and
for ℜ(
z
) > 0. Hence, by the well known formula
for ℜ(
z
) > 0 and
n
∈
, see
[1, p. 260, 6.4.1]
, the function
h
(
t
) defined by (1.3) has the following integral representation
Proposition 1.3
(Hausdorff-Bernstein-Widder Theorem
[8, p. 161, Theorem 12b]
).
A necessary and sufficient condition that f
(
x
)
should be completely monotonic for
0 <
x
< ∞
is that
where α
(
t
)
is non-decreasing and the integral converges for
0 <
x
< ∞.
Combining the complete monotonicity in Theorem 1.1 and the integral representation (1.14) with the necessary and sufficient condition in Proposition 1.3, it was revealed in
[6]
that
Replacing
by
t
in (1.16) yields
[6, Theorem 1.3]
below.
Theorem 1.4.
For t
> 0,
we have
We note that the complete monotonicity in Theorem 1.1 is the basis of the inequality (1.17) and some results in the subsequent papers
[4
,
5]
.
The aim of this paper is, with the help of the integral representation (1.14) but without using Proposition 1.3, to supply a new proof of Theorems 1.1 and 1.4 in a converse direction with that in
[4
,
5
,
6]
. In other words, Theorem 1.4 will be firstly and straightforwardly proved, and then Theorem 1.1 will be done.
I_{ν}
(
z
) in (1.10), it is easy to see that
Hence, in order to prove (1.16), it suffices to show
which is equivalent to
Consequently, the proof of the inequality (1.16), that is, Theorem 1.4, is thus complete.
Substituting the inequality (1.16) into the integral representation (1.14) leads to
h
(
t
) > 0 and for
k
∈
on (0, ∞). As a result, the function
h
(
t
) is completely monotonic on (0, ∞).
The limit (1.4) follows immediately from taking
t
→ ∞ on both sides of the integral representation (1.14). Theorem 1.1 is thus proved.
Remark 2.1.
The inequality (2.1) is equivalent to
An immediate differentiation yields
Since
Q
′′′ (
u
) and
Q
′′ (0) = 0, it follows that
Q
′′ (
u
) > 0 on (0, ∞). Owing to
Q
′(0) = 0 and
Q
′′ (
u
) > 0, it is derived that
Q
′ (
u
) > 0. Finally, since
Q
(0) = 0, the function
Q
(
u
) is positive on (0, ∞). This gives an alternative proof of the inequality (2.1).
Remark 2.2.
This is a slightly modified version of the preprint
[7]
.

complete monotonicity
;
completely monotonic function
;
integral representation
;
difference
;
exponential function
;
trigamma function
;
inequality
;
modified Bessel function of the first kind

1. INTRODUCTION

In
[3, Lemma 2]
, the inequality
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2. A NEW PROOF OF THEOREMS 1.1 AND 1.4

By the definition of the modified Bessel function
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- Q′ (u) =eu(u2− 4u+ 6) − 2 (u+ 3),
- Q′′ (u) =eu(u2− 2u+ 2) − 2,
- Q′′′ (u) =u2eu.

Abramowitz M.
,
Stegun I.A.
1970
Applied Mathematics Series
9th printing
National Bureau of Standards
Washington
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,

Guo B.-N.
,
Qi F.
2012
A completely monotonic function involving the tri-gamma function and with degree one
Appl. Math. Comput.
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(19)
9890 -
9897
** DOI : 10.1016/j.amc.2012.03.075**

Guo B.-N.
,
Qi F.
2013
Refinements of lower bounds for polygamma functions
Proc. Amer. Math. Soc.
Available online at http://dx.doi.org/10.1090/S0002-9939-2012-11387-5
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Qi F.
Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions
vailable online at http://arxiv.org/abs/1302.6731

Qi F.
,
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Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function
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Available online at http://dx.doi.org/10.1007/s00009-013-0272-2
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** DOI : 10.1007/s00009-013-0272-2**

Qi F.
,
Wang S.-H.
Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions
available online at http://arxiv.org/abs/1210.2012

Qi F.
,
Zhang X.-J.
Complete monotonicity of a difference between the exponential and trigamma functions
available online at http://arxiv.org/abs/1303.1582

Widder D.V.
1946
The Laplace Transform
Princeton University Press
Princeton

Citing 'COMPLETE MONOTONICITY OF A DIFFERENCE BETWEEN THE EXPONENTIAL AND TRIGAMMA FUNCTIONS
'

@article{ SHGHCX_2014_v21n2_141}
,title={COMPLETE MONOTONICITY OF A DIFFERENCE BETWEEN THE EXPONENTIAL AND TRIGAMMA FUNCTIONS}
,volume={2}
, url={http://dx.doi.org/10.7468/jksmeb.2014.21.2.141}, DOI={10.7468/jksmeb.2014.21.2.141}
, number= {2}
, journal={The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={QI, FENG
and
ZHANG, XIAO-JANG}
, year={2014}
, month={May}