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Some conditions for a class of functions to be completely monotonic

Journal of Inequalities and Applications20152015:11

https://doi.org/10.1186/s13660-014-0534-y

  • Received: 3 November 2014
  • Accepted: 17 December 2014
  • Published:

Abstract

In this article, we present a necessary condition and a necessary and sufficient condition for a class of functions to be completely monotonic.

Keywords

  • necessary condition
  • necessary and sufficient condition
  • completely monotonic function
  • gamma function

MSC

  • 34A40
  • 26D10
  • 26A48

1 Introduction and main results

Recall [1] that a function f is said to be completely monotonic on
$$\mathbb {R}^{+}:=(0, \infty) $$
if f has derivatives of all orders on \(\mathbb {R}^{+}\) and for all \(n\in \mathbb {N}_{0}:=\mathbb {N}\cup\{0\}\)
$$ (-1)^{n}f^{(n)}(x) \ge0,\quad x \in \mathbb {R}^{+}. $$
Here and throughout the paper, is the set of all positive integers. The set of all completely monotonic functions on \(\mathbb {R}^{+}\) is denoted by \(CM(\mathbb {R}^{+})\).
Bernstein [2] proved that a function f on the interval \(\mathbb{R}^{+}\) is completely monotonic if and only if there exists an increasing function \(\alpha(t)\) on \([0,\infty)\) such that
$$f(x)= \int_{0}^{\infty}e^{-xt}\,d\alpha(t). $$
Also recall [3] that a positive function f is said to be logarithmically completely monotonic on \(\mathbb {R}^{+}\) if f has derivatives of all orders on \(\mathbb {R}^{+}\) and for all \(n\in \mathbb {N}\)
$$ (-1)^{n}\bigl[\ln f(x)\bigr]^{(n)}\ge0,\quad x \in \mathbb {R}^{+}. $$
The class of all logarithmically completely monotonic functions on \(\mathbb {R}^{+}\) is denoted by \(LCM(\mathbb {R}^{+})\).

It was proved [4] that a logarithmically completely monotonic function is also completely monotonic.

There is a rich literature on completely monotonic, logarithmically completely monotonic functions and their applications. For more recent work, see, for example, [528].

The Euler gamma function is defined and denoted for \(\operatorname {Re}z>0\) by
$$\Gamma(z):=\int^{\infty}_{0}t^{z-1} e^{-t}\,dt. $$
The logarithmic derivative of \(\Gamma(z)\), denoted by
$$\psi(z):= \frac{\Gamma'(z)}{\Gamma(z)}, $$
is called the psi or digamma function, and the \(\psi^{(k)}\) for \(k\in \mathbb {N}\) are called the polygamma functions.
In this article, we give two necessary conditions and a necessary and sufficient condition for a class of functions
$$ f_{a,b,c}(x):=(x+a)\ln x-x-\ln\Gamma(x+b)+c,\quad x\in \mathbb {R}^{+}, $$
(1)
where \(a,c\in \mathbb {R}\), \(b \ge0\) are parameters, to be completely monotonic. The main results are as follows.

Theorem 1

A necessary condition for the function \(f_{a,b,c}(x)\) to be completely monotonic on the interval \((0, \infty)\) is that
$$\begin{aligned}& b-a=\frac{1}{2}, \end{aligned}$$
(2)
$$\begin{aligned}& 0< b\le\frac{1}{2}, \end{aligned}$$
(3)
and
$$ c\ge\ln\sqrt{2\pi}. $$
(4)

Corollary 1

A necessary condition for the function \(f_{a,b,c}(x)\) to be completely monotonic on the interval \((0, \infty)\) is that
$$ -\frac{1}{2}< a\le0. $$
(5)

Theorem 2

For
$$b\in \biggl[\frac{1}{2}-\frac{\sqrt{3}}{6},\frac{1}{2} \biggr], $$
a necessary and sufficient condition for the function \(f_{a,b,c}(x)\) to be completely monotonic on the interval \((0, \infty)\) is that
$$ b-a=\frac{1}{2} $$
(6)
and
$$ c\ge\ln\sqrt{2\pi}. $$
(7)

2 Lemmas

We need the following lemmas to prove our main results.

Let the α be real parameters, β a non-negative parameter. Define
$$ g_{\alpha,\beta}(x):=\frac{x^{x+\beta-\alpha}}{e^{x}\Gamma(x+\beta)}, \quad x\in \mathbb {R}^{+}. $$

Lemma 1

(see [11])

If
$$g_{\alpha,\beta}\in LCM\bigl(\mathbb {R}^{+}\bigr), $$
then either
$$\beta>0 \quad\textit{and}\quad \alpha\ge\max \biggl\{ \beta,\frac{1}{2} \biggr\} $$
or
$$\beta=0 \quad\textit{and}\quad \alpha\ge1. $$

Lemma 2

(see [7])

Let
$$\beta\in \biggl[\frac{1}{2}-\frac{\sqrt{3}}{6},\frac{1}{2} \biggr]. $$
If
$$\alpha\ge\frac{1}{2}, $$
then
$$g_{\alpha,\beta}\in LCM\bigl(\mathbb {R}^{+}\bigr). $$

3 Proof of the main results

Proof of Theorem 1

If
$$f_{a,b,c}\in CM\bigl(\mathbb {R}^{+}\bigr), $$
then
$$ f_{a,b,c}(x)\ge0,\quad x\in \mathbb {R}^{+}, $$
(8)
and \(f_{a,b,c}(x)\) is decreasing on \(\mathbb {R}^{+}\).
It is well known that (see [29, p.47])
$$ \ln\Gamma(x+\beta)= \biggl(x+\beta-\frac{1}{2} \biggr)\ln x-x+\frac{\ln(2\pi)}{2} +O \biggl(\frac{1}{x} \biggr), \quad\mbox{as } x\to \infty. $$
(9)
Hence
$$ f_{a,b,c}(x)= \biggl(\frac{1}{2}-b+a \biggr)\ln x-\ln \sqrt{2\pi}+c+O \biggl(\frac{1}{x} \biggr), \quad\mbox{as } x\to\infty. $$
(10)
From (8) and (10), we get
$$ \frac{1}{2}-b+a\ge\frac{\ln\sqrt{2\pi}-c+O(1/x)}{\ln x}, \quad\mbox{as } x\to\infty. $$
(11)
Since
$$ \frac{\ln\sqrt{2\pi}-c+O(1/x)}{\ln x} \to0, \quad\mbox{as } x\to\infty, $$
(12)
from (11) we have
$$ b-a \le\frac{1}{2}. $$
(13)
On the other hand, since \(f_{a,b,c}(x)\) is decreasing on \(\mathbb {R}^{+}\), from (10), we obtain
$$ \biggl(\frac{1}{2}-b+a \biggr)\ln x-\ln\sqrt{2\pi}+c+O \biggl(\frac{1}{x} \biggr) \le f_{a,b,c}(\tau), \quad\mbox{as } x\to \infty, $$
(14)
where, in (14), τ is a fixed number in \(\mathbb {R}^{+}\).
Equation (14) is equivalent to
$$ \frac{1}{2}-b+a\le\frac{\ln\sqrt{2\pi}+O(1/x)-c+f_{a,b,c}(\tau)}{\ln x}, \quad\mbox{as } x\to\infty. $$
(15)
It is easy to see that
$$ \frac{\ln\sqrt{2\pi}+O(1/x)-c+f_{a,b,c}(\tau)}{\ln x} \to 0, \quad\mbox{as } x\to\infty. $$
(16)
Then from (15) we have
$$ b-a \ge\frac{1}{2}. $$
(17)
Combining (13) and (17) gives
$$ b-a = \frac{1}{2}. $$
(18)
From (8), (10), and (18), we obtain
$$ c-\ln\sqrt{2\pi}\ge O \biggl(\frac{1}{x} \biggr), \quad\mbox{as } x\to\infty. $$
(19)
Since
$$ O \biggl(\frac{1}{x} \biggr)\to0, \quad\mbox{as } x\to\infty, $$
(20)
from (19) we have
$$ c\ge\ln\sqrt{2\pi}. $$
(21)
We note that
$$ f_{a,b,c}(x)=\ln g_{b-a,b}(x)+c. $$
(22)
If
$$f_{a,b,c}\in CM\bigl(\mathbb {R}^{+}\bigr), $$
we can verify that
$$g_{b-a,b}\in LCM\bigl(\mathbb {R}^{+}\bigr). $$
By Lemma 1, if
$$b>\frac{1}{2}, $$
then
$$ b-a\ge b>\frac{1}{2}, $$
(23)
which contradicts (18); if
$$b=0, $$
by Lemma 1, we get
$$ b-a\ge1, $$
(24)
which is another contradiction to (18). So we have proved that
$$ 0< b\le\frac{1}{2}. $$
(25)

The proof of Theorem 1 is thus completed. □

Proof of Corollary 1

This follows from (2) and (3).

The proof of Corollary 1 is completed. □

Proof of Theorem 2

By Theorem 1, the condition is necessary.

On the other hand, by Lemma 2, we see that
$$g_{b-a,b}\in LCM\bigl(\mathbb {R}^{+}\bigr). $$
Then from (22), we have, for \(n\in \mathbb {N}\),
$$ (-1)^{n} f^{(n)}_{a,b,c}(x)\ge0,\quad x\in \mathbb {R}^{+}. $$
(26)
In particular,
$$ f'_{a,b,c}(x)\le0,\quad x\in \mathbb {R}^{+}. $$
(27)
Hence \(f_{a,b,c}(x)\) is decreasing on \(\mathbb {R}^{+}\).
By (9),
$$ f_{a,b,c}(x)= \biggl(\frac{1}{2}-b+a \biggr)\ln x+c- \ln\sqrt{2\pi}+O \biggl(\frac{1}{x} \biggr),\quad \mbox{as } x\to\infty. $$
(28)
If
$$b-a=\frac{1}{2} $$
and
$$c \ge\ln\sqrt{2\pi}, $$
from (28), we obtain
$$ \lim_{x \to\infty}f_{a,b,c}(x)=c-\ln\sqrt{2\pi} \ge0. $$
(29)
Therefore
$$ f_{a,b,c}(x)\ge\lim_{x \to\infty}f_{a,b,c}(x) \ge0,\quad x\in \mathbb {R}^{+}, $$
(30)
which means that (26) is also valid for \(n=0\). Hence we have proved that
$$f_{a,b,c}\in CM\bigl(\mathbb {R}^{+}\bigr). $$

The proof of Theorem 2 is hence completed. □

Declarations

Acknowledgements

The author thanks the editor and the referees for their valuable suggestions to improve the quality of this paper.

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
Department of Mathematics, Zhongyuan University of Technology, Zhengzhou, Henan, 450007, China

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