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# Some conditions for a class of functions to be completely monotonic

*Journal of Inequalities and Applications*
**volume 2015**, Article number: 11 (2015)

## Abstract

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

## Introduction and main results

Recall [1] that a function *f* is said to be completely monotonic on

if *f* has derivatives of all orders on \(\mathbb {R}^{+}\) and for all \(n\in \mathbb {N}_{0}:=\mathbb {N}\cup\{0\}\)

Here and throughout the paper, ℕ 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

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 \(n\in \mathbb {N}\)

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, [5–28].

The Euler gamma function is defined and denoted for \(\operatorname {Re}z>0\) by

The logarithmic derivative of \(\Gamma(z)\), denoted by

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

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*

*and*

### Corollary 1

*A necessary condition for the function*
\(f_{a,b,c}(x)\)
*to be completely monotonic on the interval*
\((0, \infty)\)
*is that*

### Theorem 2

*For*

*a necessary and sufficient condition for the function*
\(f_{a,b,c}(x)\)
*to be completely monotonic on the interval*
\((0, \infty)\)
*is that*

*and*

## Lemmas

We need the following lemmas to prove our main results.

Let the *α* be real parameters, *β* a non-negative parameter. Define

### Lemma 1

(see [11])

*If*

*then either*

*or*

### Lemma 2

(see [7])

*Let*

*If*

*then*

## Proof of the main results

### Proof of Theorem 1

If

then

and \(f_{a,b,c}(x)\) is decreasing on \(\mathbb {R}^{+}\).

It is well known that (see [29, p.47])

Hence

Since

from (11) we have

On the other hand, since \(f_{a,b,c}(x)\) is decreasing on \(\mathbb {R}^{+}\), from (10), we obtain

where, in (14), *τ* is a fixed number in \(\mathbb {R}^{+}\).

Equation (14) is equivalent to

It is easy to see that

Then from (15) we have

From (8), (10), and (18), we obtain

Since

from (19) we have

We note that

If

we can verify that

By Lemma 1, if

then

which contradicts (18); if

by Lemma 1, we get

which is another contradiction to (18). So we have proved that

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

Then from (22), we have, for \(n\in \mathbb {N}\),

In particular,

Hence \(f_{a,b,c}(x)\) is decreasing on \(\mathbb {R}^{+}\).

By (9),

If

and

from (28), we obtain

Therefore

which means that (26) is also valid for \(n=0\). Hence we have proved that

The proof of Theorem 2 is hence completed. □

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## Acknowledgements

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

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### Cite this article

Guo, S. Some conditions for a class of functions to be completely monotonic.
*J Inequal Appl* **2015**, 11 (2015). https://doi.org/10.1186/s13660-014-0534-y

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

### MSC

- 34A40
- 26D10
- 26A48

### Keywords

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