Some conditions for a class of functions to be completely monotonic

Abstract

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

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, 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

$$f(x)= \int_{0}^{\infty}e^{-xt}\,d\alpha(t). $$

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, [5–28].

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. □