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# Monotonicity properties for a ratio of finite many gamma functions

## Abstract

In the paper, the authors consider a ratio of finite many gamma functions and find its monotonicity properties such as complete monotonicity, the Bernstein function property, and logarithmically complete monotonicity.

## Preliminaries

Let $$f(x)$$ be an infinite differentiable function on an infinite interval $$(0,\infty )$$.

1. (1)

If $$(-1)^{k}f^{(k)}(x)\ge 0$$ for all $$k\ge 0$$ and $$x\in (0,\infty )$$, then we call $$f(x)$$ a completely monotonic function on $$(0,\infty )$$. See the review papers [22, 31, 36] and [35, Chapter IV].

2. (2)

If $$(-1)^{k}[\ln f(x)]^{(k)}\ge 0$$ for all $$k\ge 1$$ and $$x\in (0,\infty )$$, or say, if the logarithmic derivative $$[\ln f(x)]'=\frac{f'(x)}{f(x)}$$ is a completely monotonic function on $$(0,\infty )$$, then we call $$f(x)$$ a logarithmically completely monotonic function on $$(0,\infty )$$. See the papers [3, 4, 7, 24] and [33, Chap. 5].

3. (3)

If $$f'(x)$$ is a completely monotonic function on $$(0,\infty )$$, then we call $$f(x)$$ a Bernstein function on $$(0,\infty )$$. See the paper  and the monograph .

The classical gamma function $$\varGamma (z)$$ can be defined by

$$\varGamma (z)= \int _{0}^{\infty }t^{z-1}e^{-t} \, \mathrm{d} t, \quad \Re (z)>0$$

or by

$$\varGamma (z)=\lim_{n\to \infty }\frac{n!n^{z}}{\prod_{k=0}^{n}(z+k)}, \quad z\in \mathbb{C} \setminus \{0,-1,-2,\ldots \}.$$

See [1, Chap. 6], [15, Chap. 5], the paper , and [34, Chap. 3]. In the literature, the logarithmic derivative

$$\psi (z)=\bigl[\ln \varGamma (x)\bigr]'=\frac{\varGamma '(z)}{\varGamma (z)}$$

and its first derivative $$\psi '(z)$$ are respectively called the digamma and trigamma functions. See the papers [5, 6, 10, 25, 26] and closely related references therein.

## Motivations

This paper is motivated by a series of papers [2, 11, 12, 16, 19, 21, 27, 29, 32]. For detailed review and survey, please read the papers [19, 27, 29, 32] and closely related references therein.

In the paper , motivated by [11, 12], the function

$$x\in (0,\infty )\mapsto \frac{\varGamma (nx+1)}{\varGamma (kx+1)\varGamma ((m-k)x+1)} p^{kx}(1-p)^{(m-k)x}$$
(2.1)

was considered, where $$p\in (0,1)$$ and k, m are nonnegative integers with $$0\le k\le m$$.

In [16, Theorem 2.1] and , the function

$$x\in (0,\infty )\mapsto \frac{\varGamma (1+x\sum_{i=1}^{m}\lambda _{i} )}{\prod_{i=1}^{m}\varGamma (1+x\lambda _{i})} \prod _{i=1}^{m}p_{i}^{x\lambda _{i}}$$
(2.2)

was independently studied, where $$m\ge 2$$, $$\lambda _{i}>0$$ for $$1\le i\le m$$, $$p_{i}\in (0,1)$$ for $$1\le i\le m$$, and $$\sum_{i=1}^{m}p_{i}=1$$. The q-analogue

$$x\in (0,\infty )\mapsto \frac{\varGamma _{q} (1+x\sum_{i=1}^{m}\lambda _{i} )}{\prod_{i=1}^{m}\varGamma _{q}(1+x\lambda _{i})} \prod _{i=1}^{m}p_{i}^{x\lambda _{i}}$$
(2.3)

of the function in (2.2) was investigated in , where $$q\in (0,1)$$, $$m\ge 2$$, $$\lambda _{i}>0$$ for $$1\le i\le m$$, $$p_{i}\in (0,1)$$ for $$1\le i\le m$$ with $$\sum_{i=1}^{m}p_{i}=1$$, and $$\varGamma _{q}$$ is the q-analogue of the gamma function Γ.

The functions

$$x\in (0,\infty )\mapsto \frac{\prod_{i=1}^{m}\varGamma (\nu _{i}x+1)\prod_{j=1}^{n}\varGamma (\tau _{j}x+1 )}{\prod_{i=1}^{m}\prod_{j=1}^{n}\varGamma (\lambda _{ij}x+1 )}$$
(2.4)

and

$$x\in (0,\infty )\mapsto \frac{\prod_{i=1}^{m}\varGamma (1+\nu _{i}x)\prod_{j=1}^{n}\varGamma (1+\tau _{j}x )}{ [\prod_{i=1}^{m}\prod_{j=1}^{n}\varGamma (1+\lambda _{ij}x ) ]^{\rho }}$$
(2.5)

were respectively considered in [17, Theorem 2.1] and [29, Theorem 4.1], where $$\rho \in \mathbb{R}$$ and $$\lambda _{ij}>0$$, $$\nu _{i}=\sum_{j=1}^{n}\lambda _{ij}$$, $$\tau _{j}=\sum_{i=1}^{m}\lambda _{ij}$$ for $$1\le i\le m$$ and $$1\le j\le n$$.

In , the function

$$x\in (0,\infty )\mapsto \frac{\prod_{i=1}^{m}[\varGamma (1+\nu _{i}x)]^{\nu _{i}^{\theta }} \prod_{j=1}^{n} [\varGamma (1+\tau _{j}x ) ]^{\tau _{j}^{\theta }}}{\prod_{i=1}^{m}\prod_{j=1}^{n} [\varGamma (1+\lambda _{ij}x ) ]^{\rho \lambda _{ij}^{\theta }}}$$
(2.6)

was discussed, where $$\rho ,\theta \in \mathbb{R}$$ and $$\lambda _{ij}>0$$, $$\nu _{i}=\sum_{j=1}^{n}\lambda _{ij}$$, $$\tau _{j}=\sum_{i=1}^{m}\lambda _{ij}$$ for $$1\le i\le m$$ and $$1\le j\le n$$.

In this paper, stimulated by the above six functions (2.1), (2.2), (2.3), (2.4), (2.5), and (2.6), we consider a new function

$$\mathcal{Q}(x)=\mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x) = \frac{ [\varGamma (1+x\sum_{i=1}^{m}a_{i} ) ]^{(\sum _{i=1}^{m}a_{i})^{\theta }}}{\prod_{i=1}^{m}[\varGamma (1+xa_{i})]^{\rho a_{i}^{\theta }}} \Biggl( \prod_{i=1}^{m}p_{i}^{a_{i}} \Biggr)^{\varrho x}$$
(2.7)

on $$(0,\infty )$$, where $$m\ge 2$$, $$\rho ,\varrho ,\theta \in \mathbb{R}$$, $$a=(a_{1},a_{2},\ldots ,a_{m})$$ with $$a_{i}>0$$ for $$1\le i\le m$$, and $$p=(p_{1},p_{2},\ldots ,p_{m})$$ with $$p_{i}\in (0,1)$$ for $$1\le i\le m$$ and $$\sum_{i=1}^{m}p_{i}=1$$.

## Monotonicity properties

In this section, we now start out to find and prove some monotonicity properties for the function $$\mathcal{Q}(x)=\mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x)$$ defined in (2.7). Our main results in this section can be stated in the following theorem.

### Theorem 3.1

Let$$m\ge 2$$, $$a=(a_{1},a_{2},\ldots ,a_{m})$$with$$a_{i}>0$$for$$1\le i\le m$$, and$$p=(p_{1},p_{2},\ldots , p_{m})$$with$$\sum_{i=1}^{m}p_{i}=1$$and$$p_{i}\in (0,1)$$for$$1\le i\le m$$. Then

1. (1)

when$$\rho \le 1$$and$$\theta \ge 0$$, the second logarithmic derivative

$$\bigl[\ln \mathcal{Q}(x)\bigr]''= \Biggl(\sum _{i=1}^{m}a_{i} \Biggr)^{\theta +2} \psi ' \Biggl(1+x\sum_{i=1}^{m}a_{i} \Biggr) -\rho \sum_{i=1}^{m}a_{i}^{ \theta +2} \psi '(1+a_{i}x)$$

is completely monotonic on$$(0,\infty )$$;

2. (2)

when$$\rho =1$$, $$\varrho =0$$, and$$\theta =0$$, the function

$$\mathcal{Q}_{m,a,p,1,0,0}(x)= \frac{\varGamma (1+x\sum_{i=1}^{m}a_{i} )}{\prod_{i=1}^{m}\varGamma (1+xa_{i})}$$

is increasing on$$(0,\infty )$$and its logarithmic derivative

$$\bigl[\ln \mathcal{Q}_{m,a,p,1,0,0}(x)\bigr]'= \Biggl(\sum _{i=1}^{m}a_{i} \Biggr)\psi \Biggl(1+x\sum_{i=1}^{m}a_{i} \Biggr) -\sum_{i=1}^{m}a_{i} \psi (1+a_{i}x)$$

is a Bernstein function on$$(0,\infty )$$;

3. (3)

when$$\rho =1$$, $$\varrho \ge 1$$, and$$\theta =0$$, the function$$\mathcal{Q}_{m,a,p,1,\varrho ,0}(x)$$is logarithmically completely monotonic on$$(0,\infty )$$;

4. (4)

when$$(\rho ,\varrho ,\theta )\in S$$and

$$S=\{\rho \le 1,\varrho \ge 0,\theta \ge 0\}\setminus \{\rho =1, \varrho =0, \theta =0\}\setminus \{\rho =1,\varrho \ge 1,\theta =0\},$$

the function$$\mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x)$$has a unique minimum on$$(0,\infty )$$.

### Proof

Direct calculation gives

\begin{aligned}& \ln \mathcal{Q}(x)= \Biggl(\sum_{i=1}^{m}a_{i} \Biggr)^{\theta }\ln \varGamma \Biggl(1+x\sum_{i=1}^{m}a_{i} \Biggr) -\rho \sum_{i=1}^{m}a_{i}^{\theta } \ln \varGamma (1+a_{i}x) +\varrho x\sum_{i=1}^{m}a_{i} \ln p_{i}, \\& \bigl[\ln \mathcal{Q}(x)\bigr]'= \Biggl(\sum _{i=1}^{m}a_{i} \Biggr)^{\theta +1} \psi \Biggl(1+x\sum_{i=1}^{m}a_{i} \Biggr) -\rho \sum_{i=1}^{m}a_{i}^{ \theta +1} \psi (1+a_{i}x) +\varrho \sum_{i=1}^{m}a_{i} \ln p_{i}, \end{aligned}

and

$$\bigl[\ln \mathcal{Q}(x)\bigr]''= \Biggl(\sum _{i=1}^{m}a_{i} \Biggr)^{\theta +2} \psi ' \Biggl(1+x\sum_{i=1}^{m}a_{i} \Biggr) -\rho \sum_{i=1}^{m}a_{i}^{ \theta +2} \psi '(1+a_{i}x).$$

As in [27, 29, 32], from

$$\psi '(z)= \int _{0}^{\infty }\frac{t}{1-e^{-t}}e^{-zt}\, \mathrm{d} t, \quad \Re (z)>0$$

in [1, p. 260, 6.4.1], it follows that

$$\psi '(1+\tau z)= \int _{0}^{\infty }\frac{t}{1-e^{-t}}e^{-(1+\tau z)t} \, \mathrm{d} t =\frac{1}{\tau } \int _{0}^{\infty }h \biggl(\frac{v}{\tau } \biggr)e^{-vz} \,\mathrm{d} v,$$

where $$\tau >0$$ and $$h(t)=\frac{t}{e^{t}-1}$$ is the generating function of the classical Bernoulli numbers, see [20, 23] and [34, Chap. 1]. Accordingly, we have

$$\bigl[\ln \mathcal{Q}(x)\bigr]''= \int _{0}^{\infty } \Biggl[ \Biggl(\sum _{i=1}^{m}a_{i} \Biggr)^{\theta +1}h \biggl(\frac{v}{\sum_{i=1}^{m}a_{i}} \biggr) - \rho \sum_{i=1}^{m}a_{i}^{\theta +1}h \biggl(\frac{v}{a_{i}} \biggr) \Biggr]e^{-vx}\,\mathrm{d} v.$$
(3.1)

In [27, Theorem 4.1], it was discovered that

$$\sum_{i=1}^{m} \frac{\nu _{i}^{\alpha }}{e^{x/\nu _{i}}-1}+ \sum_{j=1}^{n} \frac{\tau _{j}^{\alpha }}{e^{x/\tau _{j}}-1} \ge 2\sum_{i=1}^{m}\sum _{j=1}^{n} \frac{\lambda _{ij}^{\alpha }}{e^{x/\lambda _{ij}}-1},$$
(3.2)

where $$\alpha \ge 0$$, $$x>0$$, $$\lambda _{ij}>0$$ for $$1\le i\le m$$ and $$1\le j\le n$$, $$\nu _{i}=\sum_{j=1}^{n}\lambda _{ij}$$, and $$\tau _{j}=\sum_{i=1}^{m}\lambda _{ij}$$. As remarked in [27, Remark 4.1], setting $$n=m$$ and $$\lambda _{1k}=\lambda _{k1}=\lambda _{k}>0$$ for $$1\le k\le m$$ and letting $$\lambda _{ij}\to 0^{+}$$ for $$2\le i,j\le m$$ in inequality (3.2) result in

$$\frac{ (\sum_{k=1}^{m}\lambda _{k} )^{\alpha }}{e^{x/\sum _{k=1}^{m}\lambda _{k}}-1} \ge \sum_{k=1}^{m} \frac{\lambda _{k}^{\alpha }}{e^{x/\lambda _{k}}-1}$$
(3.3)

for $$x>0$$, $$\lambda _{k}>0$$, and $$\alpha \ge 0$$. Inequality (3.3) can be equivalently formulated as

$$\Biggl(\sum_{k=1}^{m} \lambda _{k} \Biggr)^{\alpha +1} h \biggl( \frac{x}{\sum_{k=1}^{m}\lambda _{k}} \biggr) \ge \sum_{k=1}^{m} \lambda _{k}^{\alpha +1} h \biggl(\frac{x}{\lambda _{k}} \biggr)$$
(3.4)

for $$x>0$$, $$\lambda _{k}>0$$, and $$\alpha \ge 0$$.

Combining inequality (3.4) with equation (3.1) yields that, when $$\rho \le 1$$ and $$\theta \ge 0$$, the second derivative $$[\ln \mathcal{Q}(x)]''$$ is completely monotonic on $$(0,\infty )$$.

The complete monotonicity of $$[\ln \mathcal{Q}(x)]''$$ implies that the first derivative $$[\ln \mathcal{Q}(x)]'$$ is strictly increasing on $$(0,\infty )$$. Therefore, by virtue of the limit

$$\lim_{x\to \infty }\bigl[\ln x-\psi (x)\bigr]=0$$

in [8, Theorem 1] and [9, Sect. 1.4], we have

\begin{aligned} \lim_{x\to 0^{+}}\bigl[\ln \mathcal{Q}(x)\bigr]'&=\lim _{x\to 0^{+}} \Biggl[ \Biggl(\sum_{i=1}^{m}a_{i} \Biggr)^{\theta +1}\psi \Biggl(1+x\sum_{i=1}^{m}a_{i} \Biggr) -\rho \sum_{i=1}^{m}a_{i}^{\theta +1} \psi (1+a_{i}x) \Biggr] \\ &\quad{} +\varrho \sum_{i=1}^{m}a_{i} \ln p_{i} \\ &=\psi (1) \Biggl[ \Biggl(\sum_{i=1}^{m}a_{i} \Biggr)^{\theta +1}-\rho \sum_{i=1}^{m}a_{i}^{\theta +1} \Biggr]+\varrho \sum_{i=1}^{m}a_{i} \ln p_{i} \\ &\textstyle\begin{cases} =0, & \theta =0,\rho =1,\varrho =0; \\ < 0, & \theta =0,\rho =1,\varrho >0; \\ < 0, & \theta =0,\rho < 1,\varrho \ge 0; \\ < 0, & \theta >0,\rho \le 1,\varrho \ge 0; \end{cases}\displaystyle \end{aligned}

where $$\psi (1)=-0.577\ldots$$ , and

\begin{aligned} \lim_{x\to \infty }\bigl[\ln \mathcal{Q}(x)\bigr]'&=\lim _{x\to \infty } \Biggl[ \Biggl(\sum_{i=1}^{m}a_{i} \Biggr)^{\theta +1}\psi \Biggl(1+x \sum_{i=1}^{m}a_{i} \Biggr) -\rho \sum_{i=1}^{m}a_{i}^{\theta +1} \psi (1+a_{i}x) \Biggr] \\ &\quad{} +\varrho \sum_{i=1}^{m}a_{i} \ln p_{i} \\ &=\lim_{x\to \infty } \Biggl\{ \Biggl(\sum _{i=1}^{m}a_{i} \Biggr)^{ \theta +1} \Biggl[\psi \Biggl(1+x\sum_{i=1}^{m}a_{i} \Biggr)-\ln \Biggl(1+x\sum_{i=1}^{m}a_{i} \Biggr) \Biggr] \\ &\quad{} -\rho \sum_{i=1}^{m}a_{i}^{\theta +1} \bigl[\psi (1+a_{i}x)-\ln (1+a_{i}x)\bigr] \Biggr\} +\varrho \sum_{i=1}^{m}a_{i}\ln p_{i} \\ &\quad{} +\lim_{x\to \infty } \Biggl[ \Biggl(\sum _{i=1}^{m}a_{i} \Biggr)^{ \theta +1}\ln \Biggl(1+x\sum_{i=1}^{m}a_{i} \Biggr) -\rho \sum_{i=1}^{m}a_{i}^{ \theta +1} \ln (1+a_{i}x) \Biggr] \\ &=\varrho \sum_{i=1}^{m}a_{i} \ln p_{i}+\lim_{x\to \infty }\ln \frac{ (1+x\sum_{i=1}^{m}a_{i} )^{(\sum _{i=1}^{m}a_{i})^{\theta +1}}}{\prod_{i=1}^{m}(1+a_{i}x)^{\rho a_{i}^{\theta +1}}} \\ &=\ln \lim_{x\to \infty } \frac{ (\frac{1}{x}+\sum_{i=1}^{m}a_{i} )^{(\sum _{i=1}^{m}a_{i})^{\theta +1}}}{\prod_{i=1}^{m} (\frac{1}{x}+a_{i} )^{\rho a_{i}^{\theta +1}}} \\ &\quad{} +\ln \lim_{x\to \infty }x^{(\sum _{i=1}^{m}a_{i})^{\theta +1}- \rho \sum _{i=1}^{m}a_{i}^{\theta +1}}+\varrho \sum _{i=1}^{m}a_{i} \ln p_{i} \\ &=\varrho \sum_{i=1}^{m}a_{i} \ln p_{i}+\ln \frac{ (\sum_{i=1}^{m}a_{i} )^{(\sum _{i=1}^{m}a_{i})^{\theta +1}}}{ (\prod_{i=1}^{m}a_{i}^{a_{i}^{\theta +1}} )^{\rho }} \\ &\quad{} + \textstyle\begin{cases} 0, &\rho = \frac{(\sum_{i=1}^{m}a_{i})^{\theta +1}}{\sum_{i=1}^{m}a_{i}^{\theta +1}}; \\ -\infty , &\rho > \frac{(\sum_{i=1}^{m}a_{i})^{\theta +1}}{\sum_{i=1}^{m}a_{i}^{\theta +1}}; \\ \infty , &\rho < \frac{(\sum_{i=1}^{m}a_{i})^{\theta +1}}{\sum_{i=1}^{m}a_{i}^{\theta +1}}. \end{cases}\displaystyle \end{aligned}

Let $$\xi =(\xi _{1},\xi _{2},\ldots ,\xi _{m})$$ such that $$\sum_{i=1}^{m}\xi _{i}=1$$ and $$\xi _{i}\in (0,1)$$ for $$1\le i\le m$$ and $$m\ge 2$$. Then the first derivative of the function $$f(x)=\sum_{i=1}^{m}\xi _{i}^{x}$$ is $$f'(x)=\sum_{i=1}^{m}\xi _{i}^{x}\ln \xi _{i}<0$$, which implies that the function $$f(x)$$ is strictly decreasing on $$(-\infty ,\infty )$$. Since $$f(1)=1$$, it follows that $$f(x)\lesseqqgtr 1$$ if and only if $$x\gtreqqless 1$$. This means that

$$\sum_{i=1}^{m}\xi _{i}^{x} \lesseqqgtr 1, \quad x\gtreqqless 1.$$

Replacing $$\xi _{i}=\frac{a_{i}}{\sum_{i=1}^{m}a_{i}}$$ and $$x=\theta +1$$ in the above inequality yields

$$\sum_{i=1}^{m} \biggl(\frac{a_{i}}{\sum_{i=1}^{m}a_{i}} \biggr)^{ \theta +1}\lesseqqgtr 1, \quad \theta \gtreqqless 0.$$

This can be further rewritten as

$$\sum_{i=1}^{m}a_{i}^{\theta +1} \lesseqqgtr \Biggl(\sum_{i=1}^{m}a_{i} \Biggr)^{\theta +1}, \quad \theta \gtreqqless 0, a_{i}>0, m\ge 2.$$
(3.5)

Considering inequality (3.5) reveals that

1. (1)

when $$\theta =0$$, we have

$$\lim_{x\to \infty }\bigl[\ln \mathcal{Q}(x)\bigr]'=\varrho \sum_{i=1}^{m}a_{i} \ln p_{i}+ \textstyle\begin{cases} \ln \frac{ (\sum_{i=1}^{m}a_{i} )^{\sum _{i=1}^{m}a_{i}}}{\prod_{i=1}^{m}a_{i}^{a_{i}}}+0, &\rho =1; \\ \ln \frac{ (\sum_{i=1}^{m}a_{i} )^{\sum _{i=1}^{m}a_{i}}}{ (\prod_{i=1}^{m}a_{i}^{a_{i}} )^{\rho }}+ \infty , &\rho < 1. \end{cases}$$
2. (2)

when $$\theta >0$$ and $$\rho \le 1$$, we have

$$\lim_{x\to \infty }\bigl[\ln \mathcal{Q}(x)\bigr]'=\varrho \sum_{i=1}^{m}a_{i} \ln p_{i}+ \ln \frac{ (\sum_{i=1}^{m}a_{i} )^{(\sum _{i=1}^{m}a_{i})^{\theta +1}}}{ (\prod_{i=1}^{m}a_{i}^{a_{i}^{\theta +1}} )^{\rho }}+ \infty =\infty .$$

Hence, when $$\theta =0$$ and $$\rho <1$$ or when $$\theta >0$$ and $$\rho \le 1$$, we obtain

$$\lim_{x\to \infty }\bigl[\ln \mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x) \bigr]'= \infty ;$$

when $$\theta =0$$ and $$\rho =1$$, we have

\begin{aligned} \lim_{x\to \infty }\bigl[\ln \mathcal{Q}(x)\bigr]' &= \varrho \sum_{i=1}^{m}a_{i} \ln p_{i}+ \ln \frac{ (\sum_{i=1}^{m}a_{i} )^{\sum _{i=1}^{m}a_{i}}}{\prod_{i=1}^{m}a_{i}^{a_{i}}} \\ &=(\varrho -1)\sum_{i=1}^{m}a_{i} \ln p_{i}+ \Biggl(\sum_{i=1}^{m}p_{i} \frac{a_{i}}{p_{i}} \Biggr) \ln \Biggl(\sum_{i=1}^{m}p_{i} \frac{a_{i}}{p_{i}} \Biggr) -\sum_{i=1}^{m}p_{i} \frac{a_{i}}{p_{i}} \ln \frac{a_{i}}{p_{i}}. \end{aligned}

Let f be a convex function on an interval $$I\subseteq \mathbb{R}$$, let $$m\ge 2$$ and $$x_{i}\in I$$ for $$1\le i\le m$$, and let $$q_{i}>0$$ for $$1\le i\le m$$. Then

$$f \Biggl(\frac{1}{\sum_{i=1}^{m}q_{i}}\sum_{i=1}^{m}q_{i}x_{i} \Biggr)\le \frac{1}{\sum_{i=1}^{m}q_{i}}\sum_{i=1}^{m}q_{i}f(x_{i}).$$
(3.6)

This inequality is called Jensen’s discrete inequality for convex functions in the literature [13, Sect. 1.4] and [14, Chapter I]. Applying (3.6) to $$f(x)=x\ln x$$ which is convex on $$(0,\infty )$$, $$x_{i}=\frac{a_{i}}{p_{i}}$$, and $$q_{i}=p_{i}$$ leads to

$$\Biggl(\sum_{i=1}^{m}p_{i} \frac{a_{i}}{p_{i}} \Biggr) \ln \Biggl( \sum_{i=1}^{m}p_{i} \frac{a_{i}}{p_{i}} \Biggr) \le \sum_{i=1}^{m}p_{i} \frac{a_{i}}{p_{i}}\ln \frac{a_{i}}{p_{i}}.$$

Accordingly,

$$\lim_{x\to \infty }\bigl[\ln \mathcal{Q}(x)\bigr]'\le ( \varrho -1)\sum_{i=1}^{m}a_{i} \ln p_{i}\le 0, \quad \varrho \ge 1.$$

Consequently, when $$\theta =0$$, $$\rho =1$$, and $$\varrho \ge 1$$, the function $$\mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x)$$ is logarithmically completely monotonic on $$(0,\infty )$$.

The limit

$$\lim_{x\to 0^{+}}\bigl[\ln \mathcal{Q}_{m,a,p,1,0,0}(x) \bigr]'=0$$

obtained above implies that $$[\ln \mathcal{Q}_{m,a,p,1,0,0}(x)]'\ge 0$$, $$\mathcal{Q}_{m,a,p,1,0,0}(x)$$ is increasing, and then $$[\ln \mathcal{Q}_{m,a,p,1,0,0}(x)]'$$ is a Bernstein function on $$(0,\infty )$$.

When $$(\rho ,\varrho ,\theta )\in S$$, the limits

$$\lim_{x\to 0^{+}}\bigl[\ln \mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x) \bigr]'< 0$$

and

$$\lim_{x\to \infty }\bigl[\ln \mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x) \bigr]'= \infty$$

derived above mean that the first derivative $$[\ln \mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x)]'$$ has a unique zero on $$(0,\infty )$$, that is, the functions $$\ln \mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x)$$ and $$\mathcal{Q}_{m,a,p,\rho ,\varrho ,\theta }(x)$$ have a unique minimum on $$(0,\infty )$$. The proof of Theorem 3.1 is complete. □

## An open problem

Let $$m,n\ge 2$$, $$\rho ,\varrho ,\theta \in \mathbb{R}$$, let $$\lambda =(\lambda _{ij})_{m\times n}$$ with $$\lambda _{ij}>0$$ for $$1\le i\le m$$ and $$1\le j\le n$$, let $$\nu _{i}=\sum_{j=1}^{n}\lambda _{ij}$$ and $$\tau _{j}=\sum_{i=1}^{m}\lambda _{ij}$$ for $$1\le i\le m$$ and $$1\le j\le n$$, and let $$p=(p_{ij})_{m\times n}$$ with $$\sum_{i=1}^{m}\sum_{j=1}^{n}p_{ij}=1$$ and $$p_{ij}\in (0,1)$$ for $$1\le i\le m$$ and $$1\le j\le n$$. Define

$$Q_{m,n;\lambda ;p;\rho ;\varrho ;\theta }(x)= \frac{\prod_{i=1}^{m}[\varGamma (1+\nu _{i}x)]^{\nu _{i}^{\theta }} \prod_{j=1}^{n} [\varGamma (1+\tau _{j}x ) ]^{\tau _{j}^{\theta }}}{\prod_{i=1}^{m}\prod_{j=1}^{n} [\varGamma (1+\lambda _{ij}x ) ]^{\rho \lambda _{ij}^{\theta }}} \Biggl(\prod _{i=1}^{m}p_{ij}^{\lambda _{ij}} \Biggr)^{\varrho x}$$
(4.1)

on the infinite interval $$(0,\infty )$$.

Can one find monotonicity properties for the function $$Q_{m,n;\lambda ;p;\rho ;\varrho ;\theta }(x)$$ defined in equation (4.1)?

### Remark 4.1

This paper is a slightly revised version of the electronic preprint .

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### Acknowledgements

The authors are thankful to anonymous referees for their careful corrections, helpful suggestions, and valuable comments on the original version of this paper.

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## Funding

The second author was partially supported by the National Research Foundation of Korea under Grant NRF-2018R1D1A1B07041846, South Korea.

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Correspondence to Dongkyu Lim.

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