1 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 [28] and the monograph [33].

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 [18], 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.

2 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 [2], 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 [32], 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 [19], 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 [27], 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 \(\sum_{i=1}^{m}p_{i}=1\).

3 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. □

4 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 [30].