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.
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1 Preliminaries
Let \(f(x)\) be an infinite differentiable function on an infinite interval \((0,\infty )\).
- (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)
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)
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
or by
See [1, Chap. 6], [15, Chap. 5], the paper [18], and [34, Chap. 3]. In the literature, the logarithmic derivative
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
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
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
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
and
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
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
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)
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)
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)
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)
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
and
in [1, p. 260, 6.4.1], it follows that
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
In [27, Theorem 4.1], it was discovered that
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
for \(x>0\), \(\lambda _{k}>0\), and \(\alpha \ge 0\). Inequality (3.3) can be equivalently formulated as
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
in [8, Theorem 1] and [9, Sect. 1.4], we have
where \(\psi (1)=-0.577\ldots \) , and
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
Replacing \(\xi _{i}=\frac{a_{i}}{\sum_{i=1}^{m}a_{i}}\) and \(x=\theta +1\) in the above inequality yields
This can be further rewritten as
Considering inequality (3.5) reveals that
- (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)
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
when \(\theta =0\) and \(\rho =1\), we have
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
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
Accordingly,
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
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
and
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
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].
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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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The second author was partially supported by the National Research Foundation of Korea under Grant NRF-2018R1D1A1B07041846, South Korea.
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Qi, F., Lim, D. Monotonicity properties for a ratio of finite many gamma functions. Adv Differ Equ 2020, 193 (2020). https://doi.org/10.1186/s13662-020-02655-4
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DOI: https://doi.org/10.1186/s13662-020-02655-4