We introduce completely monotonic functions of order
$r>0$ and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.
In the paper, the authors introduce a matrix-parametrized generalization of the multinomial probability mass function that involves a ratio of several multivariate gamma functions. They show the logarithmic complete monotonicity of this generalization and derive new inequalities involving ratios of multivariate gamma functions.
In the paper, the authors review origins, motivations, and generalizations of a series of inequalities involving several exponential functions and sums, establish three new inequalities involving finite exponential functions and sums by finding convexity of a function related to the generating function of the Bernoulli numbers, survey the history, backgrounds, generalizations, logarithmically complete monotonicity, and applications of a series of ratios of finite gamma functions, present complete monotonicity of a linear combination of finite trigamma functions, construct a new ratio of finite…
In this paper, we study some properties such as the monotonicity, logarithmically complete monotonicity, logarithmic convexity, and geometric convexity, of the combinations of gamma function and power function. The results we obtain generalize some related known results for parameters with specific values.
Motivated by several conjectures posed in the paper " Completely monotonic degrees for a difference between the logarithmic and psi functions",we confirm in this work some conjectures on completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function and the digamma function and we give two bounded for this degrees.
Motivated by several conjectures posed in the paper "F. Qi and A.-Q. Liu, Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math., vol. 361, pp. 366--371 (2019); available online at https://doi.org/10.1016/j.cam.2019.05.001", the authors compute several completely monotonic degrees of the remainders in the asymptotic expansions of the logarithm of the gamma function and in the asymptotic expansions of the logarithm of the digamma function.
We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of a certain explicitly written measure in the q case and of a certain elementary function in the classical q=1 case. In the latter case we further provide simple new sufficient conditions leading to many new examples of logarithmically completely monotonic gamma ratios. Finally, we apply some of our results to study monotonicity of some gamma ratios and ration…
We show that a large collection of special functions, in particular Nielsen's beta function, are generalized Stieltjes functions of order 2, and therefore logarithmically completely monotonic. This includes the Laplace transform of functions of the form $xf(x)$, where $f$ is itself the Laplace transform of a sum of dilations and translations of periodic functions. Our methods are also applied to ratios of Gamma functions, and to the remainders in asymptotic expansions of the double Gamma function of Barnes.
Exactification is the process of obtaining exact values of a function from its complete asymptotic expansion. Here Stirling's approximation for the logarithm of the gamma function or $\ln Γ(z)$ is derived completely whereby it is composed of the standard leading terms and an asymptotic series that is generally truncated. Nevertheless, to obtain values of $\ln Γ(z)$, the remainder must undergo regularization. Two regularization techniques are then applied: Borel summation and Mellin-Barnes (MB) regularization. The Borel-summed remainder possesses an infinite convergent sum of exponential integr…
For $m,n\in \mathbb{N}$, let $0 < α_i,β_j,λ_{ij} \leq 1$ be such that $\sum_{j=1}^n λ_{ij} = α_i$, $\sum_{i=1}^m λ_{ij} = β_j$, and $\sum_{i=1}^m α_i = \sum_{j=1}^n β_j \leq 1$. We prove that the ratio of gamma functions \begin{equation*} \hspace{-15mm}t \mapsto \frac{\prod_{i=1}^m Γ(α_i t + 1) \prod_{j=1}^n Γ(β_j t + 1)}{\prod_{i=1}^m \prod_{j=1}^n Γ(λ_{ij} t + 1)} \end{equation*} is logarithmically completely monotonic on $(0,\infty)$. This result complements the logarithmically complete monotonicity of multinomial probabilities shown in Ouimet (2018), Qi et al (2018), and the recent survey …
Complete monotonicity of a function involving a ratio of gamma functions and applications
本文给出了一个包含Γ函数之比的函数在代数上完全单调的充要条件。这推广和推广了[ \emph{Inequalities and monotonicity for the ratio of gamma functions}，Taiwan J. Math. \textbf{7}（2003），no.~ 2，239 \nobreakdash --247。]等人作为应用，导出了几个涉及$\mathbb{R}^n$中单位球体积的不等式，改进、推广和推广了一些已知的不等式.
Monotonicity properties of the ratio $$ \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, $$ where $f$ is an entire function are investigated. Earlier results for Euler's gamma function and other entire functions of genus 1 are generalised to entire functions of genus $p$ with negative zeros. Derivatives of order comparable to $p$ of the expression above are related to generalised Stieltjes functions of order $p+1$. Our results are applied to the Barnes multiple gamma functions. We also show how recent results on the behaviour of Euler's gamma function on vertical lines can be shar…
In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, and the Bernstein function property, of the newly introduced ratio, and construct two inequalities of multinomial coefficients and multivariate beta functions.
