Some completely monotonic functions involving the polygamma functions

多Γ函数的Sharp不等式

Sharp inequalities for polygamma functions

证明了二重不等式\frac{(k-1)!}{\Bigl\{x+\Bigl[\frac{(k-1)!}{|ψ^{(k)}(1)|}\Bigr]^{1/k}\Bigr\}^k} + \frac{k!}{x^{k+1}} < \bigl| n ^{（k）}（x）\bigr| < \frac{(k-1)!}{\bigl(x+\frac12\bigr)^k} + \frac{k!}{x^{k+1}}适用于$x>0$和$k\in\mathbb{N}$，并且常数$\Bigl[\frac{(k-1)!}{|ψ^{(k)}(1)|}\Bigr]^{1/k}$和$\frac12$是最佳可能。同时，还讨论了关于Gamma函数、psi函数和多Γ函数的一些相关不等式和（数学上）完全单调性结果.

含二、三Γ函数的均差函数完全单调的充要条件

Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic

本文给出了包含双伽玛函数和三Γ函数的均差的函数完全单调的充要条件。在此基础上，给出了包含两个Gamma函数之比的函数是代数完全单调的充要条件，并给出了多Γ函数的均差界的一些二重不等式.

A completely monotonic function involving the tri- and tetra-gamma functions

The psi function $ψ(x)$ is defined by $ψ(x)=\frac{Γ'(x)}{Γ(x)}$ and $ψ^{(i)}(x)$ for $i\in\mathbb{N}$ denote the polygamma functions, where $Γ(x)$ is the gamma function. In this paper we prove that a function involving the difference between $[ψ'(x)]^2+ψ''(x)$ and a proper fraction of $x$ is completely monotonic on $(0,\infty)$.

两个Γ函数之比的界--从Gautschi不等式和Kershaw不等式到完全单调函数

Bounds for the ratio of two gamma functions--From Gautschi's and Kershaw's inequalities to completely monotonic functions

本文以界定两个Γ函数之比为主线，回顾和分析了几个不等式，几个涉及两个伽玛函数之比或$q$ -Γ函数的函数的完全单调性，一个涉及两个Γ函数之比的函数的数学完全单调性，给出了两个Γ函数之比和多Γ函数均差的新的界，以及相关的单调性结果.

More supplements to a class of logarithmically completely monotonic functions associated with the gamma function

In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on $(0,\infty)$. As applications of the necessary and sufficient condition, some inequalities for bounding the psi and polygamma functions and the ratio of two gamma functions are derived.

Complete monotonicity of a family of functions involving the tri- and tetra-gamma functions

The psi function $ψ(x)$ is defined by $ψ(x)=\frac{Γ'(x)}{Γ(x)}$ and $ψ^{(i)}(x)$ for $i\in\mathbb{N}$ denote polygamma functions, where $Γ(x)$ is the gamma function. In this paper, we prove that the function $$ [ψ'(x)]^2+ψ"(x)-\frac{x^2+λx+12}{12x^4(x+1)^2} $$ is completely monotonic on $(0,\infty)$ if and only if $λ\le0$, and so is its negative if and only if $λ\ge4$. From this, some inequalities are refined and sharpened.

关于完全单调函数

On completely monotonic functions

本文研究了完全单调函数，给出了该函数的若干性质与新类，并提供了满足完全单调条件的特殊函数。

Some completely monotonic functions involving the $q$-gamma function

We present some completely monotonic functions involving the $q$-gamma function that are inspired by their analogues involving the gamma function.

Dunkl completely monotonic functions

We introduce the notion of Dunkl completely monotonic functions on $\left(-σ,σ\right), σ>0$. We establish a restrictive version of the analogue of Schoenberg's theorem in Dunkl setting.

On a conjecture of a logarithmically completely monotonic function

In this short note we prove a conjecture for the interval $(0,1)$, related to a logarithmically completely monotonic function, presented in \cite{BG}. Then, we extend by proving a more generalized theorem. At the end we pose an open problem on a logarithmically completely monotonic function involving $q$-Digamma function.

Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions

In the paper, the author establishes inequalities, monotonicity, convexity, and unimodality for functions concerning the modified Bessel functions of the first kind and compute the completely monotonic degrees of differences between the exponential and trigamma functions.

An application of a moment problem to completely monotonic functions

We consider the following question: if a function of the form $\int_0^{\infty}φ(t)\, e^{-xt}dt$ is completely monotonic, is it then $φ\ge0$? It turns out that the question is related to a moment problem. In the end we apply those results to answer some questions concerning complete monotonicity of certain functions raised in F. Qi and R. Agarwal, \textit{On complete monotonicity for several classes of functions related to ratios of gamma functions}, J. Inequal. Appl. (2019).

Positivity of sums and integrals for higher order nabla-convex and completely monotonic functions

We extend the definitions of $\nabla-$convex and completely monotonic functions for two variables. Some general identities of Popoviciu type for sum $\sum \sum p_{ij} f(y_i, z_j)$ and integrals $\int P(y)f(y) dy$, $\int \int P(y,z) f(y,z) dy \, dz$ are deduced. Using obtained identities, positivity of these expressions are characterized for higher order $\nabla-$convex and completely monotonic functions. Some applications in terms of generalized Cauchy means and exponential convexity are given.

A function Class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions

In this paper we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite . We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions.

Completely monotonic ratios of basic and ordinary gamma functions

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…

Partial solutions to several conjectures on completely monotonic degrees for remainders in asymptotic expansions of the digamma function

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.

Conjecture concerning a completely monotonic function

Based on a sequence of numerical computations, a conjecture is presented regarding the class of functions $H(x;a)=\exp(a)-(1+a/x)^x$, and the open problem of determining the values of $a$ for which the functions are completely monotonic with respect to $x$. The critical value of $a$ is determined here to sufficient accuracy to show that it is not a simple symbolic quantity.

A Class of logarithmically completely monotonic functions relating the $q$-gamma function and applications

In this paper, the logarithmically complete monotonicity property for a functions involving $q$-gamma function is investigated for $q\in(0,1).$ As applications of this results, some new inequalities for the $q$-gamma function are established. Furthermore, let the sequence $r_n$ be defined by $$n!=\sqrt{2πn}(n/e)^n e^{r_n}.$$ We establish new estimates for Stirling's formula remainder $r_n.$

两个Γ函数之比的界--从Wendel不等式及相关不等式到数学上完全单调的函数

Bounds for the ratio of two gamma functions--From Wendel's and related inequalities to logarithmically completely monotonic functions

本文以界定两个Γ函数之比为主线，回顾和分析了几个不等式，几个涉及两个伽玛函数或$q$ -Γ函数之比的函数的完全单调性，以及涉及两个伽玛函数或$q$ -Γ函数之比的函数在数学上完全单调的充要条件。

A completely monotonic function involving the gamma and tri-gamma functions

In the paper the author provides necessary and sufficient conditions on $a$ for the function $\frac{1}{2}\ln(2π)-x+\bigl(x-\frac{1}{2}\bigr)\ln x-\lnΓ(x)+\frac1{12}{ψ'(x+a)}$ and its negative to be completely monotonic on $(0,\infty)$, where $a\ge0$ is a real number, $Γ(x)$ is the classical gamma function, and $ψ(x)=\frac{Γ'(x)}{Γ(x)}$ is the di-gamma function. As applications, some known results and new inequalities are derived.

Some completely monotonic properties for the $(p,q )$-gamma function

It is defined $Γ_{p,q}$ function, a generalize of $Γ$ function. Also, we defined $ψ_{p,q}$-analogue of the psi function as the log derivative of $Γ_{p,q}$. For the $Γ_{p,q}$ -function, are given some properties related to convexity, log-convexity and completely monotonic function. Also, some properties of $ψ_{p,q} $ analog of the $ψ$ function have been established. As an application, when $p\to \infty, q\to 1,$ we obtain all result of \cite{Valmir1} and \cite{SHA}.

与Γ函数有关的数学上完全单调的函数

Some logarithmically completely monotonic functions related to the gamma function

本文给出了定义在$(-1,\infty)$或$(0,\infty)$中的函数$\frac1{[Γ(x+1)]^{1/x}}$，$\frac{[{Γ(x+α+1)}]^{1/(x+α)}}{[{Γ(x+1)}]^{1/x}}$，$\frac{[Γ(x+1)]^{1/x}}{(x+1)^α}$和$\frac{[Γ(x+1)]^{1/x}}{x^α}$对给定实数$α\in\tr$的数学完全单调性，恢复、推广和推广了一些已知结果.此外，还建立了代数完全单调函数的一些基本性质。

Double Inequalities for Complete Monotonicity Degrees of Remainders of Asymptotic Expansions of the Gamma and Digamma Functions

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.

幂绝对单调函数的一个性质及幂指数函数的完全单调性

A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function

引入了“代数绝对单调函数”的概念，揭示了代数绝对单调函数也是绝对单调的一个包含，证明了函数$\bigl(1+\fracαx\bigr) ^{x+β}$的代数完全单调性和代数绝对单调性，其中$α$和$β$是给定实参数，给出了一个新的证明，证明了一个代数上完全单调的函数也是完全单调的，并提出了一个公开问题。

Gamma函数和$q$ -Γ函数的单调性

Some Monotonicity Properties of Gamma and $q$-gamma Functions

证明了完全单调函数的一些性质，并应用它们得到了关于Gamma函数和$q$ -Γ函数的结果.

Completely monotonic gamma ratio and infinitely divisible H-function of Fox

We investigate conditions for logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has different scaling factor. We give necessary and sufficient conditions in terms of nonnegativity of some elementary function and more practical sufficient conditions in terms of parameters. Further, we study the representing measure in Bernstein's theorem for both equal and non-equal scaling factors. This leads to conditions on parameters under which Meijer's $G$-function or Fox's $H$-function represents an infinitely divisible probabili…

On generalized Stieltjes functions

It is shown that a function $f$ is a generalized Stieltjes function of order $λ>0$ if and only if $x^{1-λ}(x^{λ-1+k}f(x))^{(k)}$ is completely monotonic for all $k\geq 0$, thereby complementing a result due to Sokal. Furthermore, a characterization of those completely monotonic functions $f$ for which $x^{1-λ}(x^{λ-1+k}f(x))^{(k)}$ is completely monotonic for all $k\leq n$ is obtained in terms of properties of the representing measure of $f$.

Completely monotonic degree of a function involving the tri- and tetra-gamma functions

Let $ψ(x)$ be the di-gamma function, the logarithmic derivative of the classical Euler's gamma function $Γ(x)$. In the paper, the author shows that the completely monotonic degree of the function $[ψ'(x)]^2+ψ''(x)$ is $4$, surveys the history and motivation of the topic, supplies a proof for the claim that a function $f(x)$ is strongly completely monotonic if and only if the function $xf(x)$ is completely monotonic, conjectures the completely monotonic degree of a function involving $[ψ'(x)]^2+ψ''(x)$, presents the logarithmic concavity and monotonicity of an elementary function, and poses an …

正级完全单调函数与Barnes双Γ函数和Euler Γ函数的对数的渐近展开

Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function

引入$r>0$级完全单调函数，证明了Barnes双Γ函数和Euler Γ函数的对数的渐近展开的展开式可产生任意正整数级的完全单调函数.

Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions

In the paper, the authors verify the complete monotonicity of the difference $e^{1/t}-ψ'(t)$ on $(0,\infty)$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of $e^{1/z}$, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind.

A completely monotonic function used in an inequality of Alzer

The function $G(x)=(1-\ln x /\ln(1+x))x\ln x$ has been considered by Alzer, Qi and Guo. We prove that $G'$ is completely monotonic by finding an integral representation of the holomorphic extension of $G$ to the cut plane. A main difficulty is caused by the fact that $G'$ is not a Stieltjes function.

Monotonicity patterns and functional inequalities for classical and generalized Wright functions

In this paper our aim is to present the completely monotonicity and convexity properties for the Wright function. As consequences of these results, we present some functional inequalities. Moreover, we derive the monotonicity and log-convexity results for the generalized Wright functions. As applications, we present several new inequalities (like Turán type inequalities) and we prove some geometric properties for four--parametric Mittag--Leffler functions.

Complete monotonicity of a function involving the gamma function and applications

In the article we present necessary and sufficient conditions for a function involving the logarithm of the gamma function to be completely monotonic and apply these results to bound the gamma function $Γ(x)$, the $n$-th harmonic number $\sum_{k=1}^n\frac1k$, and the factorial $n!$.

涉及psi函数的函数完全单调性的两个新证明

Two new proofs of the complete monotonicity of a function involving the psi function

本文给出了函数$x^α[\ln x-ψ(x)]$在$(0,\infty)$上完全单调的充要条件$α\le1$的两个新的证明.

A logarithmically completely monotonic function involving the ratio of gamma functions

In the paper, the authors concisely survey and review some functions involving the gamma function and its various ratios, simply state their logarithmically complete monotonicity and related results, and find necessary and sufficient conditions for a new function involving the ratio of two gamma functions and originating from the coding gain to be logarithmically completely monotonic.

Complete monotonicity of a function involving the $p$-psi function and alternative proofs

In the paper the authors alternatively prove that the function $x^α\big[\ln\frac{px}{x+p+1}-ψ_p(x)\big]$ is completely monotonic on $(0,\infty)$ if and only if $α\le 1$, where $p\in\mathbb{N}$ and $ψ_p(x)$ is the $p$-analogue of the classical psi function $ψ(x)$. This generalizes a known result.

Some results on the complete monotonicity of the Mittag-Leffler functions of Le Roy type

The paper by R. Garrappa, S. Rogosin, and F. Mainardi, entitled {\em On a generalized three-parameter Wright function of the Le Roy type} and published in [Fract. Calc. Appl. Anal. {\bf 20} (2017) 1196-1215], ends up leaving the open question concerning the range of the parameters $α, β$ and $γ$ for which Mittag-Leffler functions of Le Roy type $F_{α, β}^{(γ)}$ are completely monotonic. Inspired by the 1948 seminal H. Pollard's paper which provides the proof of the complete monotonicity of the one parameter Mittag-Leffler function, the Pollard approach is used to find the Laplace transform rep…

Positivity of certain class of functions related to the Fox H-functions and applications

In this present investigation, we found a set of sufficient conditions to be imposed on the parameters of the Fox H-functions which allow us to conclude that it is non-negative. As applications, various new facts regarding the Fox-Wright functions, including complete monotonicity, logarithmic completely monotonicity and monotonicity of ratios are considered.

Logarithmically completely monotontic functions related the $q-$gamma and the $q-$digamma functions with applications

In this paper we present several new classes of logarithmically completely monotonic functions. Our functions have in common that they are defined in terms of the $q-$gamma and $q-$digamma functions. As an applications of this results, some inequalities for the $q-$gamma and the $q-$digamma functions are established. Some of the given results generalized theorems due to Alzer and Berg and C.-P. Chen and F. Qi.

Complete monotonicity of a function involving a ratio of gamma functions and applications

In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result in [\emph{Inequalities and monotonicity for the ratio of gamma functions}, Taiwanese J. Math. \textbf{7} (2003), no.~2, 239\nobreakdash--247.] and others. As applications, several inequalities involving the volume of the unit ball in $\mathbb{R}^n$ are derived, which refine, generalize and extend some known inequalities.

Complete monotonicity of functions involving the $q$-trigamma and $q$-tetragamma functions

Let $ψ_q(x)$, $ψ_q'(x)$, and $ψ_q''(x)$ for $q>0$ stand respectively for the $q$-digamma, $q$-trigamma, and $q$-tetragamma functions. In the paper, the author proves along two different approaches that the functions $[ψ'_q(x)]^2+ψ''_q(x)$ for $q>1$ and $[ψ_{q}'(x)-\ln q]^2 +ψ''_{q}(x)$ for $0<q<1$ are completely monotonic on $(0,\infty)$. Applying these results, the author derives monotonic properties of four functions involving the $q$-digamma function $ψ_q(x)$ and two double inequalities for bounding the $q$-digamma function $ψ_q(x)$.

完全单调函数与对数完全单调函数

Completely and logarithmically completely monotone functions

本研究针对完全单调与对数完全单调函数领域，通过简化定理证明、拓展适用范围、纠正错误、推广及提出新结论，研究相关Gamma函数的单调性，并提出psi函数相关公开问题，推动该理论发展。

On complete monotonicity of solution to the fractional relaxation equation with the $n$th level fractional derivative

In this paper, we first deduce the explicit formulas for the projector of the $n$th level fractional derivative and for its Laplace transform. Then the fractional relaxation equation with the $n$th level fractional derivative is discussed. It turns out that under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag-Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the 2nd level derivative.

Complete monotonicity of a ratio of gamma functions and some combinatorial inequalities for multinomial coefficients

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 …

$q$-Bernstein functions and applications

We characterize of the $q$-Bernstein functions in terms of $q$-Laplace transform. Moreover, we present several results of $q$-completely monotonic, $q$-log completely monotonic and $q$-Bernstein functions.

Complete Monotonicity of classical theta functions and applications

We produce trigonometric expansions for Jacobi theta functions\\ $θ_j(u,τ), j=1,2,3,4$\ where $τ=iπt, t > 0$. This permits us to prove that\ $\log \frac{θ_j(u, t)}{θ_j(0, t)}, j=2,3,4$ and $\log \frac{θ_1(u, t)}{πθ'_1(0, t)}$ as well as $\frac{\frac{δθ_j}{δu}}{θ_j}$ as functions of $t$ are completely monotonic. We also interested in the quotients $S_j(u,v,t) = \frac{θ_j(u/2,iπt)}{θ_j(u/2,iπt)}$. For fixed $u,v$ such that $0\leq u < v < 1$ we prove that the functions $\frac{(\fracδ{δt}S_j)}{S_j}$ for $j=1,4$ as well as the functions $-\frac{(\fracδ{δt}S_j)}{S_j}$ for $j=2,3$ are completely mono…

对数函数的完全单调性

Complete monotonicity of log-functions

本文研究涉及对数的 $s$ 元特殊函数族 $\mathcal{F}_s$ 的完全单调性，通过建立其与实多元多项式空间的线性同构，证明了该完全单调函数锥是半代数的，并给出了有限时间判定算法。

On the coefficients of an expansion of $(1+1/x)^x$ related to Carleman's inequality

In this note, we present new properties for a sequence arising in some refinements of Carleman's inequality. Our results extend some results of Yang [Approximations for constant e and their applications J. Math. Anal. Appl. 262 (2001) 651-659] and Alzer and Berg [some classes of completely monotonic functions Ann. Acad. Sci. Fennicae 27(2002) 445-460].

Nielsen's beta function and some infinitely divisible distributions

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.

Applications of the Stieltjes and Laplace transform representations of the hypergeometric functions

In our previous work we found sufficient conditions to be imposed on the parameters of the generalized hypergeometric function in order that it be completely monotonic or of Stieltjes class. In this paper we collect a number of consequences of these properties. In particular, we find new integral representations of the generalized hypergeometric functions, evaluate a number of integrals of their products, compute the jump and the average value of the the generalized hypergeometric function over the branch cut, establish new inequalities for this function in the half plane Re(z)<1. Furthermore,…

关于 1/(1-cos(x)) 及其导数的不等式

Inequalities for 1/(1-cos(x)) and its derivatives

本文研究函数 $1/(1-\cos x)$ 的单调性质，并给出其各阶导数在 $x,y,x+y$ 组合下的最佳常数界。

A note on analytic continuation of characteristic functions

We derive necessary and sufficient conditions for a continuous bounded function $f: R\to C$ to be a characteristic function of a probability measure. The Cauchy transform $K_f$ of $f$ is used as analytic continuation of $f$ to the upper and lower half-planes in $C$. The conditions depend on the behavior of $K_f(z)$ and its derivatives on the imaginary axis in $C$. The main results are given in terms of completely monotonic and absolutely monotonic functions.

A family of entire functions connecting the Bessel function $J_1$ and the Lambert $W$ function

Motivated by the problem of determining the values of $α>0$ for which $f_α(x)=e^α- (1+1/x)^{αx},\ x>0$ is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family $φ_α$, $α>0$, of entire functions such that $f_α(x) =\int_0^\infty e^{-sx}φ_α(s)\,ds, \ x>0.$ We show that each function $φ_α$ has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions $φ_α$, which turn out to be related to the well known Bessel function $J_1$ and the Lambert $W$ function. On …

随机集的部分可分性与完全单调函数的幂

Partial divisibility of random sets and powers of completely monotone functions

本文刻画有限格上完全单调函数保持完全单调性的幂指数阈值，证明其在有限分配格上尖锐，并应用于随机闭集空泛函及 m-可分集逼近。