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 rational 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.
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.
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.
In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma 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).
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.
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.
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.
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.
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 $(0,\infty)$.
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 $r_n.$
In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, the complete monotonicity of several functions involving ratios of two gamma or $q$-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio of two gamma functions and divided differences of polygamma functions, and related monotonicity results.
In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, several complete monotonicity of functions involving ratios of two gamma or $q$-gamma functions, and necessary and sufficient conditions for functions involving ratios of two gamma or $q$-gamma functions to be logarithmically completely monotonic.
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.
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.
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}.
In this article, logarithmically complete monotonicity properties of some functions such as $\frac1{[Γ(x+1)]^{1/x}}$, $\frac{[{Γ(x+α+1)}]^{1/(x+α)}}{[{Γ(x+1)}]^{1/x}}$, $\frac{[Γ(x+1)]^{1/x}}{(x+1)^α}$ and $\frac{[Γ(x+1)]^{1/x}}{x^α}$ defined in $(-1,\infty)$ or $(0,\infty)$ for given real number $α\in\tr$ are obtained, some known results are recovered, extended and generalized. Moreover, some basic properties of the logarithmically completely monotonic functions are established.
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 …
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.
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 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.
In the article, a notion "logarithmically absolutely monotonic function" is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity and the logarithmically absolute monotonicity of the function $\bigl(1+\fracαx\bigr) ^{x+β}$ are proved, where $α$ and $β$ are given real parameters, a new proof for the inclusion that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed.
A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the …
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.
We present a simple proof of monotonicity for cooperative systems of linear ordinary differential equations, without having recourse to approximation by strongly cooperative systems.
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…
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 $f$.
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.
We prove the conjecture stated 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), 1-42, that the function $1/\arctan$ is logarithmically completely monotonic on $(0,\infty)$, but not a Stieltjes transform.
For $m,n\in\mathbb{N}$, let $f_{m,n}(x)=\bigr[ψ^{(m)}(x)\bigl]^2+ψ^{(n)}(x)$ on $(0,\infty)$. In the present paper, we prove using two methods that, among all $f_{m,n}(x)$ for $m,n\in\mathbb{N}$, only $f_{1,2}(x)$ is nontrivially completely monotonic on $(0,\infty)$. Accurately, the functions $f_{1,2}(x)$ and $f_{m,2n-1}(x)$ are completely monotonic on $(0,\infty)$, but the functions $f_{m,2n}(x)$ for $(m,n)\ne(1,1)$ are not monotonic and does not keep the same sign on $(0,\infty)$.
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.
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 $n!$.
In the present paper, we give two new proofs for the necessary and sufficient condition $α\le1$ such that the function $x^α[\ln x-ψ(x)]$ is completely monotonic on $(0,\infty)$.
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.
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…
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.
In the present paper, we establish necessary and sufficient conditions for the functions $x^α\bigl\lvertψ^{(i)}(x+β)\bigr\lvert$ and $α\bigl\lvertψ^{(i)}(x+β)\bigr\lvert-x\bigl\lvertψ^{(i+1)}(x+β)\bigr\lvert$ respectively to be monotonic and completely monotonic on $(0,\infty)$, where $i\in\mathbb{N}$, $α>0$ and $β\ge0$ are scalars, and $ψ^{(i)}(x)$ are polygamma functions.
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.
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.
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.
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.
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 $ψ_q(x)$.
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…
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.
We consider two operations on the Mittag-Leffler function which cancel the exponential term in the expansion at infinity, and generate a completely monotonic function. The first one is the action of a certain differential-difference operator, and leads to a characterization via some necktie domain. The second one is the subtraction of the exponential term itself multiplied by an incomplete Gamma function. These results extend previous works by various authors.
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 …
We propose a necessary and sufficient condition for a real-valued function on the real line to be a characteristic function of a probability measures. The statement is given in terms of harmonic functions and completely monotonic 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.
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 …
We prove the complete monotonicity on $(0,\infty)^n$ for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result of Szego and answers, among other things, a long-standing question of Lewy and Askey concerning the positivity of Taylor coefficients for certain rational functions. Our proofs are based on two_ab initio_ methods for proving that $P^{-β}$ is completely monotone on a convex cone $C$: the determinantal method and the quadratic-form method. These methods are clo…
