Complete Monotonicity Research Articles

Complete monotonicity of solutions of the Abel equation $F(e^x)=F(x)+1$

Complete monotonicity of solutions of the Abel equation $F(e^x)=F(x)+1$

Two Approximation Formulas for Bateman’s G-Function with Bounded Monotonic Errors

Two new approximation formulas for Bateman’s G-function are presented with strictly monotonic error functions and we deduced their sharp bounds. We also studied the completely monotonic (CM) degrees of two functions involving G(r), deducing two of its inequalities and improving some of the recently published results.

Convolution of beta prime distribution

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae’s relationships for 3 F 2 ( 1 ) {}_3F_2(1) . Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turán’s inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill’s ratio is discussed in detail.

Complete monotonicity of time-changed Lévy processes at first passage

We consider the class of (possibly killed) spectrally positive Lévy process that have been time-changed by the inverse of an integral functional. Within this class we characterize the family of those processes which satisfy the following property: as functions of point of issue, the Laplace transforms of their first-passage times downwards are completely monotone. A wide (dense, in a sense) subfamily of this family admits closed form expressions for said Laplace transforms.

Complete monotonicity of the remainder of the asymptotic series for the ratio of two gamma functions

For u>0 with u≠1,2 and m∈N and letfm(x;u)=1u−1ln⁡Γ(x+u)Γ(x+1)−ln⁡(x+u2)−∑k=2mck(u)xk. We prove the complete monotonicity of fm(x;u) for m=2,3 and f4n−1(x;1/2), −f4n−2(x;1/2) for n∈N in x on (0,∞). From these several new sharp bounds for the ratio of gamma functions and divided difference of polygamma functions are established. Lastly, a conjecture is posed.

A deep look into the dagum family of isotropic covariance functions

AbstractThe Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in $\mathbb{R}^d$ of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.

Decreasing property and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

Abstract With the aid of convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the author presents decreasing property of a ratio constituted via three derivatives of a sum involving trigamma function and discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic.

Convexity of ratios of the modified Bessel functions of the first kind with applications.

Let I_{nu }left( xright) be the modified Bessel function of the first kind of order nu . Motivated by a conjecture on the convexity of the ratio W_{nu }left( xright) =xI_{nu }left( xright) /I_{nu +1}left( xright) for nu >-2, using the monotonicity rules for a ratio of two power series and an elementary technique, we present fully the convexity of the functions W_{nu }left( xright) , W_{nu }left( xright) -x^{2}/left( 2nu +4right) and W_{nu }left( x^{1/theta }right) for theta ge 2 on left( 0,infty right) in different value ranges of nu , which give an answer to the conjecture and extend known results. As consequences, some monotonicity results and new functional inequalities for W_{nu }left( xright) are established. As applications, an open problem and a conjectures are settled. Finally, a conjecture on the complete monotonicity of W_{nu }left( x^{1/theta }right) for theta ge 2 is proposed.

Complete Monotonicity for a New Ratio of Finitely Many Gamma Functions

In this paper, by deriving an inequality involving the generating function of the Bernoulli numbers, the author introduces a new ratio of finitely many gamma functions, finds complete monotonicity of the second logarithmic derivative of the ratio, and simply reviews the complete monotonicity of several linear combinations of finitely many digamma or trigamma functions.

Complete Monotonicity of Functions Related to Trigamma and Tetragamma Functions

In this paper, we study the completely monotonic property of two functions involving the function △(x) = and deduce the double inequality , x > 0 which improve some recent results, where ψ(x) is the logarithmic derivative of the Gamma function. Also, we deduce the completely monotonic degree of a function involving ψ'(x).

Detecting p‐Hacking

We theoretically analyze the problem of testing for p‐hacking based on distributions of p‐values across multiple studies. We provide general results for when such distributions have testable restrictions (are non‐increasing) under the null of no p‐hacking. We find novel additional testable restrictions for p‐values based on t‐tests. Specifically, the shape of the power functions results in both complete monotonicity as well as bounds on the distribution of p‐values. These testable restrictions result in more powerful tests for the null hypothesis of no p‐hacking. When there is also publication bias, our tests are joint tests for p‐hacking and publication bias. A reanalysis of two prominent data sets shows the usefulness of our new tests.

A Laplace Mixture Representation of the Horseshoe and Some Implications

The horseshoe prior, defined as a half Cauchy scale mixture of normal, provides a state of the art approach to Bayesian sparse signal recovery. We provide a new representation of the horseshoe density as a scale mixture of the Laplace density, explicitly identifying the mixing measure. Using the celebrated Bernstein--Widder theorem and a result due to Bochner, our representation immediately establishes the complete monotonicity of the horseshoe density and strong concavity of the corresponding penalty. Consequently, the equivalence between local linear approximation and expectation--maximization algorithms for finding the posterior mode under the horseshoe penalized regression is established. Further, the resultant estimate is shown to be sparse.

Logarithmically complete monotonicity of a matrix-parametrized analogue of the multinomial distribution

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.

Monotonicity and inequalities involving the modified Bessel functions of the second kind

Let Kν(x) be the modified Bessel functions of the second kind. In this paper, we give the monotonicity and complete monotonicity results for several functions involving Kν(x), and establish several new sharp double inequalities for Kν(x). In particular, the double inequalities(x+a1)ν−1/2<2πxνexKν(x)<(x+b1)ν−1/2(1+a2x)ν−1/2<21−νΓ(ν)xνexKν(x)<(1+b2x)ν−1/2hold for x>0 and ν≥1 with the best constantsa1=min⁡{c0,12ν+14} and b1=max⁡{c0,12ν+14},a2=1max⁡{c0,ν−1/2} and b2=1min⁡{c0,ν−1/2},where c0=2(Γ(ν)/π)2/(2ν−1).

Logarithmically complete monotonicity of ratios of q-gamma functions

For u,v,r,s∈R and 0<q≠1, let Γq, ψq be the q-gamma, -psi functions and let Wq;u,v be defined on (−min⁡{u,v},∞) byWq;u,v(x)=(Γq(x+u)Γq(x+v))1/(u−v) if u≠v and Wq;u,u(x)=eψq(x+u). In this paper, by a new way we present the necessary and sufficient conditions for the ratio (Wq;u,v/Wq;r,s) to be logarithmically completely monotonic on (−ρ,∞), where ρ=min⁡{u,v,r,s}. This extends and generalizes some existing results.

A Solution to Qi&amp;#039;s Eighth Open Problem on Complete Monotonicity

A Solution to Qi&amp;#039;s Eighth Open Problem on Complete Monotonicity

Lower Bound of Sectional Curvature of Fisher–Rao Manifold of Beta Distributions and Complete Monotonicity of Functions Involving Polygamma Functions

In the paper, by virtue of convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the Fisher–Rao manifold of beta distributions.

Moment-sequence transforms

We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, with possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.

Using Complete Monotonicity to Deduce Local Error Estimates for Discretisations of a Multi-Term Time-Fractional Diffusion Equation

Abstract Time-fractional initial-boundary problems of parabolic type are considered. Previously, global error bounds for computed numerical solutions to such problems have been provided by Liao et al. (SIAM J. Numer. Anal. 2018, 2019) and Stynes et al. (SIAM J. Numer. Anal. 2017). In the present work we show how the concept of complete monotonicity can be combined with these older analyses to derive local error bounds (i.e., error bounds that are sharper than global bounds when one is not close to the initial time t = 0 {t=0} ). Furthermore, we show that the error analyses of the above papers are essentially the same – their key stability parameters, which seem superficially different from each other, become identical after a simple rescaling. Our new approach is used to bound the global and local errors in the numerical solution of a multi-term time-fractional diffusion equation, using the L1 scheme for the temporal discretisation of each fractional derivative. These error bounds are α-robust. Numerical results show they are sharp.

Turán Inequalities and Complete Monotonicity for a Class of Entire Functions

Turán Inequalities and Complete Monotonicity for a Class of Entire Functions