Complete Monotonicity Properties Research Articles

New Monotonicity and Infinite Divisibility Properties for the Mittag-Leffler Function and for Stable Distributions

Hyperbolic complete monotonicity property (HCM) is a way to check if a distribution is a generalized gamma (GGC), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions Eα,α∈(0,2], enjoy the HCM property, and then intervene deeply in the probabilistic context. We prove that for suitable α and complex numbers z, the real and imaginary part of the functions x↦Eαzx, are tightly linked to the stable distributions and to the generalized Cauchy kernel.

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.

New Integral Representations for the Fox–Wright Functions and Its Applications II

In this paper our aim is to establish new integral representations for the Fox–Wright function $${}_{p}\Psi_{q}[^{(\alpha_{p},A_{p})}_{(\beta_{q},B_{q})}|z]$$ when $$\mu=\sum_{j=1}^{q}\beta_{j}-\sum_{k=1}^{p}\alpha_{k}+\frac{p-q}{2}=-m,\;\;m\in\mathbb{N}_{0}.$$ In particular, closed-form integral expressions are derived for the four parameter Wright function under a special restriction on parameters. Exponential bounding inequalities are derived for a class of the Fox–Wright function. Moreover, complete monotonicity property is presented for these functions.

Complete monotonicity and inequalities related to generalized k-gamma and k-polygamma functions

In this paper, we prove new complete monotonicity properties of some functions related to generalized k-gamma and k-polygamma functions. Applications of the results yield various new inequalities. In the end, double inequalities are constructed involving the k-generalized digamma and polygamma functions.

Logarithmically Complete Monotonicity of Certain Ratios Involving the \(k\)-Gamma Function

In this paper, we prove logarithmically complete monotonicity properties of certain ratios of the \(k\)-gamma function. As a consequence, we deduce some inequalities involving the \(k\)-gamma function and the \(k\)-trigamma function.

Extension of complete monotonicity results involving the digamma function

Abstract By using some analytical techniques, we prove a complete monotonicity property of a certain function involving the (p, k)-digamma function. Subsequently, we derive some inequalities for the (p, k)- digamma function. As special cases of the established results, we deduce some new results concerning the p-digamma and the k-digamma functions. Our results are extensions of some previous results due to Qiu and Vuorinen, Mortici, and Merovci.

Complete Monotonicity Properties of a Function Involving the Polygamma Function

Complete Monotonicity Properties of a Function Involving the Polygamma Function

On Modifications of the Exponential Integral with the Mittag-Leffler Function

Abstract In this paper we survey the properties of the Schelkunoff modification of the Exponential integral and we generalize it with the Mittag-Leffler function. So doing we get a new special function (as far as we know) that may be relevant in linear viscoelasticity because of its complete monotonicity properties in the time domain. We also consider the generalized sine and cosine integral functions.

Monotonicity, concavity, and inequalities related to the generalized digamma function

In this paper, we establish a concave theorem and some inequalities for the generalized digamma function. Hence, we give complete monotonicity property of a determinant function involving all kinds of derivatives of the generalized digamma function.

Some monotonicity properties and inequalities for the (p, k)-gamma function

In this paper, the authors present some complete monotonicity properties and some inequalities involving the (p, k)-analogue of the Gamma function. The established results provide the (p, k)-generalizations for some results known in the literature.

A real distinct poles rational approximation of generalized Mittag-Leffler functions and their inverses: Applications to fractional calculus

The generalized Mittag-Leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives, respectively. However, their computational complexities have made them difficult to deal with numerically. We propose a real distinct pole rational approximation of the generalized Mittag-Leffler function. Under some mild conditions, this approximation is proven and empirically shown to be L-Acceptable. Due to the complete monotonicity property of the Mittag-Leffler function, we derive a rational approximation for the inverse generalized Mittag-Leffler function. These approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order. Several applications are presented such as complementary error function, solution of fractional differential equations, and the ultraslow diffusion model using the structural derivative.

Monotonicity and absolute monotonicity for the two-parameter hyperbolic and trigonometric functions with applications

In this paper, we present the monotonicity and absolute monotonicity properties for the two-parameter hyperbolic and trigonometric functions. As applications, we find several complete monotonicity properties for the functions involving the gamma function and provide the bounds for the error function.

Certain approximations to achieve sharp lower and upper bounds for the Mills' ratio of the inverse Gaussian distribution

In this note, we prove new complete monotonicity properties of some functions associated with the inverse Gaussian distribution. Based on these properties, we present some lower and upper bounds with explicit expressions for the Mills' ratio of the inverse Gaussian distribution. Finally, for demonstrating the efficiency of our estimates, some numerical computations are provided.

A class of logarithmically completely monotonic functions related to 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 $$n!=\sqrt{2\pi n}(n/e)^n e^{r_n}.$$ We establish new estimates for Stirling's formula remainder $r_n.$

Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function

In the paper, the authors find integral representations, complete monotonicity, limits, and other properties of remainders of the Binet and Stirling formulas for the gamma function and their derivatives. These properties strengthen almost all results in three papers published in the Journal of Computational and Applied Mathematics, Applied Mathematics Letters, and Applied Mathematics and Computation in the years 2006, 2011, and 2014 by seven mathematicians. The proofs in the paper unify and are simpler than those in the three papers.

Asymptotic expansions and completely monotonic functions associated with the gamma, psi and polygamma functions

We present new asymptotic expansions of the logarithm of the gamma function in terms of the psi and polygamma functions. Based on these expansions, we prove new complete monotonicity properties of some functions involving the gamma, psi and polygamma functions.

Some completely monotonic functions associated with the q-gamma and the q-polygamma functions

In this paper, the q-analogue of the Stirling formula for the q-gamma function (Moak formula) is exploited to prove the complete monotonicity properties of some functions involving the q-gamma and the q-polygamma functions for all real number q > 0. The monotonicity of these functions is used to establish sharp inequalities for the q-gamma and the q-polygamma functions and the q-Harmonic number. Our results are shown to be a generalization of results which were obtained by Selvi and Batir [23].

Symmetric polynomials in information theory: Entropy and subentropy

Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore, we see that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials, we also derive a series of further properties of H and Q.

Completely monotone functions and some classes of fractional evolution equations

The abstract Cauchy problem for the distributed order fractional evolution equation in the Caputo and in the Riemann–Liouville sense is studied for operators generating a strongly continuous one-parameter semigroup on a Banach space. Continuous as well as discrete distribution of fractional time-derivatives of order less than one is considered. The problem is reformulated as an abstract Volterra integral equation. It is proven that its kernel satisfies certain complete monotonicity properties. Based on these properties, the well-posedness of the problem is established and a series expansion of the solution is obtained. In case of ordered Banach space this representation implies positivity of the solution operator. In addition, a subordination formula is obtained.

Monotonicity properties of [formula omitted]-digamma and [formula omitted]-trigamma functions

Some complete monotonicity results for q-polygamma functions are proved. Our results extend positivity of some functions containing q-polygamma functions to complete monotonicity property. Also, we give two new inequalities for q-trigamma function.