Complete Monotonicity Research Articles

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.

Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex

Let d∈N and let γi∈[0,∞), xi∈(0,1) be such that ∑i=1d+1γi=M∈(0,∞) and ∑i=1d+1xi=1. We prove thata↦Γ(aM+1)∏i=1d+1Γ(aγi+1)∏i=1d+1xiaγi is completely monotonic on (0,∞). This result generalizes the one found by Alzer [2] for binomial probabilities (d=1). As a consequence of the log-convexity, we obtain some combinatorial inequalities for multinomial coefficients. We also show how the main result can be used to derive asymptotic formulas for quantities of interest in the context of statistical density estimation based on Bernstein polynomials on the d-dimensional simplex.

Generalization of inclusion–exclusion integral for nondiscrete monotone measure space

We generalize the inclusion–exclusion integral for nondiscrete monotone measure spaces. The inclusion–exclusion integral for a monotone measure space on a finite space is discussed in previous papers. This is an integral with respect to a nonadditive monotone measure, which includes the Lebesgue and Choquet integrals. We are concerned with the monotonicity of the integral, and a sufficient condition for the monotonicity of the t-norm based inclusion–exclusion integral is given by introducing the complete monotonicity of the t-norm based interaction operator. Moreover concrete examples of the interaction operators that satisfy the complete monotonicity are shown.

Complete Monotonicity of a Function Involving the (P, K)-Digamma Function

Complete Monotonicity of a Function Involving the (P, K)-Digamma Function

Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions

In the paper, the authors extend a function arising from the Bernoulli trials in probability and involving the gamma function to its largest ranges, find logarithmically complete monotonicity of these extended functions, and, in light of logarithmically complete monotonicity of these extended functions, derive some inequalities for multinomial coefficients and multivariate beta functions. These results recover, extend, and generalize some known conclusions.

A comparison theorem for two divided differences and applications to special functions

In this paper, we present a general comparison theorem for two divided differences of a three times differentiable function. This gives a unified treatment for (logarithmically) complete monotonicity, monotonicity and inequalities involving some special functions including gamma, psi and polygamma functions. As their consequences, we not only refine and generalize some important results, but also present simple and interesting alternative proofs of certain earlier results.

Positive Definiteness of Complex Piecewise Linear Functions and Some of Its Applications

Given α ∈ (0, 1) and c = h + iβ, h, β ∈ R, the function fα,c: R → C defined as follows is considered: (1) fα,c is Hermitian, i.e., $${f_{\alpha ,c}}\left( { - x} \right)\overline {{f_{\alpha ,c}}\left( x \right)} ,x \in \mathbb{R};$$ , x ∈ R; (2) fα,c(x) = 0 for x > 1; moreover, on each of the closed intervals [0, α] and [α, 1], the function fα,c is linear and satisfies the conditions fα,c(0) = 1, fα,c(α) = c, and fα,c(1) = 0. It is proved that the complex piecewise linear function fα,c is positive definite on R if and only if m(α) ≤ h ≤ 1 − α and |β| ≤ γ(α, h), $$where m\left( \alpha \right) = \left\{ \begin{gathered} 0if1/\alpha \notin \mathbb{N}, \hfill \\ - \alpha if1/\alpha \in \mathbb{N}. \hfill \\ \end{gathered} \right.$$ If m(α) ≤ h ≤ 1 − α and α ∈ Q, then γ(α, h) > 0; otherwise, γ(α, h) = 0. This result is used to obtain a criterion for the complete monotonicity of functions of a special form and prove a sharp inequality for trigonometric polynomials.

A diagonal recurrence relation for the Stirling numbers of the first kind

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

An integral representation, complete monotonicity, and inequalities of the Catalan numbers

In the paper, by virtue of the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the generating function of the Catalan numbers in combinatorics. From this, the authors derive an alternative integral representation, complete monotonicity, determinantal and product inequalities for the Catalan numbers.

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.

Training set selection for monotonic ordinal classification

In recent years, monotonic ordinal classification has increased the focus of attention for machine learning community. Real life problems frequently have monotonicity constraints. Many of the monotonic classifiers require that the input data sets satisfy the monotonicity relationships between its samples. To address this, a conventional strategy consists of relabeling the input data to achieve complete monotonicity. As an alternative, we explore the use of preprocessing algorithms without modifying the class label of the input data.In this paper we propose the use of training set selection to choose the most effective instances which lead the monotonic classifiers to obtain more accurate and efficient models, fulfilling the monotonic constraints. To show the benefits of our proposed training set selection algorithm, called MonTSS, we carry out an experimentation over 30 data sets related to ordinal classification problems.

Complete monotonicity involving some ratios of gamma functions

In this paper, by using the properties of an auxiliary function, we mainly present the necessary and sufficient conditions for various ratios constructed by gamma functions to be respectively completely and logarithmically completely monotonic. As consequences, these not only unify and improve certain known results including Qi’s and Ismail’s conclusions, but also can generate some new inequalities.

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.

Some inequalities for the Bell numbers

In this paper, we present derivatives of the generating functions for the Bell numbers by induction and by the Faa di Bruno formula, recover an explicit formula in terms of the Stirling numbers of the second kind, find the (logarithmically) absolute and complete monotonicity of the generating functions, and construct some inequalities for the Bell numbers. From these inequalities, we derive the logarithmic convexity of the sequence of the Bell numbers.

(p, q)-Extended Bessel and Modified Bessel Functions of the First Kind

Inspired by certain recent extensions of the Euler’s beta, Gaus hypergeometric and confluent hypergeometric functions (Choi et al. in Honam Math 36(2):339–367, 2014), we introduce (p, q)-extended Bessel function \(J_{\nu ,p,q}\), the (p, q)-extended modified Bessel function \(I_{\nu ,p,q}\) of the first kind of order \(\nu \) by making use two additional parameters in the integrand, as well as the (p, q)-extended Struve \(\mathbf{H}_{\nu ,p,q}\) and the modified Struve \(\mathbf{L}_{\nu ,p,q}\) functions. Systematic investigation of its properties, among others integral representations, bounding inequalites Mellin transforms (for all newly defined Bessel and Struve functions), complete monotonicity, Turan type inequality, associated non-homogeneous differential-difference equations (exclusively for extended Bessel functions) are presented. Brief presentation of another members of Bessel functions family: spherical, ultraspherical, Delerue hyper-Bessel and their modified counterparts and the Wright generalized Bessel function with links to their (p, q)-extensions are proposed.

Some families of generalized Mathieu-type power series, associated probability distributions and related inequalities involving complete monotonicity and log-convexity

Some families of generalized Mathieu-type power series, associated probability distributions and related inequalities involving complete monotonicity and log-convexity

Some properties of the Schröder numbers

In the paper, the authors present some properties, including convexity, complete monotonicity, product inequalities, and determinantal inequalities, of the large Schroder numbers and find three relations between the Schroder numbers and central Delannoy numbers. Moreover, the authors sketch generalizing the Schroder numbers and central Delannoy numbers and their generating functions.

Schur-convexity of the Catalan–Qi function related to the Catalan numbers

In the paper, the authors present the Schur-convexity of the absolute of the logarithm of the Catalan–Qi function and prove the logarithmically complete monotonicity of the Catalan–Qi function.

Complete monotonicity of some entropies

It is well-known that the Shannon entropies of some parameterized probability distributions are concave functions with respect to the parameter. In this paper we consider a family of such distributions (including the binomial, Poisson, and negative binomial distributions) and investigate their Shannon, Renyi, and Tsallis entropies with respect to complete monotonicity.