105 Results Found

In this paper, we show that the functions $x^m\left|{\beta }^{\left(m\right)}\left(x\right)\right|$ are not completely monotonic on $(0,\infty )$ for all $m\in \mathbb{N},$ where $\beta \left(x\right)$ is the Nielsen’s $\beta$ -function and we...

We extend the classical Euler-type integral representations for the Appell functions $F_1$, $F_2$, and $F_3$, to the appropriate Kampé de Fériet functions by using integration against the Meijer–Nørlund G-function. In particular, th...

In this paper, we present some new symmetric bounds for the gamma and polygamma functions. For this goal, we present two functions involving gamma and polygamma functions and we investigate their complete monotonicity. Also, we investigate their comp...

Hyperbolic complete monotonicity property ($\mathrm{HCM}$) is a way to check if a distribution is a generalized gamma ($\mathrm{GGC}$), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions $E_{\alpha },\alpha \in (0,2]$...

Our starting point is an integral inequality that involves convex, concave and monotonically increasing functions. We provide some interpretations of the inequality, in terms of both probability and terms of linear functionals, from which we further...

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of t...

In the paper, the authors derive an integral representation, present a double inequality, supply an asymptotic formula, find an inequality, and verify complete monotonicity of a function involving the gamma function and originating from geometric pro...

We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at $-\infty$, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy functi...

In this paper, we find a solution of an open problem posed by Alzer, Berg, and Koumandos: determine $(\alpha ,m)\in {\mathbb{R}}^{+}\times \mathbb{N}$ such that the function $x^{\alpha }\left|{\psi }^{\left(m\right)}\left(x\right)\right|$ is completely monotonic on ...

We study the projection of an element of fractional Gaussian noise onto its neighbouring elements. We prove some analytic results for the coefficients of this projection. In particular, we obtain recurrence relations for them. We also make several co...

In this paper, the authors review and survey some results published since 2020 about (complete) monotonicity, inequalities, and their necessary and sufficient conditions for several newly introduced functions involving polygamma functions and origina...

This paper delves into the extension and characterization of radial positive definite functions into non-integer dimensions. We provide a thorough investigation by employing the Riemann–Liouville fractional integral and fractional Caputo deriva...

In the paper, the authors introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers, and the Catalan–Qi function, and discover some properties of the unified generalization, including a product-...

The fractional Zener constitutive law is frequently used as a model of solid-like viscoelastic behavior. In this work, a class of linear viscoelastic models of Zener type, which generalize the fractional Zener model, is studied by the use of Bernstei...

In the paper, the authors find a sufficient and necessary condition for the power-exponential function ${\left(1+\frac{1}{x}\right)}^{\alpha x}$ to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions ${\left(1+\frac{1}{x}\right)}^{\alpha x}$ and ${(1}^{}$...

In this paper, we first deduce the explicit formulas for the projector of the nth level fractional derivative and for its Laplace transform. Afterwards, the fractional relaxation equation with the nth level fractional derivative is discussed. It turn...

We prove that the function $\sigma \left(s\right)$ defined by $\beta \left(s\right)=\frac{6s^2+12s+5}{3s^2(2s+3)}-\frac{{\psi }^{\prime }\left(s\right)}{2}-\frac{\sigma \left(s\right)}{2s^5},s>0$, is strictly increasing with the sharp bounds $0<\sigma \left(s\right)<\frac{49}{120}$, where $\beta \left(s\right)$ is Nielsen’s beta function a...

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In this paper,...

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In the paper,...

In the paper, the authors present a brief overview and survey of the scientific work by Chinese mathematician Feng Qi and his coauthors.

A class of generalized fractional Zener-type viscoelastic models with general fractional derivatives is considered. Two integral representations are derived for the corresponding relaxation modulus. The first representation is established by applying...

In this paper, we present some new symmetric bounds for Bateman’s G-function and its derivatives, in terms of the digamma and polygamma functions, which are better than some recent results.

The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, $W_0\left(t\right)$, in a peculiar...

The fractional order differential equation \(u'(t)=Au(t)+\gamma D_t^{\alpha} Au(t)+f(t), \ t>0\), \(u(0)=a\in X\) is studied, where \(A\) is an operator generating a strongly continuous one-parameter semigroup on a Banach space \(X\), \(D_t^{\alp...

In this paper, a new approximation formula for the gamma function and some of its symmetric inequalities are established. We prove the superiority of our results over Yang and Tian’s approximation formula for the gamma function of order $v^{\&minus}$...

In this paper, we consider the differential equation $y^{″}+{\omega }^2\rho \left(x\right)y=0$, where $\omega$ is a positive parameter. The principal concern here is to find conditions on the function ${\rho }^{-1/2}\left(x\right)$ which ensure that the consecutive difference...

Both the stretched exponential function and the Prony series are prominent mathematical frameworks extensively employed to characterize monotonically decaying phenomena across physics and engineering disciplines. Despite being fundamentally distinct...

Superspirals include a very broad family of monotonic curvature curves, whose radius of curvature is defined by a completely monotonic Gauss hypergeometric function. They are generalizations of log-aesthetic curves, and other curves whose radius of c...

In the paper, the author gives a solution to a conjecture on a double inequality for a function involving the tri- and tetra-gamma functions, which was first posed in Remark 6 of the paper “Complete monotonicity of a function involving the tri-...

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 $G\left(r\right)$, deducing two...

Recently, some new types of monotonicity—in particular, weak monotonicity and directional monotonicity of an n-ary real function—were introduced and successfully applied. Inspired by these generalizations of monotonicity, we introduce a new notion fo...

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to paramet...

In this paper, we introduce fractal interpolation on complete semi-vector spaces. This approach is motivated by the requirements of the preservation of positivity or monotonicity of functions for some models in approximation and interpolation theory....

The paper proposes to consider individual heuristics as unreliably operating parts of the information processing system. In a separate case, several different heuristics are adopted to solve the same problem, and the results obtained are adjusted in...

This paper presents a unified framework for constructing two-branched fuzzy implications and families of copulas based on the same composition principles involving monotone and convex functions. The proposed methodology yields operators with a genuin...

The symmetric patterns that inequalities contain are reflected in researchers’ studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function ${\mathsf{\Gamma }}_{\mu }\left(v\right)$ and study the completely mon...

We survey the results on representations of committees and constitutions by game forms that possess some kind of equilibrium strategies for each profile of preferences of the players. The survey is restricted to discrete models, that is, we deal with...

The process of machine learning is to find parameters that minimize the cost function constructed by learning the data. This is called optimization and the parameters at that time are called the optimal parameters in neural networks. In the process o...

Aggregation operators are central to multi-criteria decision-making under complex fuzzy information, where the additional phase dimension of complex fuzzy sets captures periodic or cyclical uncertainty beyond the reach of classical fuzzy sets. Existi...

We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorem...

In this paper, a completely smooth lower-order penalty method for solving a second-order cone mixed complementarity problem (SOCMCP) is studied. Four distinct types of smoothing functions are taken into account. According to this method, SOCMCP is ap...

We develop a model of cheap talk with transparent and monotone motives from a seller to an informed buyer. By transparent and monotone motives, we mean that the seller’s preference does not depend on the state of the world and is increasing in...

To provide a unified description of the monotonic and cyclic compressive responses of concrete after freeze–thaw exposure, a one-dimensional phenomenological damage–plasticity constitutive framework is proposed. Freeze–thaw-induced...

Suppose there is a collection of n simple polygons in the plane, none of which overlap each other. The polygons are interlocked if no subset can be separated arbitrarily far from the rest. It is natural to ask the characterization of the subsets that...

We compare the classical viscoelastic models due to Becker and Lomnitz with respect to a recent viscoelastic model based on the Lambert W function. We take advantage of this comparison to derive new analytical expressions for the relaxation spectrum...

A crucial issue in applying the ordered weighted averaging (OWA) operator for decision making is the determination of the associated weights. This paper proposes a general least convex deviation model for OWA operators which attempts to obtain the de...

In this paper, we introduce and study a normalized four-parameter Le Roy-type q-Mittag–Leffler (LR-q-ML) function defined by means of the q-Pochhammer symbol, the q-Gamma function, and the q-factorial, whose parameters directly influence the as...

We study a class of evolutionary variational inequalities in a Hilbert space that models the long-run balanced-growth equilibrium of a competitive economy with transaction costs, time-dependent production and infrastructure constraints, and exogenous...

In this paper, we establish a fundamental lemma based on symmetric quantum monotonicity within the framework of symmetric quantum calculus. Using this lemma, we derive the symmetric quantum Favard inequality and its Berwald-type extensions, together...

For a series system with heterogeneous proportional reversed-hazard-rate components coupled through an Archimedean survival copula, we determine the exact admissible range of power-mean indices for a universal homogeneous stochastic upper envelope. T...