1,228 Results Found

Monotonicity points and the modulus of monotonicity are studied in b-Orlicz function spaces over finite nonatomic measure spaces, equipped with the Luxemburg quasi-norm. Lower strictly monotone points and lower and upper locally uniformly monotone po...

In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the...

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 $E_{\alpha },\alpha \in (0,2]$...

We show that there is a one-to-one correspondence between positive operator-monotone functions on the positive reals, monotone Riemannian metrics, and finite positive Borel measures on the unit interval. This correspondence appears as an integral rep...

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...

In this paper, two new approximation formulas with monotonic remainders for the gamma function have been presented. Also, we present some numerical comparisons between our new approximation formulas and some known ones, which demonstrate the superior...

In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function $\mathcal{F}$. Also...

Functional inequalities involving special functions are very useful in mathematical analysis, and several interesting results have been obtained in this topic. Several methods have been used by many authors in order to derive upper or lower bounds of...

In this paper, we establish some new dynamic inequalities involving C-monotonic functions with $C\ge 1,$ on time scales. As a special case of our results when $C=1,$ we obtain the inequalities involving increasing or decreasing functions (where for $C=1,$...

Through the paper, we present several inequalities involving C-monotonic functions with $C\ge 1,$ on nabla calculus via time scales. It is known that dynamic inequalities generate many different inequalities in different calculus. The main results will...

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 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...

Prescribed performance control (PPC) guarantees that tracking errors evolve within user-defined envelopes, yet the design of the performance certificate $\rho \left(t\right)$ itself has been restricted to hand-tuned parametric forms since the theory’s incep...

Prostate cancer is the second most frequent malignancy (after lung cancer). Preoperative staging of PCa is the basis for the selection of adequate treatment tactics. In particular, an urgent problem is the classification of indolent and aggressive fo...

We present new results in interval analysis (IA) and in the calculus for interval-valued functions of a single real variable. Starting with a recently proposed comparison index, we develop a new general setting for partial order in the (semi linear)...

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...

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces $L^{\Phi }$ generated by the b-Orlicz function $\Phi$ equipped with a Luxemburg quasi-norm contain both classical spaces $L^p(p\&$...

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 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...

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...

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...

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...

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...

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...

Since the monotonicity of the best approximant is crucial to establish partial ordering methods, in this paper, we, respectively, characterize the best approximants in Banach function spaces and Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$, in which we especially focus...

Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consi...

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, 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 ...

Developing stability analysis methods for modern dynamical system solutions has been a significant challenge in the field. This study aims to formulate a qualitative analysis approach for the monotone stability region of a specific solution to a sing...

In the paper, by virtue of a derivative formula for the ratio of two differentiable functions and with the help of a monotonicity rule, the authors expand a logarithmic expression involving the sine function into the Maclaurin power series in terms o...

In this paper, we establish new generalized Mitrinović-Adamović-type inequalities in a wider range $(0,\pi )$ by using the monotonicity of certain functions. These inequalities contain sharp and tractable bounds for the function ${\left({\displaystyle \frac{\sin x}{x}}\right)}^3.$ All t...

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently...

In the paper, the authors establish the monotonicity results of the ratios between normalized tails of the Maclaurin power series expansions of the sine and cosine functions and restate them in terms of the generalized hypergeometric functions.

As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoret...

In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementar...

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...

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, 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...

Fractional-order differential operators accentuate weak edges and fine texture more effectively than their integer-order counterparts, but in image enhancement the fractional machinery is confined almost entirely to the spatial stage, while the tonal...

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-...

This paper develops two novel arcsine-type analytic approximation formulas for $\mathrm{arcsl}\left(x\right)$, the Gauss lemniscate sine function, each equipped with a monotonic and bounded remainder term of order $x^{29}$ as $x\to 0$, which demonstrates the high accuracy of t...

Due to the classifying theorems by Petz and Kubo–Ando, we know that there are bijective correspondences between Quantum Fisher Information(s), operator means, and the class of symmetric, normalized operator monotone functions on the positive ha...

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...

This paper proposes a framework for generating symmetric two-branched copulas directly from fuzzy implications via monotone function composition. The proposed approach is based on a two-branched composition principle induced by fuzzy implications and...

In this paper, we first revisit and bring together as a sort of survey, properties of Bernstein polynomials of one variable. Secondly, we extend the results from one variable to several ones, namely—uniform convergence, uniform convergence of t...

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 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...

This paper addresses the theoretical issues in k-valued logic, which are crucial for developing solutions in various fields of science and technology. One of the fundamental issues is a complete description of the closed classes of functions of three...

As a key step to endow the neural network with nonlinear factors, the activation function is crucial to the performance of the network. This paper proposes an Efficient Asymmetric Nonlinear Activation Function (EANAF) for deep neural networks. Compar...