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This contributed volume focuses on various important areas of mathematics in which approximation methods play an essential role. It features cutting-edge research on a wide spectrum of analytic inequalities with emphasis on differential and integral inequalities in the spirit of functional analysis, operator theory, nonlinear analysis, variational calculus, featuring a plethora of applications, making this work a valuable resource. The reader will be exposed to convexity theory, polynomial inequalities, extremal problems, prediction theory, fixed point theory for operators, PDEs, fractional integral inequalities, multidimensional numerical integration, Gauss–Jacobi and Hermite–Hadamard type inequalities, Hilbert-type inequalities, and Ulam’s stability of functional equations. Contributions have been written by eminent researchers, providing up-to-date information and several results which may be useful to a wide readership including graduate students and researchers working in mathematics, physics, economics, operational research, and their interconnections.



Harmonic Hermite–Hadamard Inequalities Involving Mittag-Leffler Function

The main objective of this paper is to establish some new refinements of Hermite–Hadamard like inequalities via harmonic convex functions on the co-ordinates with a kernel involving generalized Mittag-Leffler function. Several special cases are also discussed as applications of our main results. The techniques of this paper may be starting point for further research in this dynamic field.
Muhammad Uzair Awan, Marcela V. Mihai, Khalida Inayat Noor, Muhammad Aslam Noor

Two-Dimensional Trapezium Inequalities via pq-Convex Functions

We establish some new two-dimensional trapezium-like inequalities involving partial differentiable pq-convex functions on rectangle. The concept of pq-convex functions also includes the harmonic convex functions and convex functions as special cases. These results represent refinement and improvement of the known results. Some cases are discussed, which can be obtained as applications of the results. The ideas and techniques of this chapter may be a starting point for further research.
Muhammad Uzair Awan, Muhammad Aslam Noor, Khalida Inayat Noor, Themistocles M. Rassias

New k-Conformable Fractional Integral Inequalities

A new integral identity using the concepts of k-conformable fractional calculus is obtained. Utilizing the preinvexity property of the functions associated upper bounds is also obtained. Some special cases of the obtained results are also discussed.
Muhammad Uzair Awan, Muhammad Aslam Noor, Sadia Talib, Khalida Inayat Noor, Themistocles M. Rassias

On the Hyers–Ulam–Rassias Approximately Ternary Cubic Higher Derivations

In this paper, we prove the generalized Hyers–Ulam–Rassias stability of ternary cubic higher derivations by using a version of the fixed point theorem.
H. Azadi Kenary, Themistocles M. Rassias

Hyers–Ulam Stability for Differential Equations and Partial Differential Equations via Gronwall Lemma

In this paper, we will study Hyers–Ulam stability for Bernoulli differential equations, Riccati differential equations, and quasi-linear partial differential equations of first order, using Gronwall Lemma, following a method given by Rus.
Sorina Anamaria Ciplea, Daniela Marian, Nicolaie Lungu, Themistocles M. Rassias

On b-Metric Spaces and Brower and Schauder Fixed Point Principles

In the paper, we present the basic ideas in b-metric spaces (and b-normed spaces). The main result is the Schauder fixed point principle. For the proof, we use the method presented by Dugundji and Granas in their book [4].
Stefan Czerwik

Deterministic Prediction Theory

We give a general method for predicting spatio-temporal regions with “strange” systemic occurrences. To do so, we consider systemic indices and their measurements into the under consideration fixed spatio-temporal region. Given a set of preselected future points, the magnitude of the (Euclidean or not) distance between the surface of these systemic indices and a parametrized surface that interpolates or passes very close to the points of systemic measurements and given preselected vector values may be viewed as a measure for assessing the appearance of peculiar systemic incidents over the region under consideration; so, depending on these preselected points, we provide a general algorithmic framework for predicting spatio-temporal regions into which crucial systemic events are expected.
Nicholas J. Daras

Accurate Approximations of the Weighted Exponential Beta Function

In this chapter, we provide several error bounds in approximating the Weighted Exponential Beta function
$$\displaystyle F\left ( \alpha ,\beta ;\gamma \right ) :=\int _{0}^{1}\exp \left [ \gamma x^{\alpha }\left ( 1-x\right ) ^{\beta }\right ] dx, $$
where α, β and γ are positive numbers, with some simple quadrature rules of Beta-Taylor, Ostrowski and Trapezoid type.
Silvestru Sever Dragomir, Farzad Khosrowshahi

On the Multiplicity of the Zeros of Polynomials with Constrained Coefficients

We survey a few recent results focusing on the multiplicity of the zero at 1 of polynomials with constrained coefficients. Some closely related problems and results are also discussed.
Tamás Erdélyi

Generalized Barycentric Coordinates and Sharp Strongly Negative Definite Multidimensional Numerical Integration

This paper is devoted to study and construct a family of multidimensional numerical integration formulas (cubature formulas), which approximate all strongly convex functions from above. We call them strongly negative definite cubature formulas (or for brevity snd-formulas). We attempt to quantify their sharp approximation errors when using continuously differentiable functions with Lipschitz continuous gradients. We show that the error estimates based on such cubature formulas are always controlled by the Lipschitz constants of the gradients and the error associated with using the quadratic function. Moreover, assuming the integrand is itself strongly convex, we establish sharp upper as well as lower refined bounds for their error estimates. Based on the concepts of barycentric coordinates with respect to an arbitrary polytope P, we provide a necessary and sufficient condition for the existence of a class of snd-formulas on P. It consists of checking that such coordinates exist on P. Then, the Delaunay triangulation is used as a convenient partition of the integration domain for constructing the best piecewise snd-formulas in L 1 metric. Finally, we present numerical examples illustrating the proposed method.
Allal Guessab, Tahere Azimi Roushan

Further Results on Continuous Random Variables via Fractional Integrals

In this paper, some new fractional weighted inequalities related to Čebyšev, Ostrowski, and Lupaş inequalities are established, and some of their applications for continuous random variables having the probability density function (p.d.f.) defined on a finite interval are derived. Furthermore, some upper bounds for fractional expectation and fractional variance are given.
Ibrahim Slimane, Zoubir Damani, Shilpi Jain, Praveen Agarwal

Nonunique Fixed Points on Partial Metric Spaces Via Control Functions

In this note, we aim to emphasize the significance of the nonunique fixed point results in an abstract space: partial metric space. Indeed, partial metric is a natural extension of the standard metric from the aspect of computer science. The presented results aim to cover and unify several results on the topic in the related literature. We also indicate the validity of the results by a concrete example.
Erdal Karapınar

Some New Refinement of Gauss–Jacobi and Hermite–Hadamard Type Integral Inequalities

In this paper, the authors discover two interesting identities regarding Gauss–Jacobi and Hermite–Hadamard type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss–Jacobi type integral inequalities are established. Also, using the second lemma, some new estimates with respect to Hermite–Hadamard type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different positive real numbers and new error estimates for the trapezoidal formula are provided as well. These results give us the generalizations, refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.
Artion Kashuri, Rozana Liko

New Trapezium Type Inequalities for Preinvex Functions Via Generalized Fractional Integral Operators and Their Applications

The authors have proved an identity for trapezium type inequalities of differentiable preinvex functions with respect to another function via generalized integral operator. The obtained results provide unifying inequalities of trapezium type. Various special cases have been identified. Also, some applications of presented results to special means and new error estimates for the trapezium formula have been analyzed. The ideas and techniques of this paper may stimulate further research in the field of integral inequalities.
Artion Kashuri, Themistocles M. Rassias

New Trapezoid Type Inequalities for Generalized Exponentially Strongly Convex Functions

By using a new general identity and introducing some very general new notions of generalized exponentially strongly convex functions, new trapezoid type inequalities are established. We apply these inequalities to provide approximations for the integral of a real valued function. Approximations for some new weighted means of two positive numbers are also obtained.
Kuang Jichang

Additive-Quadratic ρ-Functional Equations in β-Homogeneous Normed Spaces

Let \(M_1f(x,y) : = \frac {3}{4} f(x+y) - \frac {1}{4}f(-x-y) + \frac {1}{4} f(x-y) + \frac {1}{4} f(y-x) - f(x) - f(y)\) and \(M_2 f(x,y): = 2 f\left ( \frac {x+y}{2} \right ) + f\left ( \frac {x-y}{2}\right ) + f\left ( \frac {y-x}{2}\right ) - f(x) - f(y).\) We solve the additive-quadratic ρ-functional inequalities
$$\displaystyle \begin{array}{@{}rcl@{}} {} {}\| M_1 f(x,y)\| \le \|\rho M_2f(x,y)\|, \end{array} $$
where ρ is a fixed complex number with \(|\rho |<\frac {1}{2}\), and
$$\displaystyle \begin{array}{@{}rcl@{}} {} {}\| M_2 f(x,y)\| \le \|\rho M_1 f(x,y)\| , \end{array} $$
where ρ is a fixed complex number with |ρ| < 1. Using the direct method, we prove the Hyers–Ulam stability of the additive-quadratic ρ-functional inequalities (1) and (2) in β-homogeneous complex Banach spaces.
Jung Rye Lee, Choonkil Park, Themistocles M. Rassias, Sungsik Yun

Stability of Bi-additive s-Functional Inequalities and Quasi-multipliers

Park et al. (Rocky Mt J Math 49, 593–607 (2019)) solved the following bi-additive s-functional inequalities:
$$\displaystyle \begin{array}{@{}rcl@{}} {} && \| f(x{+}y, z{-}w) {+} f(x{-}y, z{+}w) {-}2f(x,z){+}2 f(y, w)\| \\ && \quad \le \left \|s \left (2f\left (\frac {x{+}y}{2}, z{-}w\right ) {+} 2f\left (\frac {x{-}y}{2}, z{+}w\right ) {-} 2f(x,z ){+} 2 f(y, w)\right )\right \| , \end{array} $$
$$\displaystyle \begin{array}{@{}rcl@{}} {} && \left \|2f\left (\frac {x+y}{2}, z-w\right ) +2 f\left (\frac {x-y}{2}, z+w\right ) -2 f(x,z )+2 f(y, w)\right \| \\ && \quad \le \left \|s \left ( f(x+y, z-w) + f(x-y, z+w) -2f(x,z) +2 f(y, w) \right )\right \| , \end{array} $$
where s is a fixed nonzero complex number with |s| < 1. Using the direct method, we prove the Hyers–Ulam stability of quasi-multipliers on Banach algebras, associated with the bi-additive s-functional inequalities (1) and (2).
Jung Rye Lee, Choonkil Park, Themistocles M. Rassias, Sungsik Yun

On the Stability of Some Functional Equations and s-Functional Inequalities

In this work, the Hyers–Ulam type stability and the hyperstability of the following functional equations
$$\displaystyle \begin {aligned}{} f(x+y)+f(x-y)&=f(2x)+f(y)+f(-y),\\ f(ax+y)+f(ax-y)&=f(ax)+af(x),\\ f(ax+y)+f(ax-y)&=f(ax)+af(x)+f(y)+f(-y) \end {aligned} $$
are proved. We also introduce and solve some s-functional inequalities, and we prove their Hyers–Ulam stabilities.
B. Noori, M. B. Moghimi, A. Najati, Themistocles M. Rassias

Stability of the Cosine–Sine Functional Equation on Amenable Groups

In this paper, we establish the stability of the functional equation
$$\displaystyle f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y) $$
on amenable groups.
Ajebbar Omar, Elqorachi Elhoucien

Introduction to Halanay Lemma, via Weakly Picard Operator Theory

In this paper, we present an introduction to Halanay lemma, from the weakly Picard operator theory point of view.
A. Petruşel, I. A. Rus

An Inequality Related to Möbius Transformations

The open unit ball \(\mathbb {B} = \{\mathbf {v}\in \mathbb {R}^n\colon \|\mathbf {v}\|<1\}\) is endowed with Möbius addition ⊕M defined by
$$\displaystyle \mathbf {u}\oplus _M\mathbf {v} = \dfrac {(1 + 2\langle \mathbf {u},\mathbf {v}\rangle + \|\mathbf {v}\|{ }^2)\mathbf {u} + (1 - \|\mathbf {u}\|{ }^2)\mathbf {v}}{1 + 2\langle \mathbf {u},\mathbf {v}\rangle + \|\mathbf {u}\|{ }^2\|\mathbf {v}\|{ }^2}, $$
for all \(\mathbf {u},\mathbf {v}\in \mathbb {B}\). In this article, we prove the inequality
$$\displaystyle \dfrac {\|\mathbf {u}\|-\|\mathbf {v}\|}{1+\|\mathbf {u}\|\|\mathbf {v}\|}\leq \|\mathbf {u}\oplus _M \mathbf {v}\| \leq \dfrac {\|\mathbf {u}\|+\|\mathbf {v}\|}{1-\|\mathbf {u}\|\|\mathbf {v}\|} $$
in \(\mathbb {B}\). This leads to a new metric on \(\mathbb {B}\) defined by
$$\displaystyle d_T(\mathbf {u},\mathbf {v}) = \tan ^{-1}{\|-\mathbf {u}\oplus _M\mathbf {v}\|}, $$
which turns out to be an invariant of Möbius transformations on \(\mathbb {R}^n\) carrying \(\mathbb {B}\) onto itself. We also compute the isometry group of \((\mathbb {B}, d_T)\) and give a parametrization of the isometry group by vectors and rotations.
Themistocles M. Rassias, Teerapong Suksumran

On a Half-Discrete Hilbert-Type Inequality in the Whole Plane with the Hyperbolic Tangent Function and Parameters

In this paper, introducing multi-parameters and using properties of series, we prove a half-discrete Hilbert-type inequality in the whole plane with kernel in terms of the hyperbolic tangent function. The constant factor related to the Riemann zeta function and the gamma function is proved to be the best possible. In the form of applications, we also present equivalent forms, a few particular inequalities, operator expressions and reverses.
Michael Th. Rassias, Bicheng Yang, Andrei Raigorodskii

Analysis of Apostol-Type Numbers and Polynomials with Their Approximations and Asymptotic Behavior

In this chapter, using the methods and techniques of approximation of some classical polynomials and numbers including the Apostol–Bernoulli numbers and polynomials, we survey and investigate various properties of the Boole type combinatorial numbers and polynomials. By applying the p-adic q-integrals including the bosonic and fermionic p-adic integrals on p-adic integers, we study on generating functions for the generalized Boole type combinatorial numbers and polynomials attached to the Dirichlet character. These numbers and polynomials are related to the generalized Apostol–Bernoulli numbers and polynomials, the generalized Apostol–Euler numbers and polynomials, generalized Apostol–Daehee numbers and polynomials, and also generalized Apostol–Changhee numbers and polynomials. With the help of these generating functions, PDEs and their functional equation, many formulas, identities and relations involving the generalized Apostol–Daehee and Apostol–Changhee numbers and polynomials, the Stirling numbers, the Bernoulli numbers of the second kind, the generalized Bernoulli numbers and the generalized Euler numbers, and the Frobenius–Euler polynomials are given. Finally, by using asymptotic estimates for the Apostol–Bernoulli polynomials, asymptotic estimates for Boole type combinatorial numbers and polynomials are given.
Yilmaz Simsek

A General Lower Bound for the Asymptotic Convergence Factor

We provide a rather general and very simple to compute lower bound for the asymptotic convergence factor of compact subsets of ℂ with connected complement and finitely many connected components.
N. Tsirivas

Orlicz Version of Mixed Mean Dual Affifine Quermassintegrals

In this paper, our main aim is to generalize the mixed mean dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new geometric operator by calculating the first Orlicz variation of the mixed mean dual affine quermassintegrals and call it the Orlicz mixed mean dual affine quermassintegrals. The fundamental notions and conclusions of the mixed mean dual affine quermassintegrals, and the Minkowski and Brunn–Minkowski inequalities for the mixed mean dual affine quermassintegrals are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual quermassintegrals are also included in our conclusions. The new Orlicz isoperimetric inequalities in special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn–Minkowski inequality, which also imply the L p-dual Minkowski inequality and Brunn–Minkowski inequality for the mixed mean dual affine quermassintegrals.
C. -J. Zhao, W. -S. Cheung

A Reduced-Basis Polynomial-Chaos Approach with a Multi-parametric Truncation Scheme for Problems with Uncertainties

Polynomial-chaos (PC) expansions constitute an invaluable tool for the investigation of uncertainty quantification problems, yet minimizing the consequences of the so-called curse of dimensionality requires methodologies that ensure reliable performance with a set of basis functions with reduced cardinality. In this work, we propose the construction of the PC basis set using a multi-parametric truncation scheme that generalizes standard ones and enables the derivation of anisotropic surrogates in a flexible fashion. The specification of the truncation rule’s design parameters relies on a preliminary variance analysis, which entails only a fraction of the overall computational cost and enables a sufficient screening of the input variables. Despite its simplicity, the proposed approach is capable of deriving as credible results as the original PC method with fewer basis functions, due to the elimination of unnecessary terms, thus providing a more efficient framework for the study of demanding stochastic problems.
Theodoros T. Zygiridis
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