Mohammad W. Alomari

Jadara University · Mathematics

In this work, we define a generalization of the Davis–Wielandt radius of Hilbert space operators for an arbitrary norm and obtain some applications for Hilbert–Schmidt numerical radius inequalities. For an operator \(T\in \mathscr {B} \left( \mathscr {H}\right) \), the Davis–Wielandt radius is defined as Using the generalization of the Davis–Wielan...

This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by Taylor’s approach. Specifically, we present an enhanced variant achieved by accelerating the expansion of the Obreschkoff formula. This results in a higher-order implicit corrected method that out...

The main goal of this article is to develop a general method for improving some new power inequalities for log-concave functions, which extends and unifies some recent results. As a consequence, we prove generalized multiple term reverses refinements of the difference between the arithmetic and power means inequality for scalars and operators.

We apply Riemann-Liouville fractional integral to get generalisation of companion of Ostrowski's type integral inequality for differentiable mappings whose 1st derivatives are bounded. The present article recapture all results of M. W. Alomari's article and also for one more article of different authors. Applications are also deduced for numerical...

This study systematically develops error estimates tailored to a specific set of general quadrature rules that exclusively incorporate first derivatives. Moreover, it introduces refined versions of select generalized Ostrowski’s type inequalities, enhancing their applicability. Through the skillful application of Hayashi’s celebrated inequality to...

The paper presents a novel approach to generalize the two-point weighted Ostrowski’s formula for Riemann-Stieltjes integrals by utilizing a unique class of functions of bounded rvariation. The proposed approach yields several results that exhibit sharp and better bounds compared to existing established results by using parameters and weights. Addit...

In this paper, we introduce the f -operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the q -operator radius. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.

Complex fuzzy sets (CFSs) have recently emerged as a potent tool for expanding the scope of fuzzy sets to encompass wider ranges within the unit disk in the complex plane. This study explores complex fuzzy numbers and introduces their application for the first time in the literature to address a complex fuzzy partial differential equation that invo...

Complex fuzzy sets (CFSs) have recently emerged as a potent tool for expanding the scope of fuzzy sets to encompass wider ranges within the unit disk in the complex plane. This study explores complex fuzzy numbers and introduces their application for the first time in the literature to address a complex fuzzy partial differential equation that invo...

Perturbed Milne's Quadrature Rule for n-Times Differentiable Abstract: In this work, a perturbed Milne's quadrature rule for n-times differentiable functions with L p-error estimates is derived. One of the most important advantages of our result is that it is verified for p-variation and Lipschitz functions. Several error estimates involving L p-bo...

This paper introduces several generalized extensions of some recent numerical radius inequalities of Hilbert space operators. More preciously, these inequalities refine the recent inequalities that were proved in literature. It has already been demonstrated that some inequalities can be improved or restored by concatenating some into one inequality...

The goal of this study is to refine some numerical radius inequalities in a novel way. The new improvements and refinements purify some famous inequalities pertaining to Hilbert space operators numerical radii. The inequalities that have been demonstrated in this work are not only an improvement over old inequalities but also stronger than them. Se...

In this work, in spite of Milne’s recommendation using the three-point Newton–Cotes open formula (Milne’s rule) as a predictor rule and three-point Newton–Cotes closed formula (Simpson’s rule) as a corrector rule for 4-th differentiable functions with bounded derivatives. There is still a great need to introduce such formulas in other Lp spaces. Of...

This paper proves several new inequalities for the Euclidean operator radius, which refine some recent results. It is shown that the new results are much more accurate than the related, recently published results. Moreover, inequalities for both symmetric and non-symmetric Hilbert space operators are studied.

In this work, new refinements of some numerical radius inequalities are proved. Namely, new improvements and refinements purify the recent inequalities of some famous inequalities concerning the numerical radius of Hilbert space operators. The proven inequalities in this work are not only an improvement over old inequalities, but rather they are st...

We use the recent generalization of the numerical radius introduced by Abu-Omar and Kittaneh, to develop a new generalization of the Berezin transform. The definition of the Berezin trace classes is also introduced. Based on that, several (2x2) Berezin radius inequalities are established.

In this work, the concept of the Davis-Wielandt Berezin number is introduced. Some upper and lower bounds for the Davis-Wielandt Berezin number are introduced. A connection between norm-parallelism to the identity operator and an equality condition for the Davis-Wielandt Berezin number is also discussed. Some bounds for the Davis-Wielandt Berezin n...

In this note, we give full complete positive proof of the celebrated unsolved Erd¨os–Straus conjecture. Similarly, the Sierpinski conjecture follows. A relaxed extension of the restricted Hagedorn equation is presented.

In diverse branches of mathematics, several inequalities have been studied and applied. In this article, we improve Furuta's inequality. Subsequently, we apply this improvement to obtain new radius inequalities that have not been reported in the current literature. Numerical examples illustrate the main findings.

In this work, an operator superquadratic function (in the operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. A general Bohr’s inequality for positive operators is thus deduced. A Jensen-type ineq...

In this paper, we generalize and refine some Berezin number inequalities for Hilbert space operators. Namely, we refine the Hermite-Hadamard inequality and some other recent results by using the concept of superquadraticity and convexity. Then we extend these inequalities for the Berezin number. Among other inequalities, it is shown that if S, T ∈...

In this paper, we introduce the $f-$operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the $q-$operator radius. Properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.

In this work, the fuzzy fractional two-point boundary value problems (FFTBVPs) are analyzed and solved using the fuzzy fractional homotopy analysis method (FF-HAM). Fuzzy set theory mixed with Caputo fractional derivative properties is utilized to produce a new formulation of the standard HAM in the fuzzy domain for the persistence of approximation...

We show that if f is a non-negative superquadratic function, then \(A\mapsto \mathrm {Tr}f(A)\) is a superquadratic function on the matrix algebra. In particular, $$\begin{aligned} \mathrm {Tr} f\left( {\frac{{A + B}}{2}} \right) +\mathrm {Tr} f\left( \left| {\frac{{A - B}}{2}}\right| \right) \le \frac{{\mathrm {Tr} {f\left( A \right) } + \mathrm {...

The Berezin transform $\widetilde{A}$ and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set $Q$ with normalized reproducing kernel $k_{\eta}:=\dfrac{K_{\eta}}{\left\Vert K_{\eta}\right\Vert}$ are defined, respectively, by $\widetilde{A}(\eta)=\left\langle {A}k_{\eta},k_{\eta}\right\rangle$, $\eta\in Q$ and...

In this work, some numerical radius inequalities based on the recent Dragomir extension of Furuta’s inequality are obtained. Some particular cases are also provided. Among others, the celebrated Kittaneh inequality reads: wT≤12T+T*. It is proved that wT≤12T+T*−12infx=1Tx,x12−T*x,x122, which improves on the Kittaneh inequality for symmetric and non-...

Different types of mathematical inequalities have been largely analyzed and employed. In this paper, we introduce improvements to some Ostrowski type inequalities and present their corresponding proofs. The presented proofs are based on applying the celebrated Hayashi inequality to certain functions. We provide examples that show these improvements...

In this work, some generalized Euclidean operator radius inequalities are established. Refinements of some well-known results are provided. Among others, some bounds in terms of the Cartesian decomposition of a given Hilbert space operator are proved.

In this work, we introduce and investigate a new subclass of analytic bi-univalent functions based on subordination conditions between the zero-truncated Poisson distribution and Gegenbauer polynomials. More precisely, we will estimate the first two initial Taylor-Maclaurin coefficients and solve the Fekete-Szegö functional problem for functions be...

In this work, some new upper and lower bounds for the Davis-Wielandt radius are introduced. Generalizations of some presented results are obtained. Some bounds for the Davis-Wielandt radius for n × n operator matrices are established. An extension of the Davis-Wielandt radius to the Euclidean operator radius is introduced.

In this work, some new inequalities for the numerical radius of block $n$-by-$n$ matrices are presented. As an application, the bounding of zeros of polynomials using the Frobenius companion matrix partitioned by the Cartesian decomposition method is proved. We highlight several numerical examples showing that our approach to bounding zeros of poly...

In this work, an extension of two-point Ostrowski's formula for $n$-times differentiable functions is proved. A generalization of Taylor formula is deduced. An identity of Fink type for this extension is provided. Error estimates for the considered formulas are also given. Two-point Ostrowski-Gruss type inequalities are pointed out. An expansion of...

International Workshop on Functional Analysis and Topological Structures

Grüss-type inequalities have been widely studied and applied in different contexts. In this work, we provide and prove vectorial versions of Grüss-type inequalities involving vector-valued functions defined on Rn for inner- and cross-products.

Some companions of Ostrowski's integral inequality for the Riemann-Stieltjes integral b a f (t) du (t), where f is assumed to be of r-H-Hölder type on [a, b] and u is of bounded variation on [a, b], are proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out. MSC: 2...

1 In this work, we generalize and extend the concept of superquadratic functions by introducing h-superquadratic functions. Some basic properties of h-superquadraticity are elaborated 3 and investigated. A connection between h-convexity and h-superquadraticity is provided. 4 Inequalities of Jensen's and Jensen-Mercer's types with their converses ar...

In this work, a refinement of the Cauchy–Schwarz inequality in inner product space is proved. A more general refinement of the Kato’s inequality or the so called mixed Schwarz inequality is established. Refinements of some famous numerical radius inequalities are also pointed out. As shown in this work, these refinements generalize and refine some...

In this work, some numerical radius inequalities for Hilbert space operators are introduced. Namely, we applying the Hermite-Hadamard inequality and some other recent results by using the concept of operator convexity and superquadracity.

In this work, we offer new applications of Hayashi’s inequality for differentiable functions by proving new error estimates of the Ostrowski- and trapezoid-type quadrature rules.

Two new inequalities for Riemann--Stieltjes integral are introduced for functions of bounded $p$-variation and H\"{o}lder continuous integrators.

In this paper, an inequality of Simpson type for quasi-convex mappings is proved. The constant in the classical Simpson's inequality is improved. Furthermore, the obtained bounds can be (much) better than some recent obtained bounds. Application to Simpson's quadrature rule is also given. MSC: Primary 26D15; Secondary 65D30; 65D32

In this work, we improve and refine some numerical radius inequalities. In particular, for all Hilbert space operators $T$, the celebrated Kittaneh inequality reads: \begin{align*} \frac{1}{4}\left\| T^*T + TT^*\right\|\le w^{2 }\left(T \right) \le \frac{1}{2}\left\| T^*T + TT^*\right\|. \end{align*} In this work we provide some important refinemen...

In this work, several inequalities of Popoviciu type for h-MN-convex functions are proved, where M or N are denote to Arithmetic, Geometric and Harmonic means and $h$ is a non-negative superadditive or subadditive function.

In this note, we give an alternative proof of the celebrated Dini's theorem regarding uniform convergence of monotonic a decreasing sequence of continuous functions defined on a compact set K.

In this work, an improvement of Hölder–McCarty inequality is established. Based on that, several refinements of the generalized mixed Schwarz inequality are obtained. Consequently, some new numerical radius inequalities are proved. New inequalities for the numerical radius of n × n matrix of Hilbert space operators are proved as well. Some refineme...

In this work, Some new inequalities for the numerical radius of block $n$-by-$n$ matrices are presented. As an application, bounding of zeros of polynomials using the Frobenius companion matrix partitioned by the Cartesian decomposition approach is proved and affirmed by a numerical example showing that our approach of bounding zeros of polynomials...

For a bounded linear operator, acting in the reproducing kernel Hilbert space \({\mathcal {H}}={\mathcal {H}}\left( \Omega \right) \) over some set \(\Omega \), its Berezin symbol (or Berezin transform)\(\widetilde{\text { }A}\) is defined by $$\begin{aligned} {\widetilde{A}}\left( \lambda \right) :=\left\langle A{\widehat{k}}_{\lambda },{\widehat{...

In this work, a refinement of the Cauchy--Schwarz inequality in inner product space is proved. A more general refinement of the Kato's inequality or the so called mixed Schwarz inequality is established. Refinements of some famous numerical radius inequalities are also pointed out. As shown in this work, these refinements generalize and refine some...

In this work, some new upper and lower bounds of the Davis-Wielandt radius are introduced. Generalizations of some presented results are obtained. Some bounds of the Davis-Wielandt radius for $n\times n$ operator matrices are established. An extension of the Davis-Wielandt radius to the Euclidean operator radius is introduced.

In this work, the mixed Schwarz inequality for semi-Hilbertian space operators is proved. Namely, for every positive Hilbert space operator $A$. If $f$ and $g$ are nonnegative continuous functions on $\left[0,\infty\right)$ satisfying $f(t)g(t) =t$ $(t\ge0)$, then \begin{align*} \left| {\left\langle {T x,y} \right\rangle_A } \right| \le \left\| {f\...

In this work, a pre-Gr\"{u}ss inequality for positive Hilbert space operators is proved. So that, some numerical radius inequalities are proved. On the other hand, based on a non-commutative Binomial formula, a non-commutative upper bound for the numerical radius of the summand of two bounded linear Hilbert space operators is proved. A commutative...

We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{{A + B}}{2}} \right) +\tr f\left(\left| {\frac{{A - B}}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*}...

There are many criterion to generalize the concept of numerical radius; one of the most recent interesting generalization is what so-called the generalized Euclidean operator radius. Simply, it is the numerical radius of multivariable operators. In this work, several new inequalities, refinements, and generalizations are established for this kind o...

In this work, we discuss the continuity of h -convex functions by introducing the concepts of h -convex curves ( h -cord). Geometric interpretation of h -convexity is given. The fact that for a h -continuous function f , is being h -convex if and only if is h -midconvex is proved. Generally, we prove that if f is h -convex then f is h -continuous....

In this work, an operator superquadratic function (in operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. Equivalent statements of a non-commutative version of Jensen's inequality for operator sup...

In this work, some numerical radius inequalities based on the recent Dragomir extension of Furuta's inequality are obtained. Some particular cases are also provided.

In this work, we improve and refine some numerical radius inequalities. In particular, for all Hilbert space operators $T$, the celebrated Kittaneh inequality reads: \begin{align*} \frac{1}{4}\left\| T^*T + TT^*\right\|\le w^{2 }\left(T \right) \le \frac{1}{2}\left\| T^*T + TT^*\right\|. \end{align*} In this work we provide some important refinemen...

In this work, we introduce the class of h-MN-convex functions by generalizing the concept of MN-convexity and combining it with h-convexity. Namely, Let I, J be two intervals subset of (0, ∞) such that (0, 1) ⊆ J and [a, b] ⊆ I. Consider a non-negative function h : (0, ∞) → (0, ∞) and let M : [0, 1] → [a, b] (0 < a < b) be a Mean function given by...

In this work, we discuss the continuity of h-convex functions by introducing the concepts of h-convex curves (h-cord). Geometric interpretation of h-convexity is given. The fact that for a h-continuous function f, is being h-convex if and only if is h-midconvex is proved. Generally, we prove that if f is h-convex then f is h-continuous. A discussio...

In this work, we introduce the class of h-MN-convex functions by generalizing the concept of MN-convexity and combining it with h-convexity. Some analytic properties for each class of functions are explored and investigated. Characterizations of each type are given. Various Jensen's type inequalities and their converses are proved.

In this work, a generalization of pre-Gruss inequality is established. Several bounds for the difference between two Chebyshev functional are proved.

In this paper, new upper and lower bounds for the Trapezoid inequality of absolutely continuous functions are obtained. Applications to some special means are provided as well.

A weighted companion of Ostrowski-Midpoint type inequality is established. Application to a composite quadrature rule is provided.

In this work, sharp Wirtinger type inequalities for double integrals are established. As applications, two sharp \v{C}eby\v{s}ev type inequalities for absolutely continuous functions whose second partial derivatives belong to $L^2$ space are proved.

In this work, some generalizations and refinements inequalities for the numerical radius of the product of Hilbert space operators are proved. New inequalities for the numerical radius of block matrices of Hilbert space operators are also established.

In this paper, Rolle’s, MVT, Cauchy MVT, Pompeiu’s MVT, and Cauchy–Pompeiu’s MVT on hypercuboids are proved and investigated.

In this work, a generalization of Chebyshev functional is presented. New inequalities of Gruss type via Pompeiu’s mean value theorem are established. Improvements of some old inequalities are proved. A generalization of pre-Gruss inequality is elaborated. Some remarks to further generalization of Chebyshev functional are presented. As applications,...

In this work, an operator superquadratic function (in operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. General Bohr's inequality for positive operators is deduceed. A Jensen type inequality is...

Given any ${\bf{a}}: = \left( {a_1 ,a_2 , \ldots ,a_n } \right)$ and ${\bf{b}}: = \left( {b_1 ,b_2 , \ldots ,b_n } \right)$ in $\mathbb{R}^n$. The $\textbf{n}$-fold convex function defined on $\left[ {{\bf{a}},{\bf{b}}} \right]$, ${\bf{a}},{\bf{b}} \in \mathbb{R}^n$ with ${\bf{a}}<{\bf{b}}$ is a convex function in each variable separately. In this...

In this work, an operator version of Popoviciu’s inequality for positive operators on Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, using the same technique, an operator version of Popoviciu’s inequality for convex functions is obtained. Some other related inequalities are also presented.

In this work, we construct a new general Two-point quadratre rules for the Riemann--Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$, where the integrand $f$ is assumed to be satisfied the H\"{o}lder condition on $[a,b]$ and the integrator $u$ is of bounded variation on $[a,b]$. The dula formulas under the same assumption are pro...

In this work, Lp-error estimates of general two and three point quadrature rules for Riemann-Stieltjes integrals are give n. The presented proofs depend on new triangle type inequalities of Riemann-Stieltjes integrals

In this work, sharp Wirtinger type inequalities for double integrals are established. As applications, two sharp \v{C}eby\v{s}ev type inequalities for absolutely continuous functions whose second partial derivatives belong to $L^2$ space are proved.

In this work, generalizations of some inequalities for continuous synchronous (h-asynchronous) functions of linear bounded selfadjoint operators under positive linear maps in Hilbert spaces are proved.

In this work, generalizations of some inequalities for continuous $h$-synchronous ($h$-asynchronous) functions of linear bounded selfadjoint operators under positive linear maps in Hilbert spaces are proved.

In this work, an extension of the generalized mixed Schwarz inequality is proved. A companion of the generalized mixed Schwarz inequality (or Kittaneh inequality) in which the Cartesian decomposition of operators is replaced by the polar decomposition is also given. Based on that some numerical radius inequalities are proved.

In this paper, three-point quadrature rules for the Riemann-Stieltjes integral are introduced. Some inequalities of Ostrowski's type are also obtained.

In this work, an improvement of H\"{o}lder-McCarty inequality is established. Based on that, several refinements of the generalized mixed Schwarz inequality are obtained. Consequently, some new numerical radius inequalities are proved. Some of the presented results are refined and it shown to be better than earlier results were proved in literature...

In this work, we offer new applications of Hayashi's inequality for differentiable functions by proving new error estimates of the Ostrowski and trapezoid type quadrature rules.

In this work, operator version of Popoviciu's inequality for positive selfadjoint operators in Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, operator version of Popoviciu's inequality for convex functions is presented. Some other related inequalities are also deduced.

A generalization of Mercer inequality for h-convex function is presented. As application, a weighted generalization of triangle inequality is given.

In this work, L p -error estimates of general two and three point quadrature rules for Riemann-Stieltjes integrals are given. The presented proofs depend on new triangle type inequalities of Riemann-Stieltjes integrals.

In this work, the q-analogue of Bernoulli inequality is proved. Some other related results are presented.

For absolutely continuous functions whose first derivatives in absolute value are M N-convex several inequalities of Ostrowski's type are introduced. Other related results by applying Hölder integral inequality are also provided.

In this work, generalizations of some inequalities for continuous h-synchronous (h-asynchronous) functions of selfadjoint linear operators in Hilbert spaces are proved.

In this work, inequalities of Beesack–Wirtinger type for absolutely continuous functions whose derivatives belong to Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L...

In this work, a general two-point Ostrowski’s formula from an analytic point of view is presented. New triangle type inequalities for Riemann–Stieltjes integrals are established. Sharp two-point Ostrowski’s type inequalities for functions of bounded p-variation and functions satisfy Lipschitz condition involving Lp\documentclass[12pt]{minimal} \use...

In this work, several inequalities of Popoviciu type for h-MN-convex functions are proved, where M and N are denote to Arithmetic, Geometric and Harmonic means and h is a non-negative superadditive or subadditive function.

In this work, we introduce the class of h-MN-convex functions by generalizing the concept of MN-convexity and combining it with h-convexity. Namely, let M : [0, 1] → [a, b] be a Mean function given by M (t) = M (t; a, b); where by M (t; a, b) we mean one of the following functions: At (a, b) := (1 − t) a + tb, Gt (a, b) = a1−tbt and Ht (a, b) := ab...

In this paper, new inequalities connected with the celebrated Steffensen's integral inequality are proved.

In this work, a generalization of pre-Gr\"{u}ss inequality is established. Several bounds for the difference between two \v{C}eby\v{s}ev functional are proved.

Extensions and generalizations of Alzer inequality; which is of Wirtinger type are proved. All obtained bounds are sharp. As applications, a sharp trapezoid type inequality is provided.

In this work, an expansion of Guessab--Schmeisser two points formula for n-times differentiable functions via Fink type identity is established. Generalization of the main result for harmonic sequence of polynomials is established. Several bounds of the presented results are proved. As applications, some quadrature rules are elaborated and discusse...

In literature the Hermite-Hadamard inequality was eligible for many reasons, one of the most surprising and interesting that the Hermite-Hadamard inequality combine the midpoint and trapezoid formulae in an inequality. In this work, a Hermite-Hadamard like inequality that combines the composite trapezoid and composite midpoint formulae is proved. S...

In this work, several Apery's type inequalities for n-times differentiable functions are proved. Apery's type Inequalities Via Harmonic sequence of polynomials are elaborated and discussed.

In thiswork, several bounds for the difference between two \v{C}eby\v{s}ev type functional under various assumptions for the functions involved are proved.