A sharp bound for the Chebyshev functional
In this work a sharp bound of the Čebyšev functional for absolutely continuous functions $f,g$ whose derivatives $f'\in L_{\infty}[a,b]$ and $g'\in L_{1}[a,b]$ is obtained.
math.CA↗
arXiv subjects
Mohammad W. Alomari
Publications and source records attributed to Mohammad W. Alomari.
$L_p$-Bounds for the Čebyšev functional
In this paper several new bounds for the Čebyšev functional involving $L_p$-norm are presented.
math.CA↗
On two inequalities of Čebyšev
In this work, several sharp bounds for the Čebyšev functional involving various type of functions are proved. In particular, for the Čebyšev functional of two absolutely continuous functions whose first derivatives are both convex, convex and belong to $L_p$-spaces, convex and bounded variation, convex and Lipschitz mappings new sharp bounds are presented. Other related results regarding two convex and concave functions are given.
math.CA↗
Generalized Euclidean Operator Radius
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.
math.FA↗
Bounding the zeros of polynomials using the Frobenius companion matrix partitioned by the Cartesian decomposition
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 could be very effective in comparison with the most famous and some recent results presented in the field.
math.FA↗
On Cauchy-Schwarz type inequalities and applications to numerical radius inequalities
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 recent and old results obtained in literature. Among others, it is proved that if $T\in\mathscr{B}\left(\mathscr{H}\right)$, then \begin{align*} ω^{2}\left(T\right) &\le \frac{1}{12} \left\| \left| T \right|+\left| {T^* } \right|\right\|^2 + \frac{1}{3} ω\left(T\right)\left\| \left| T \right|+\left| {T^* } \right|\right\| \\ &\le\frac{1}{6} \left\| \left| T \right|^2+ \left| {T^* } \right|^2 \right\| + \frac{1}{3} ω\left(T\right)\left\| \left| T \right|+\left| {T^* } \right|\right\|, \end{align*} which refines the recent inequality obtained by Kittaneh and Moradi in \cite{KM}.
math.FA↗
On the Davis-Wielandt radius inequalities of Hilbert space operators
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.
math.FA↗
The generalized Schwarz inequality for semi-Hilbertian space operators and Some $A$-numerical radius inequalities
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\left( {\left| T \right|_A x} \right)} \right\|_A \left\| {g\left( {\left| {T^{\sharp_A } } \right|_A y} \right)} \right\|_A \end{align*} for every Hilbert space operator $T$ such that the range of $T^* A$ is a subset in the range of $A$, such that $A$ commutes with $T$, and for all vectors $x,y\in \mathscr{H}$, where $\left| T \right|_A = \left(AT^{\sharp_A}T\right)^{1/2}$ such that $T^{\sharp_A}=A^\dagger T^*A$, where $A^\dagger$ is the Moore-Penrose inverse of $A$. Based on that, some inequalities for the $A$-numerical radius are introduced.
math.FA↗
Klein's trace inequality and superquadratic trace functions
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*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$ for all positive matrices $A,B$. It gives in particular an improvement of the Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.
math.FA↗
Some properties of h-MN-convexity and Jensen's type inequalities
In this work, we introduce the class of $h$-${\rm{MN}}$-convex functions by generalizing the concept of ${\rm{MN}}$-convexity and combining it with $h$-convexity. Namely, Let $I,J$ be two intervals subset of $\left(0,\infty\right)$ such that $\left(0,1\right)\subseteq J$ and $\left[a,b\right]\subseteq I$. Consider a non-negative function $h: (0,\infty)\to \left(0,\infty\right)$ and let ${\rm{M}}:\left[0,1\right]\to \left[a,b\right] $ $(0<a<b)$ be a Mean function given by ${\rm{\rm{M}}}\left(t\right)={\rm{\rm{M}}}\left( {h(t);a,b} \right)$; where by ${\rm{\rm{M}}}\left( {h(t);a,b} \right)$ we mean one of the following functions: $A_h\left( {a,b} \right):=h\left( {1 - t} \right)a + h(t) b$, $G_h\left( {a,b} \right)=a^{h(1-t)} b^{h(t)}$ and $H_h\left( {a,b} \right):=\frac{ab}{h(t) a + h\left( {1 - t} \right)b} = \frac{1}{A_h\left( {\frac{1}{a},\frac{1}{b}} \right)}$; with the property that ${\rm{\rm{M}}}\left( {h(0);a,b} \right)=a$ and ${\rm{M}}\left( {h(1);a,b} \right)=b$. A function $f : I \to \left(0,\infty\right)$ is said to be $h$-${\rm{\rm{MN}}}$-convex (concave) if the inequality \begin{align*} f \left({\rm{M}}\left(t;x, y\right)\right) \le (\ge) \, {\rm{N}}\left(h(t);f (x), f (y)\right), \end{align*} holds for all $x,y \in I$ and $t\in [0,1]$, where M and N are two mean functions. In this way, nine classes of $h$-${\rm{MN}}$-convex functions are established and some of their analytic properties are explored and investigated. Characterizations of each type are given. Various Jensen's type inequalities and their converses are proved.
math.CA↗
Improvements of Some Numerical radius inequalities
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 refinements for the upper bound of the Kittaned inequality. Indeed, we establish \begin{align*} w^{2 }\left(T \right) \le \frac{1}{2}\left\| T^*T + TT^*\right\| - \frac{1}{4} \mathop {\inf }\limits_{\left\| x \right\| = 1} \left( {\left\langle {\left| T \right|x,x} \right\rangle - \left\langle {\left| T^* \right|x,x} \right\rangle } \right)^2, \end{align*} which also refined and improved as \begin{align*} w^{2 }\left(T \right) \le \frac{1}{2}\left\| T^*T + TT^*\right\| - \frac{1}{2} \mathop {\inf }\limits_{\left\| x \right\| = 1} \left( {\left\langle {\left| T \right|x,x} \right\rangle - \left\langle {\left| T^* \right|x,x} \right\rangle } \right)^2, \end{align*} and \begin{align*} w^{2 }\left(T \right) \le \frac{1}{2} \left\|T^*T+TT^* \right\| -\frac{1}{2} \mathop {\inf }\limits_{\left\| x \right\| = 1} \left(\left\langle {\left| T \right|^{2 }x,x} \right\rangle^{\frac{1}{2}} - \left\langle {\left| T^* \right|^{2 } x,x} \right\rangle^{\frac{1}{2}}\right)^2, \end{align*} with third improvement \begin{align*} w^2 \left( {T } \right) \le \frac{1}{4 }\left\| {\left| T \right| + \left| {T^* } \right| } \right\|^{2} - \frac{1}{4 }\mathop {\inf }\limits_{\left\| x \right\| = 1} \left( {\left\langle {\left| T \right| x,x} \right\rangle - \left\langle {\left| {T^* } \right| x,x} \right\rangle } \right)^2. \end{align*} Other general related results are also considered.
math.FA↗
Numerous approximations of Riemann-Stieltjes double integrals
The concept of Riemann-Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$; where $f$ is called the integrand, $u$ is called the integrator, plays an important role in Mathematics. The approximation problem of the Riemann-Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$ in terms of the Riemann-Stieltjes sums have been considered recently by many authors. However, a small attention and a few works have been considered for mappings of two variables; i.e., The approximation problem of the Riemann-Stieltjes double integral $\int_a^b {\int_c^d {f\left( {t,s} \right)d_s d_t u\left( {t,s} \right)} }$ in terms of the Riemann-Stieltjes double sums. This study is devoted to obtain several bounds for $\int_a^b {\int_c^d {f\left( {t,s} \right)d_s d_t u\left( {t,s} \right)} }$ under various assumptions on the integrand $f$ and the integrator $u$. Mainly, the concepts of bounded variation and bi-variation are used at large in the thesis. Several proposed cubature formula are introduced to approximate such double integrals. For mappings of two variables several inequalities of Trapezoid, Grüss and Ostrowski type for mappings of bounded variation, bounded bi-variation, Lipschitzian and monotonic are introduced and discussed. Namely, Trapezoid-type rules for $\mathcal{RS}$-Double integrals are proved, and therefore the classical Hermite-Hadamard inequality for mappings of two variables is established. A Korkine type identity is used to obtain several Grüss type inequalities for integrable functions. Finally, approximating real functions of two variables which possess $n$-th partial derivatives of bounded bi-variation, Lipschitzian and absolutely continuous are established and investigated.
math.CA↗
Bounds for the difference between two Čebyšev functionals
In this work, a generalization of pre-Grüss inequality is established. Several bounds for the difference between two Čebyšev functional are proved.
math.CA↗
Refinements of Some Numerical radius inequalities for Hilbert Space Operators
In this work, some generalizations and refinements inequalities for numerical radius of the product of Hilbert space operators are proved. New inequalities for numerical radius of block matrices of Hilbert space operators are also established.
math.FA↗
Two-Point Quadrature Rules for Riemann-Stieltjes Integrals with Lp-error estimates
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 with the Hölder condition on $[a,b]$ and the integrator $u$ is of bounded variation on $[a,b]$. The dual formulas under the same assumption are proved. Some sharp error $L^p$--Error estimates for the proposed quadrature rules are also obtained.
math.CA↗
Numerical radius inequalities for Hilbert Space Operators
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 numerical radius of $n\times n$ matrix of Hilbert space operators are proved as well. Some refinements of some earlier results were proved in literature are also given. Some of the presented results are refined and it shown to be better than earlier results were proved in literature.
math.FA↗
Popoviciu's type inequalityies for h-MN-convex functions
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.
math.CA↗
Two-point Ostrowski and Ostrowski-Grüss type inequalities with applications
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 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.
math.CA↗