S. S. Dragomir

• Professor of Mathematics
• Victoria University Melbourne at Victoria University

This paper investigates a generalization of the spherical numerical radius for a pair (B,C) of bounded linear operators on a complex Hilbert space H. The generalized spherical numerical radius is defined as wp(B,C):=supx∈H,∥x∥=1|〈Bx,x〉|p+|〈Cx,x〉|p1p, p≥1. We derive lower bounds for wp2(B,C) involving combinations of B and C, where p>1. Additionally...

The article examines inequalities for norms and numerical radii of bounded linear operators on complex Hilbert spaces. It focuses on scenarios where three operators are involved, with one being positive, and investigates their sums or products. Some of our findings extend existing inequalities established in the literature.

The concept of the weighted $\mathcal{A}$-numerical radius was recently defined, where $\mathcal{A}$ is assumed to be a positive operator. In this paper, we introduce another weighted $\mathcal{A}$-numerical radius, denoted by $\omega_{(\varepsilon, \mathcal{A})}\left( \cdot \right)$, for operators in semi-Hilbert spaces. We establish some basic pr...

This in-depth study looks at symmetric four-point Newton-Cotes-type inequalities with a focus on error estimates for numerical integration. The precision of these estimates is explored across various classes of functions, including those with bounded variation, bounded derivatives, Lipschitzian derivatives, convex derivatives, and others. The resea...

In this article, we would like to introduce another generalized class of convex functions which we call as (α, η, γ, δ)−p convex functions. This new class contains another two new classes namely, (α, η) − p convex functions of the 1st and 2nd kinds. Further, we also generalize some results related to famous Hermite-Hadamard type inequality stated i...

The primary objective of this study is to unveil new upper bounds for the numerical radius of operators in Hilbert spaces, utilizing an enhanced version of the McCarthy inequality. These newly derived estimates refine existing inequalities for bounded linear operators.

This paper presents new weighted lower and upper bounds for the Euclidean numerical radius of pairs of operators in Hilbert spaces. We show that some of these bounds improve on recent results in the literature. We also find new inequalities for the numerical radius and the Davis–Wielandt radius. The lower and upper bounds we obtain are not symmetri...

In the paper, we define the novel orthogonality based on the Hermite–Hadamard (HH) type under multiple integrals and consider its connections with Birkhoff–James orthogonality and convex function. The orthogonality is analyzed to investigate its properties and some new characterizations of real inner product spaces. At the same time, we investigate...

Assume that h : G → ℂ {h:G\rightarrow\mathbb{C}} is analytic on the convex domain G and x ∈ ℒ ⁢ ( ℬ ; E , 𝒜 , μ ) {x\in\mathcal{L}(\mathcal{B};E,\mathcal{A},\mu)} , the set of Bochner-integrable functions on a measurable space ( E , 𝒜 , μ ) {(E,\mathcal{A},\mu)} endowed with a countably-additive scalar measure μ on a σ-algebra 𝒜 {\mathcal{A}} of su...

This study explores the power vector inequalities for a pair of operators ( B , C ) \left(B,C) in a Hilbert space. By utilizing a Mitrinović-Pečarić-Fink-type inequality for inner products and norms, we derive various power vector inequalities. Specifically, we consider the cases where ( B , C ) \left(B,C) is equal to ( A , A * ) \left(A,{A}^{* })...

In this paper we obtain some bounds concerning the operator perspectives of selfadjoint operators in Hilbert spaces by making use of Ostrowski type inequalities. Applications for weighted operator geometric mean and relative operator entropy are also provided.

We consider trapezoid type inequalities for twice differentiable convex functions, perturbed by a non-negative weight. Applications on a normed space \( (X, \lVert \,\cdot\, \rVert) \) are considered, by establishing bounds for the term \[ \begin{multline*} \frac{1}{2} \left[\lVert \frac{x+y}{2} \rVert^p + \frac{\lVert x \rVert^p + \lVert y \rVert^...

The intent of the current study is to explore convex stochastic processes within a broader context. We introduce the concept of unified stochastic processes to analyze both convex and non-convex stochastic processes simultaneously. We employ weighted quasi-mean, non-negative mapping γ, and center-radius ordering relations to establish a class of ex...

In this study, we introduce a novel local fractional integral identity related to the Gaussian two-point left Radau rule. Based on this identity, we establish some new fractal inequalities for functions whose first-order local fractional derivatives are generalized convex and concave. The obtained results not only represent an extension of certain...

This paper presents new lower and upper bounds for the Euclidean numerical radius of operator pairs in Hilbert spaces, demonstrating improvements over recent results by other authors. Additionally, we derive new inequalities for the numerical radius and the Davis–Wielandt radius as natural consequences of our findings.

In this paper, we employ a generalization of the Boas–Bellman inequality for inner products, as developed by Mitrinović–Pečarić–Fink, to derive several upper bounds for the 2p-th power with p≥1 of the numerical radius of the off-diagonal operator matrix 0AB*0 for any bounded linear operators A and B on a complex Hilbert space H. While the general m...

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Let H be a Hilbert space and Ω a locally compact Hausdorff space endowed with a Radon measure μ with ∫_{Ω}1dμ(t)=1. In this paper we show among others that, if f is continuous differentiable convex on the open interval I, (A_{τ})_{τ∈Ω} is a continuous field of positive operators in B(H) such that Sp(A_{τ}) ⊂I for each τ∈Ω and B and operator such th...

In this paper, we present various inequalities regarding the linear combinations of orthogonal projections. These results aim to generalize and refine well-known inequalities, such as those due to Buzano and Ostrowski. Additionally, we investigate a specific case of these linear combinations and introduce new refinements of the Cauchy–Schwarz inequ...

This paper aims to establish new upper bounds for the Euclidean operator radius concerning pairs of bounded linear operators in a complex Hilbert space. To achieve this objective, we utilize some Boas-Bellman type inequalities as proof tools. Furthermore, we extend our findings to derive novel upper bounds for the numerical radius of operators in H...

This book covers contemporary topics in mathematical analysis and its applications and relevance in other areas of research. It provides a better understanding of methods, problems, and applications in mathematical analysis. It also covers applications and uses of operator theory, approximation theory, optimization, variable exponent analysis, ineq...

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s inequality, Newton-type inequal...

In this article, we establish Hermite-Hadamard-type inequalities for the two classes of functions X ± λ ( Ω ) = { f ∈ C 2 ( Ω ) : Δ f ± λ f ≥ 0 } {X}_{\pm \lambda }\left(\Omega )=\{f\in {C}^{2}\left(\Omega ):\Delta f\pm \lambda f\ge 0\} , where λ > 0 \lambda \gt 0 and Ω \Omega is an open subset of R 2 {{\mathbb{R}}}^{2} . We also obtain a character...

In this paper, our goal is to establish novel inequalities for the Euclidean operator radius of a pair of bounded linear operators defined on a complex Hilbert space. Additionally, we use the Heilbronn inequality to derive further inequalities relevant to both single and pairs of Hilbert space operators.

Some new inequalities of Grüss' type for functions of selfadjoint operators in Hilbert spaces, under suitable assumptions for the involved operators, are given. Several examples for particular functions of interest are provided as well. 2000 Mathematics Subject Classification. 47A63, 47A99

Some power inequalities for the numerical radius of a product of two operators in Hilbert spaces with applications for commutators and self-commutators are given. 2000 Mathematics Subject Classification. 47A12, 47A30, 47A63, 47B15

Several Simpson 1 8 tensorial type inequalities for selfadjoint operators have been obtained with variation depending on the conditions imposed on the function f||1/8[f(A)⊗1 + 6f(A⊗1 + 1⊗B/2) + 1⊗f(B)] − ∫01f(λ1⊗B + (1−λ)A⊗1)dλ|| ≤ 5||1⊗B − A⊗1||/32 ||f’||I,+∞.

Some new Jensen's type inequalities for differentiable log-convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided. 2000 Mathematics Subject Classification. 47A63, 47A99

For μ ∈ C 1 ( I ) \mu \in {C}^{1}\left(I) , μ > 0 \mu \gt 0 , and λ ∈ C ( I ) \lambda \in C\left(I) , where I I is an open interval of R {\mathbb{R}} , we consider the set of functions f ∈ C 2 ( I ) f\in {C}^{2}\left(I) satisfying the second-order differential inequality d d t μ d f d t + λ f ≥ 0 \frac{{\rm{d}}}{{\rm{d}}t}\left(\phantom{\rule[-0.75...

Let $f\left( \lambda \right) =\sum_{n=0}^{\infty }\alpha _{n}\lambda ^{n}$ be a function defined by power series with complex coefficients and convergent on the open disk $D\left( 0,R\right) \subset \mathbb{C}$, $R>0$ and $x,y\in \mathcal{B}$, a Banach algebra, with $xy=yx.$ In this paper we establish some new upper bounds for the norm of the Čebyš...

Let $\mathcal{B}_A(\mathbb{H})$ denote the algebra of bounded linear operators on a complex Hilbert space $\mathbb{H}$ that admit $A$-adjoint operators, where $A$ is a non-zero positive semi-definite operator on $\mathbb{H}$. A commuting operator tuple $\mathbf{T}=(T_1,\ldots,T_d)\in \mathcal{B}_A(\mathbb{H})^d$ is called jointly $A$-normaloid if $...

In mathematical analysis theory of inequalities has considerable influence due to its massive utility in various fields of physical sciences. These are investigated via multiple approaches to acquire more precise and rectified forms of already celebrated consequences. Integral inequalities are investigated to compute the error bounds for quadrature...

In this article, a very effective technique called generalized auxiliary equation mapping method has been employed to investigate some very important nonlinear equations in optical fibers such as Fokas system and (2 + 1) Davey-Stewartson (DS) system. Under different situations, the obtained solutions exhibit various wave pattern like bright and dar...

The main focus of this paper is on establishing inequalities for the norm and numerical radius of various operators applied to a power series with the complex coefficients h(λ)=∑k=0∞akλk and its modified version ha(λ)=∑k=0∞|ak|λk. The convergence of h(λ) is assumed on the open disk D(0,R), where R is the radius of convergence. Additionally, we expl...

Consider the power series with complex coefficients h(z)=∑k=0∞akzk and its modified version ha(z)=∑k=0∞|ak|zk. In this article, we explore the application of certain Hölder-type inequalities for deriving various inequalities for operators acting on the aforementioned power series. We establish these inequalities under the assumption of the converge...

Let $H$ be a complex Hilbert space. Assume that the power series with complex coefficients $f(z):=\sum\nolimits_{k=0}^{\infty }a_{k}z^{k}$ is convergent on the open disk $D(0,R),~f_{a}(z):=\sum\nolimits_{k=0}^{\infty}\left\vert a_{k}\right\vert z^{k}$ that has the same radius of convergence $R$ and $A,~B,~C\in B(H)$ with $\left\Vert A\right\Vert $...

Let $H$ be a Hilbert space. In this paper we show among others that, if the selfadjoint operators $A$ and $B$ satisfy the condition $0$ $<$ $m\leq A,$ $B\leq M,$ for some constants $m,$ $M,$ then \begin{align*} 0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes 1+1\otimes B^{2}}{2}-A\otimes B\right) \\ & \leq \left( 1-\nu \ri...

Motivated by the results previously reported, the current work aims at developing new numerical radius upper bounds of Hilbert space operators by offering new improvements to the well-known Cauchy-Schwarz inequality. In particular, a novel Lemma (3.1) is given, which is utilized to further generalize several vector and numerical radius type inequal...

Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and studied various properties o...

In this paper we introduce some new f-divergence measures that we call t-asymmetric/symmetric divergence measure and integral divergence measure, establish their joint convexity and provide some inequalities that connect these f-divergences to the classical one intyroduced by Csiszar in 1963. Applications for the dichotomy class of convex functions...

In this paper we obtain several operator inequalities providing upper bounds for the Davis-Choi-Jensen's Difference Ph (f (A)) - f (Ph (A)) for any convex function f : I → R, any selfadjoint operator A in H with the spectrum Sp (A) ⊂ I and any linear, positive and normalized map Ph : B (H) → B (K), where H and K are Hilbert spaces. Some examples of...

Let H be a Hilbert space. Assume that f : [0,?) ? R is continuous and A, B > 0. We define the tensorial perspective for the function f and the pair of operators (A, B) by Pf,? (A,B) := (1 ? B) f (A ? B?1). In this paper we show among others that, if f is differentiable convex, then Pf,? (A, B) ?[ f (u)?f? (u)u](1 ? B) + f?(u)(A?1), for A, B > 0 and...

Let $r_e(\mathbf{S})$ be the Euclidean spectral radius associated with a $q$-tuple $\mathbf{S}=(S_1,\ldots,S_q)$ of bounded linear operators on a complex Hilbert space. The principal objective of our study is to establish various compelling upper bounds involving $r_e(\cdot)$. In particular, our findings demonstrate that, for all $t\in \left[ 0,1\r...

Let H be a Hilbert space. In this paper we show among others that, if f, g are continuous on the interval I with 0 <γ ≤ f (t)/g (t)≤ Γ for t ∈ I and if A and B are selfadjoint operators with Sp (A), Sp (B) ⊂ I, then [f1−ν(A)g ν(A)] ⊗ [f ν(B)g 1−ν(B)] ≤ (1 − ν) f(A) ⊗ g (B) + νg(A) ⊗ f(B) ≤[(γ + Γ)2/4γΓ ]R [f1−ν (A) g ν(A)] ⊗ [f ν(B) g1−ν (B)]. The...

In this paper, we obtain some reverses of Callebaut and Hölder inequalities for isotonic functionals via a reverse of Young’s inequality we have established recently. Applications for integrals and n-tuples of real numbers are provided as well.

For a continuous and positive function w ( λ ), λ > 0 and µ a positive measure on (0, ∞ ) we consider the following integral transform 𝒟 ( w , μ ) ( T ) : = ∫ 0 ∞ w ( λ ) ( λ + T ) − 1 d μ ( λ ) , \mathcal{D}\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right){{\left( {\lambda + T} \right)}^{ - 1}}d\mu \left( \lambda \r...

Let H be a Hilbert space. Assume that f is continuously differentiable on I with ‖f′‖_{I,∞}:=sup_{t∈I}|f′(t)

In this paper we introduce some f-divergence measures that are related to the Jensen's divergence introduced by Burbea and Rao in 1982. We establish their joint convexity and provide some inequalities between these measures and a combination of Csiszár's f-divergence, f-midpoint divergence and f-integral divergence measures.

In this paper we introduce the concept of pre-Schur convex functions defined on general domains from plane. Then, by making use of Green’s identity for double integrals, we establish some integral inequalities for this class of functions that naturally generalize the case of Schur convex functions. Some exmples for rectangles and disks are also pro...

In this paper several tensorial norm inequalities of Ostrowski type for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f.

This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the s-convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson’s, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications of our main outcom...

Let $H$ be a Hilbert space. In this paper we show among others that, if $f,$ $g$ are synchronous and continuous on $I$ and $A,$ $B$ are selfadjoint with spectra ${Sp}\left( A\right) ,$ ${Sp}\left( B\right) \subset I,$ then% \begin{equation*} \left( f\left( A\right) g\left( A\right) \right) \otimes 1+1\otimes \left( f\left( B\right) g\left( B\right)...

The main focus of this paper is the study of the Selberg operator. It aims to establish appropriate bounds for the norm and numerical radius of the product of three bounded operators, with one of them being a Selberg operator. Moreover, it offers several bounds involving the summation of operators, notably the Selberg operator. Through the examinat...

In this paper we establish a natural extension of the Wirtinger inequality to the case of complex integral of analytic functions. Applications related to the trapezoid inequalities are also provided. Examples for logarithmic and exponential complex functions are given as well.

In this chapter we establish some Ostrowski and generalized trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Lipschitzian functions. Applications for mid-point and trapezoid inequalities are provided as well. They generalize the know results holding for the classical Riemann integra...

Let $\mathbf{T}=(T_1,\ldots,T_d)$ be an arbitrary $d$-tuple of bounded linear operators on a complex Hilbert space $\mathbb{H}$. For $0 \le t \le 1$, the generalized spherical Aluthge transform of $\mathbf{T}$ was recently defined by Benhida et al. as \begin{equation*} \Delta_t(\mathbf{T}):=\big(P^t V_1P^{1-t}, \ldots, P^t V_dP^{1-t}\big), \end{equ...

For a continuous and positive function w ( λ ) , λ > 0 and µ a positive measure on (0, ∞) we consider the following integral transform 𝒟 ( w , μ ) ( T ) : = ∫ 0 ∞ w ( λ ) ( λ + T ) - 1 d μ ( λ ) , \mathcal{D}\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right){{\left( {\lambda + T} \right)}^{ - 1}}d\mu \left( \lambda \r...

This paper introduces several refinements of the classical Selberg inequality, which is considered a significant result in the study of the spectral theory of symmetric spaces, a central topic in the field of symmetry studies. By utilizing the contraction property of the Selberg operator, we derive improved versions of the classical Selberg inequal...

For a continuous and positive function $w\left( \lambda \right) ,$ $\lambda >0$ and $\mu $ a positive measure on $(0,\infty )$ we consider the following integral transform % \begin{equation*} \mathcal{D}\left( w,\mu \right) \left( T\right) :=\int_{0}^{\infty }w\left( \lambda \right) \left( \lambda +T\right) ^{-1}d\mu \left( \lambda \right) , \end{e...

Some inequalities involving the distance in metric spaces are provided.

In this article, we investigate new findings on Boas–Bellman-type inequalities in semi-Hilbert spaces. These spaces are generated by semi-inner products induced by positive and positive semidefinite operators. Our objective is to reveal significant properties of such spaces and apply these results to the field of multivariable operator theory. Spec...

This paper presents new results related to Bombieri’s generalization of Bessel’s inequality in a semi-inner product space induced by a positive semidefinite operator A. Specifically, we establish new inequalities that generalize the classical Bessel inequality and extend previous results in this area. Furthermore, our findings have applications to...

In this paper, we study p-tuples of bounded linear operators on a complex Hilbert space with adjoint operators defined with respect to a non-zero positive operator A. Our main objective is to investigate the joint A-numerical radius of the p-tuple.We established several upper bounds for it, some of which extend and improve upon a previous work of t...

In this paper we prove among others that, if ( A j ) j =1,..., m are positive definite matrices of order n ≥ 2 and q j ≥ 0, j = 1, ..., m with ∑ j = 1 m q j = 1 $$\sum\nolimits_{j = 1}^m {{q_j} = 1} $$ , then 0 ≤ 1 1 − min i ∈ { 1 , … , m } { q i } × [ ∑ i = 1 m q i ( 1 − q i ) [ det ( A i ) ] − 1 − 2 n + 1 ∑ 1 ≤ i < j ≤ m q i q j [ det ( A i + A j...

Some fundamental inequalities for Lerch transcendent function with positive terms by utilising certain classical results due to Hölder, Čebyšev, Grüss and others, are established. Some particular cases of interest for Polylogarithm function, Hurwitz zeta function and Legendre chi function are also given.

In this paper, we aim to establish several estimates concerning the generalized Euclidean operator radius of d-tuples of A-bounded linear operators acting on a complex Hilbert space H, which leads to the special case of the well-known A-numerical radius for d=1. Here, A is a positive operator on H. Some inequalities related to the Euclidean operato...

Assume that A j , j ∈ {1, … , m } are positive definite matrices of order n . In this paper we prove among others that, if 0 < l I n ≤ A j , j ∈ {1, … , m } in the operator order, for some positive constant l , and I n is the unity matrix of order n , then where Pk ≥ 0 for k ϵ {1, …, m } and .

In this paper, we extract variety of new exact traveling wave solutions of space–time fractional nonlinear Telegraph equation for transmission lines by using improved generalized Riccati equation mapping (IGREM) method. The aforementioned equation has been solved for the first time using conformable fractional derivative. The nonlinear Telegraph eq...

Let Ɓ be a unital Banach algebra, let a ∈ Ɓ, G be a convex domain of ℂ with σ(a) ⊂ G, let α, β ∈ G, and let f : G → ℂ be analytic on G. By using the analytic functional calculus, we obtain (among others) the following result: ‖f(a)-12∑k=0n1k![f(k)(α)(a-α)k+(-1)kf(k)(β)(β-a)k]‖≤12(n+1)![‖a-α‖n+1+‖β-a‖n+1]×max{sups∈[0,1]‖f(n+1)[(1-s)α+sa]‖,sups∈[0,1]...

The method of Lyapunov is one of the most effective methods for the analysis of the partial stability of dynamical systems. Different authors develop the problem of partial practical stability based on Lyapunov techniques. In this paper, we investigate the partial practical stability of linear time-invariant perturbed systems based on the integral...

In this note we prove among others that ∑1≤i<j≤npipjdS(xi,xj)≤ {2s−1 infx∈X [ Σ k=1n pk (1 − pk) ds (xk, x)], s ≥ 1; infx∈X [ Σ k=1n pk (1 − pk) ds (xk, x)] , 0 < s < 1, where (X, d) is a metric space, xi ∈ X, pi ≥ 0, i ∈ {1, ..., n} with Σ i=1n pi = 1 and s > 0. This generalizes and improves some early upper bounds for the sum Σ1≤i<j≤n pipjd (xi,...

In this paper we prove among others that, if the positive definite matrices A, B of order n satisfy the condition 0 < mIn ≤ B − A ≤ M In, for some constants 0 < m < M, where In is the identity matrix, then 0 ≤ (1 − t) [det (A)]−1 + t [det (A + mIn)]−1 − [det (A + mtIn)]−1 ≤ (1 − t) [det (A)]−1 + t [det (B)]−1 − [det ((1 − t) A + tB)]−1 ≤ (1 − t) [d...

For a continuous and positive function w (λ), λ > 0 and µ a positive measure on (0, ∞) we consider the following integral transform D (w, µ) (T ) := ∫0∞w (λ) (λ + T ) −1 dµ (λ) , where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among others that, if A ≥ m 1 > 0, B ≥ m 2 > 0, then ||D (w, µ) (B)...