Search

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}^{* }) or (Re(A),Im(A))(\hspace{0.1em}\text{Re}\hspace{0.1em}\left(A),\hspace{0.1em}\text{Im}\hspace{0.1em}\left(A)) for an operator AA in B(H)B\left(H), where HH is a Hilbert space. This leads to the derivation of vector, norm, and numerical radius inequalities for a single operator. Furthermore, we obtain power inequalities for the ss-rr-norm and ss-rr -numerical radius of the operator pair (B,C)∈B(H)\left(B,C)\in B\left(H), which generalizes the Euclidean norm and Euclidean numerical radius. Finally, we apply these results to derive the corresponding inequalities for a single operator A∈B(H)A\in B\left(H).

In this article, we establish Hermite-Hadamard-type inequalities for the two classes of functions X±λ(Ω)={f∈C2(Ω):Δf±λf≥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 R2{{\mathbb{R}}}^{2}. We also obtain a characterization of the set X−λ(Ω){X}_{-\lambda }\left(\Omega ). Notice that in the one-dimensional case, if Ω=I\Omega =I (an open interval of R{\mathbb{R}}) and λ=ρ2\lambda ={\rho }^{2}, ρ>0\rho \gt 0, then Xλ(Ω){X}_{\lambda }\left(\Omega ) (resp. X−λ(Ω){X}_{-\lambda }\left(\Omega )) reduces to the class of functions f∈C2(I)f\in {C}^{2}\left(I) such that ff is trigonometrically ρ\rho -convex (resp. hyperbolic ρ\rho -convex) on II.

For μ∈C1(I)\mu \in {C}^{1}\left(I), μ>0\mu \gt 0, and λ∈C(I)\lambda \in C\left(I), where II is an open interval of R{\mathbb{R}}, we consider the set of functions f∈C2(I)f\in {C}^{2}\left(I) satisfying the second-order differential inequality ddtμdfdt+λf≥0\frac{{\rm{d}}}{{\rm{d}}t}\left(\phantom{\rule[-0.75em]{}{0ex}},\mu \frac{{\rm{d}}f}{{\rm{d}}t}\right)+\lambda f\ge 0 in II. The considered set includes several classes of generalized convex functions from the literature. In particular, if μ≡1\mu \equiv 1 and λ=k2\lambda ={k}^{2}, k>0k\gt 0, we obtain the class of trigonometrically kk-convex functions, while if μ≡1\mu \equiv 1 and λ=−k2\lambda =-{k}^{2}, k>0k\gt 0, we obtain the class of hyperbolic kk-convex functions. In this article, we establish a Fejér-type inequality for the introduced set of functions without any symmetry condition imposed on the weight function and discuss some special cases of weight functions. Moreover, we provide characterizations of the classes of trigonometrically and hyperbolic kk-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 convex and operator convex functions are also provided.

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 provided.

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)-1dμ(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 \right),} where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.

In this paper we prove among others that, if (Aj)j=1,...,m are positive definite matrices of order n ≥ 2 and qj ≥ 0, j = 1, ..., m with ∑j=1mqj=1$$\sum\nolimits_{j = 1}^m {{q_j} = 1} $$, then 0≤11−mini∈{1,…,m}{qi}×[∑​i=1mqi(1−qi)[det(Ai)]−1−2n+1∑​1≤i

In this paper we show that, if that the function f : [0, ∞) → 𝔾 is operator monotone in [0, ∞) then there exist b ≥ 0 and a positive measure m on [0, ∞) such that [f(B)-f(A)](B-A)==b(B-A)2+∫0∞s2[∫01[((1-t)A+tB+s)-1(B-A)]2dt]dm(s)\matrix{ {\left[ {f\left( B \right) - f\left( A \right)} \right]\left( {B - A} \right) = } \hfill \cr { = b{{\left( {B - A} \right)}^2} + \int_0^\infty {{s^2}\left[ {\int_0^1 {{{\left[ {{{\left( {\left( {1 - t} \right)A + tB + s} \right)}^{ - 1}}\left( {B - A} \right)} \right]}^2}dt} } \right]dm\left( s \right)} } \hfill \cr } for all A, B > 0. Some necessary and sufficient conditions for the operators A, B > 0 such that the inequality f(B)B+f(A)A≥f(A)B+f(B)Af\left( B \right)B + f\left( A \right)A \ge f\left( A \right)B + f\left( B \right)A holds for any operator monotone function f on [0, ∞) are also given.

For a continuous and positive function ω (λ); λ> 0 and μ a positive measure on [0; ∞) we consider the following 𝒟-logarithmic integral transform𝒟ℒog(w,μ)(T):=∫0∞w(λ)1n(λ+Tλ)dμ(λ),\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right)1{\rm{n}}\left( {{{\lambda + T} \over \lambda }} \right)d\mu \left( \lambda \right),} where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.

In this paper we establish some two point weighted Taylor’s expansions for analytic functions f : D ⊆ ℂ→ ℂ defined on a convex domain D. Some error bounds for these expansions are also provided. Examples for the complex logarithm and the complex exponential are also given.

In this paper we obtain some new inequalities for a generalized finite Hilbert transform of convex functions. Applications for particular instances of finite Hilbert transforms are given as well.

In this paper we consider the hypo-q-operator norm and hypo-q-numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.

Let f be an operator monotonic function on I and A, B∈I (H), the class of all selfadjoint operators with spectra in I. Assume that p : [0.1], →ℝ is non-decreasing on [0, 1]. In this paper we obtained, among others, that for A ≤ B and f an operator monotonic function on I,

In this paper we establish some trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Hölder continuous functions. Applications for the g-mean of two numbers are provided as well. Some particular cases for Hadamard fractional integrals are also provided.

In this paper we obtain some new inequalities for Heinz operator mean.

In this paper by using the notion of sesquilinear form we introduce a new class of numerical range and numerical radius in normed space 𝒱, also its various characterizations are given. We apply our results to get some inequalities.

Some operator inequalities for synchronous functions that are related to the čebyšev inequality are given. Among other inequalities for synchronous functions it is shown that ∥ø(f(A)g(A)) - ø(f(A))ø(g(A))∥ ≤ max{║ø(f2(A)) - ø2(f(A))║, ║ø)G2(A)) - ø2(g(A))║} where A is a self-adjoint and compact operator on B(ℋ ), f, g ∈ C (sp (A)) continuous and non-negative functions and ø: B(ℋ ) → B(ℋ ) be a n-normalized bounded positive linear map. In addition, by using the concept of quadruple D-synchronous functions which is generalizes the concept of a pair of synchronous functions, we establish an inequality similar to čebyšev inequality.

In this paper, some generalizations of Pompeiu's inequality for two complex-valued absolutely continuous functions are provided. They are applied to obtain some new Ostrowski type results. Reverses for the integral Cauchy-Bunyakovsky-Schwarz inequality are provided as well.

Perturbed companions of Ostrowski’s inequality for absolutely continuous functions whose derivatives are either bounded or of bounded variation and applications are given.

We establish in this paper some Jensen’s type inequalities for functions defined by power series with nonnegative coefficients. Applications for functions of selfadjoint operators on complex Hilbert spaces are provided as well.

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 research synthesizes and extends existing knowledge, providing a nuanced understanding of how error bounds depend on the characteristics of integrated functions. Through a systematic review of seminal works, the study contributes to the practical application of numerical integration techniques, offering insight for researchers and practitioners to make informed choices based on the specific features of the functions involved. [ABSTRACT FROM AUTHOR]

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 symmetrical. [ABSTRACT FROM AUTHOR]