Best approximation of $(\mathcal{G}_{1},\mathcal{G}_{2})$-random operator inequality in matrix Menger Banach algebras with application of stochastic Mittag-Leffler and $\mathbb{H}$-Fox control functions

We stabilize pseudostochastic $(\mathcal{G}_{1},\mathcal{G}_{2})$ ( G 1 , G 2 ) -random operator inequality using a class of stochastic matrix control functions in matrix Menger Banach algebras. We get an approximation for stochastic $(\mathcal{G}_{1},\mathcal{G}_{2})$ ( G 1 , G 2 ) -random operator inequality by means of both direct and fixed point methods. As an application, we apply both stochastic Mittag-Leffler and $\mathbb{H}$ H -fox control functions to get a better approximation in a random operator inequality.

Some norm and operator inequalities to improved AM-GM inequality with Specht’s ratio

In this paper, we present a refinement of the well-known arithmetic-geometric mean inequality. As application of our result, we obtain an operator inequality. We give an improvement of the inequality presented by Kittaneh for the numerical radius.

Some Refinements of the Numerical Radius Inequalities via Young Inequality

In this paper, we get an improvement of the Hölder-McCarthy operator inequality in the case when r ≥ 1 and refine generalized inequalities involving powers of the numerical radius for sums and products of Hilbert space operators.

REFINEMENTS AND REVERSES OF HÖLDER-MCCARTHY OPERATOR INEQUALITY VIA A CARTWRIGHT-FIELD RESULT

By the use of a classical result of Cartwright and Field we obtain in this paper some new refinements and reverses of Hölder-McCarthy operator inequality in the case p 2 (0; 1). A comparison for the two upper bounds obtained showing that neither of them is better in general, is also performed.

Some operator inequalities for Hermitian Banach $*$-algebras

In this paper, we extend the Kubo-Ando theory from operator means on C$^{*}$-algebras to a Hermitian Banach $*$-algebra $\mathcal {A}$ with a continuous involution. For this purpose, we show that if $a$ and $b$ are self-adjoint elements in $\mathcal {A}$ with spectra in an interval $J$ such that $a \leq b$, then $f(a) \leq f(b)$ for every operator monotone function $f$ on $J$, where $f(a)$ and $f(b)$ are defined by the Riesz-Dunford integral. Moreover, we show that some convexity properties of the usual operator convex functions are preserved in the setting of Hermitian Banach $*$-algebras. In particular, Jensen's operator inequality is presented in these cases.

Approximation of an Additive ϱ1,ϱ2-Random Operator Inequality

We solve the additive ϱ1,ϱ2-random operator inequality ξtTω,u+v−Tω,u−Tω,v≥κMξtϱ1Tω,u+v+Tω,u−v−2Tω,u,ξtϱ22Tω,u+v/2−Tω,u−Tω,v, in which ϱ1,ϱ2∈ℂ are fixed and max2ϱ1,ϱ2<1. Finally, we get an approximation of the mentioned additive ϱ1,ϱ2-random operator inequality by direct technique.