Mario Krnić

Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators

The main objective of this paper is an improvement of the original weighted operator arithmetic-geometric mean inequality in Hilbert space. We define the difference operator between the arithmetic and geometric means and investigate its properties. Due to the derived properties, we obtain a refinement and a converse of the observed operator mean inequality. As an application, we establish one significant operator mean, which interpolates the arithmetic and geometric means, that is, the Heinz operator mean. We also obtain an improvement of this interpolation.

Multiple Hilbert and Hardy-Hilbert inequalities with non-conjugate parameters

The main objective of this paper is a study of some new generalisations of Hilbert and Hardy-Hilbert type inequalities involving non-conjugate parameters. We prove general forms of multiple Hilbert-type inequalities, and we also introduce multiple inequalities of Hardy-Hilbert type with non-conjugate parameters.

Half-discrete Hilbert-type inequalities with mean operators, the best constants, and applications

In this article we derive several new half-discrete Hilbert-type inequalities with a general homogeneous kernel, involving arithmetic, geometric and harmonic mean operators. The main results are proved for the case of non-conjugate exponents. A special emphasis is given to determining conditions under which these inequalities include the best possible constants. As an application, we consider some operator expressions closely connected to established inequalities.

Jessen's functional, its properties and applications

Abstract In this paper we consider Jessens functional, defined by means of a positive isotonic linear functional, and investigate its properties. Derived results are then applied to weighted generalized power means, which yields extensions of some recent results, known from the literature. In particular, we obtain the whole series of refinements and converses of numerous classical inequalities such as the arithmetic-geometric mean inequality, Youngs inequality and Hölders inequality

On the properties of McShane’s functional and their applications

In this paper we derive some remarkable properties of McShane’s functional, defined by means of positive isotonic linear functionals. These properties are then applied to weighted generalized means. A series of consequences among additive and multiplicative type mean inequalities is given, as well as a special consideration of Hölder’s inequality, in view of the new results.

A HILBERT INEQUALITY AND AN EULER-MACLAURIN SUMMATION FORMULA

We obtain a generalized discrete Hilbert and Hardy-Hilbert inequality with non-conjugate parameters by means of an Euler-Maclaurin summation formula. We derive some general results for homogeneous functions and compare our findings with existing results. We improve some earlier results and apply the results to some special homogeneous functions.

Reverse Young-type inequalities for matrices and operators

We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if A,B ∈ B(H) are positive operators and r ≥ 0, A∇−rB + 2r(A∇B − A♯B) ≤ A♯−rB. We also prove that equality holds if and only if A = B. In addition, we establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if A and B are positive definite matrices and r ≥ 0, then tr ((1 + r)A− rB) ≤ tr|A1+rB−r| − r( √ trA− √ trB)2.

Refined Heinz operator inequalities

Motivated by the well-known Heinz norm inequalities, in this article we study the corresponding Heinz operator inequalities. We derive the whole series of refinements of these operator inequalities, first with the help of the well-known Hermite–Hadamard inequality, and then, utilizing the parametrized family of the so-called Heron means. In such a way, we obtain improvements of some recent results, known from the literature.

Improved Heinz inequalities via the Jensen functional

By virtue of convexity of Heinz means, in this paper we derive several refinements of Heinz norm inequalities with the help of the Jensen functional and its properties. In addition, we discuss another approach to Heinz operator means which is more convenient for obtaining the corresponding operator inequalities for positive invertible operators.

Weighted Montgomery identity for the fractional integral of a function with respect to another function

Abstract. We present a weighted Montgomery identity for the fractional integral of a function f with respect to another function g and use it to obtain weighted Ostrowski type inequalities for fractional integrals involving functions whose first derivatives belong to Lp spaces. These inequalities are generally sharp in case p>1