Neda Lovričević

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.

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.

Zipf-Mandelbrot law, f-divergences and the Jensen-type interpolating inequalities

Motivated by the method of interpolating inequalities that makes use of the improved Jensen-type inequalities, in this paper we integrate this approach with the well known Zipf–Mandelbrot law applied to various types of f-divergences and distances, such are Kullback–Leibler divergence, Hellinger distance, Bhattacharyya distance (via coefficient), χ2

Superadditivity of the Levinson functional and applications

\chi^{2}

Jensen's Operator and Applications to Mean Inequalities for Operators in Hilbert Space

-divergence, total variation distance and triangular discrimination. Addressing these applications, we firstly deduce general results of the type for the Csiszár divergence functional from which the listed divergences originate. When presenting the analyzed inequalities for the Zipf–Mandelbrot law, we accentuate its special form, the Zipf law with its specific role in linguistics. We introduce this aspect through the Zipfian word distribution associated to the English and Russian languages, using the obtained bounds for the Kullback–Leibler divergence.

Eigenvalue inequalities for differences of means of Hilbert space operators

We study the Levinson functional, constructed as a difference between the right-hand side and the left-hand side of the Levinson inequality. We show that it possesses the properties of superadditivity and monotonicity. As a consequence, we obtain mutual bounds for this functional, expressed via the non-weighted functional of the same type. In this way, a refinement and a converse of the Levinson inequality in a difference form is obtained.