Kristina Krulić

GENERALIZED NONCOMMUTATIVE HARDY AND HARDY–HILBERT TYPE INEQUALITIES

We extend and unify several Hardy type inequalities to functions whose values are positive semi-definite operators. In particular, our methods lead to the operator versions of Hardy–Hilberts and Godunovas inequalities. While classical Hardy type inequalities hold for parameter values p > 1, it is typical that the operator versions hold only for 1 < p ≤ 2, even for functions with values in 2 × 2 matrices.

On an Inequality of H. G. Hardy

We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. Finally, we apply our main result to multidimensional settings to obtain new results involving mixed Riemann-Liouville fractional integrals.

Fractional Steffensen type inequalities

In this article, we state, prove and discuss new Steffensen type fractional inequalities involving fractional derivatives of Canavati, Riemann–Liouville and Caputo types. Finally, we define functionals as a difference between the right- and left-hand sides of the new fractional Steffensen type inequalities and obtain related analogous of the Lagrange and the Cauchy mean value theorems. Furthermore, exponential convexity is proven.