# More accurate numerical radius inequalities (II)

## TL;DR

This paper introduces refined and generalized inequalities for the numerical radius of operators, extending previous results and providing tighter bounds using convex functions and operator decompositions.

## Contribution

It presents new inequalities that refine and generalize existing numerical radius bounds, including extensions of Kittaneh's results, using convex functions and operator decompositions.

## Key findings

- New bounds for the numerical radius involving convex functions
- Extension and refinement of Kittaneh's inequalities
- Inequalities applicable to operators with Cartesian decompositions

## Abstract

In a recent work of the authors, we showed some general inequalities governing numerical radius inequalities using convex functions. In this article, we present results that complement the aforementioned inequalities. In particular, the new versions can be looked at as refined and generalized forms of some well known numerical radius inequalities. Among many other results, we show that \[\left\| f\left( \frac{{{A}^{*}}A+A{{A}^{*}}}{4} \right) \right\|\le \left\| \int_{0}^{1}{f\left( \left( 1-t \right){{B}^{2}}+t{{C}^{2}} \right)dt} \right\|\le f\left( {{w}^{2}}\left( A \right) \right),\] when $A$ is a bounded linear operator on a Hilbert space having the Cartesian decomposition $A=B+iC.$ This result, for example, extends and refines a celebrated result by kittaneh.