On numerical radius inequalities via McCarthy inequality

Muhammad Fazeel Anwar
Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan
Saira Iqbal
Department of Mathematics, Ghazi University, Dera Ghazi Khan 32200, Pakistan
Muhammad Saeed Akram
Department of Mathematics, Ghazi University, Dera Ghazi Khan 32200, Pakistan

Article ID: 4409

Keywords: numerical radius; McCarthy inequality; operator inequalities; Hilbert space; refinement

Abstract

This paper aims to establish new inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space. By employing a generalized form of the McCarthy inequality, we derive several upper bounds for the numerical radius of a single operator as well as for expressions involving sums and products of operators. The obtained results extend and refine a number of existing inequalities in the literature. In particular, many known numerical radius inequalities are recovered as special cases of our results, thereby providing a unified framework for their analysis. The refinement is based on the Akkouchi-Ighachane version of the Hölder-McCarthy inequality, which allows the usual McCarthy term to be replaced by a smaller parameter-dependent expression before taking the supremum. This gives a common refinement mechanism for estimates involving a single operator, Cartesian decompositions, finite sums, block matrices and the Euclidean operator radius. We also state explicitly the parameter choices which recover the earlier inequalities and include simple finite-dimensional comparisons showing that, for suitable non-normal matrices, the refined bounds may be strictly sharper than the corresponding classical estimates. These comparisons confirm that the refinement provides a usable quantitative improvement, rather than only a formal parameter extension of existing bounds in concrete cases.

Published

2026-07-14

How to Cite

Anwar, M. F., Iqbal, S., & Akram, M. S. (2026). On numerical radius inequalities via McCarthy inequality. Advances in Differential Equations and Control Processes, 33(3). https://doi.org/10.59400/adecp4409

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Vol. 33 No. 3 (2026): In Progress

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This work is licensed under a Creative Commons Attribution 4.0 International License.

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