Skip to content
DraconiticResearch
Search Sources API
197,788 studies31 sourcesUpdated 27 min ago
Peer-reviewedCC BY 4.0Mathematics

Reversed and Refined Berezin Number Inequalities via Buzano’s Inequality and Polar Decomposition

  • Reem K. Alhefthi1
  • Silvestru Sever Dragomir2
  1. 1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
  2. 2Department of Mathematical and Geospatial Sciences, School of Science, RMIT University, GPO Box 2476, Melbourne, VIC 3001, Australia
Open at sourceDOICite

Abstract

The Berezin number and Berezin norm have become indispensable tools for analyzing the structure of operators on reproducing kernel Hilbert spaces, yet the gap between these quantities and the operator norm is still far from fully understood. In this paper, we narrow that gap by establishing several new reverse-type inequalities for ber(V) and ∥V∥ber˜ under natural disc-localization and accretivity conditions. Combining a refinement of the Schwarz inequality due to Dragomir with Buzano’s inequality, we further derive sharp bounds for ber(A*TA) and for products TA, together with their power-type extensions. Applying the polar decomposition, we also obtain companion inequalities for |T|α and the Aluthge transform. Finally, for operators with closed range, we use the Moore–Penrose inverse to derive new reverse-type Berezin inequalities that refine and complement the results obtained via the polar decomposition, and we illustrate the sharpness of these bounds with an explicit example.

Related

  1. Berezin--Li--Yau inequalities for Aharonov--Bohm operators
    Michael Levitin, Iosif Polterovich, Romain Speciel
    arXivPreprintCC BY-NC-SA 4.0Full textAnalysis & PDEsMathematical physicsarXiv
  2. Inequalities for the Berezin Number and Norms of a Product of Three Operators
    Silvestru Sever Dragomir, Ghada N. Alnemer, Ahad Hamoud Alotaibi
    AxiomsPeer-reviewedCC BY 4.0General mathematicsCrossref
  3. Kernel Methods for Refined Prophet Inequalities
    Patrick Loiseau, Mathieu Molina, Vianney Perchet +2
    HALPreprintLicence not statedGeneral mathematicsHAL
  4. Refined Bohr Inequalities and Bohr Radii Involving Schwarz Mappings for Holomorphic Functions in Banach Spaces and JB*-Triples
    Nabadwip Sarkar, Pradip Das
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  5. Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality
    Emanuel Milman, Shohei Nakamura, Hiroshi Tsuji
    arXivPreprintarXiv licenceFull textAnalysis & PDEsProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@article{alhefthi2026reversed,
  title = {{Reversed and Refined Berezin Number Inequalities via Buzano’s Inequality and Polar Decomposition}},
  author = {Reem K. Alhefthi and Silvestru Sever Dragomir},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14203827},
  url = {https://www.mdpi.com/2227-7390/14/20/3827},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Reversed and Refined Berezin Number Inequalities via Buzano’s Inequality and Polar Decomposition
AU  - Reem K. Alhefthi
AU  - Silvestru Sever Dragomir
PY  - 2026
DA  - 2026/10/21
JO  - Mathematics
DO  - 10.3390/math14203827
UR  - https://www.mdpi.com/2227-7390/14/20/3827
AB  - The Berezin number and Berezin norm have become indispensable tools for analyzing the structure of operators on reproducing kernel Hilbert spaces, yet the gap between these quantities and the operator norm is still far from fully understood. In this paper, we narrow that gap by establishing several new reverse-type inequalities for ber(V) and ∥V∥ber˜ under natural disc-localization and accretivity conditions. Combining a refinement of the Schwarz inequality due to Dragomir with Buzano’s inequality, we further derive sharp bounds for ber(A*TA) and for products TA, together with their power-type extensions. Applying the polar decomposition, we also obtain companion inequalities for |T|α and the Aluthge transform. Finally, for operators with closed range, we use the Moore–Penrose inverse to derive new reverse-type Berezin inequalities that refine and complement the results obtained via the polar decomposition, and we illustrate the sharpness of these bounds with an explicit example.
SN  - 2227-7390
ER  -