Abramovich, Shoshana; Barić, Josipa; Pečarić, Josip A new proof of an inequality of Bohr for Hilbert space operators. (English) Zbl 1162.47016 Linear Algebra Appl. 430, No. 4, 1432-1435 (2009). The authors give a simple proof of Bohr’s inequality for Hilbert space operators, i.e., \[ |A-B|^{2}+|(1-p)A-B|^{2}\leq p|A|^{2}+q|B|^{2} \] for \(\frac{1}{p}+\frac{1}{q}=1\) and \(A,B\in B(H)\). In the proof, which simplifies the one in [W.–S.Cheung and J.Pečarić, J. Math.Anal.Appl.323, 403–412 (2006; Zbl 1108.26018)], the authors point out that the equality \[ p|A|^{2}+q|B|^{2}=|A-B|^{2}+\frac{1}{p-1}|(p-1)A+B|^{2} \] is important. Reviewer: Takeaki Yamazaki (Yokohama) Cited in 4 Documents MSC: 47A63 Linear operator inequalities 26D15 Inequalities for sums, series and integrals 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) Keywords:Bohr’s inequality Citations:Zbl 1108.26018 PDFBibTeX XMLCite \textit{S. Abramovich} et al., Linear Algebra Appl. 430, No. 4, 1432--1435 (2009; Zbl 1162.47016) Full Text: DOI References: [1] Bohr, H., Zur Theorie der fastperiodischen Funktionen I, Acta Math., 45, 29-127 (1924) · JFM 50.0196.01 [2] W.-S. Cheung, J. Pecaric, D. Zhao, On Bohr’s inequalities, in: Themistocles M. Rassias (Ed.), Inequalities and Applications, in press.; W.-S. Cheung, J. Pecaric, D. Zhao, On Bohr’s inequalities, in: Themistocles M. Rassias (Ed.), Inequalities and Applications, in press. · Zbl 1262.47025 [3] Cheung, W.-S.; Pecaric, Josip, Bohr’s inequalities for Hilbert space operators, J. Math. Anal. Appl., 323, 403-412 (2006) · Zbl 1108.26018 [4] Hirzalla, O., Non-commutative operator Bohr inequality, J. Math. Anal. Appl., 282, 578-583 (2003) · Zbl 1028.47009 [5] Mitrinović, D. S.; Pecaric, J. E.; Fink, A. M., Classical and New Inequalities in Analysis (1993), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0771.26009 [6] Pečarić, J. E.; Dragomir, S. S., A refinement of Jensen inequality and applications, Studia Univ, Babes-Bolyai Math., 34, 15-19 (1989) · Zbl 0687.26009 [7] Pečarić, J. E.; Rassias, Th. M., Variations and generalizations of Bohr’s inequality, J. Math. Anal. Appl., 174, 138-146 (1993) · Zbl 0777.26018 [8] Rassias, Th. M., On generalization of a triangle-like inequality due to H. Bohr, Abstracts Amer. Math. Soc., 5 (1985) [9] Rassias, Th. M., On characterizations of inner spaces and generalizations of the H. Bohr inequality, (Rassias, Th. M., Topics in Mathematical Analysis (1989), World Scientific: World Scientific Singapore) · Zbl 0777.26018 [10] Vasić, P. M.; Kečkić, J. D., Some inequalities for complex numbers, Math. Balkanica, 1, 282-286 (1971) · Zbl 0219.26012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.