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A new proof of an inequality of Bohr for Hilbert space operators. (English) Zbl 1162.47016

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.

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)

Citations:

Zbl 1108.26018
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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
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