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Bohr’s inequalities in \(n\)-inner product spaces. (English) Zbl 1185.46012

The classical Bohr inequality states that, for any \(z, w\in \mathbb C\) and conjugate exponents \(p,q >1\), \(|z + w|^2\leq p|z|^2 + q|w|^2\).
In the past few decades, various generalizations of this inequality have been obtained [W.-S.Cheung and J.Pečarić, J. Math.Anal.Appl.323, No.1, 403–412 (2006; Zbl 1108.26018); O.Hirzallah, J. Math.Anal.Appl.282, No.2, 578–583 (2003; Zbl 1028.47009)], without restriction to the conjugate components \(p,q\).
Here, Bohr’s inequality is generalised to the context of \(n\)-inner-product spaces for all positive conjugate exponents \(p, q\). As an application of these Bohr-type inequalities, an interesting inequality on operators in an \(n\)-inner product space is presented. In particular, the parallelogram law is recovered.

MSC:

46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
26D15 Inequalities for sums, series and integrals
30A10 Inequalities in the complex plane
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