Abstract
We obtain an inequality complementary to the Cauchy-Schwarz inequality in Hilbert space. The inequalities involving first three powers of a self-adjoint operator are derived. The inequalities include the bounds for the third central moment, as a special case. It is shown that an upper bound for the spectral radius of a matrix is a root of a particular cubic equation, provided all eigenvalues are positive.
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Acknowledgement
The first two authors wish to express their gratitude to Prof. Rajendra Bhatia for his helpful guidance and suggestions, and also thank Indian Statistical Institute for sponsoring their visit to New Delhi in January 2009, when this work had begun. The authors acknowledge the support of the UGC-SAP.
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Sharma, R., Bhandari, R. & Gupta, M. Inequalities related to the Cauchy-Schwarz inequality. Sankhya A 74, 101–111 (2012). https://doi.org/10.1007/s13171-012-0013-9
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DOI: https://doi.org/10.1007/s13171-012-0013-9