Abstract
We present some matrix versions of covariance, harmonic mean and other inequalities for positive definite random matrices and present some open problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ando T (1979) Concavity of certain maps on positive matrices and applications to hadamard products. Linear Alg Appl 26:203–241
Ando T (1983) On the arithmetic-geometric-harmonic mean inequalities for positive definite matrices. Linear Alg Appl 52(53):31–37
Bajraktarevic M (1958) Sur une equation fonctionnelle aux valeurs moyennes. Period Math-Phys Astron II Ser 13:243–247
Bobrovsky BZ, Mayer-Wolf E, Zakai M (1987) Some classes of global cramer-rao bounds. Ann Stat 15:1401–1420
Davis C (1963) Notions generalizing convexity for functions defined on spaces of matrices. In: Proceeding of Symposium in Pure Mathematics, vol 8, Convexity, pp 187–201
Dey A, Hande S, Tiku ML (1994) Statistical proofs of some matrix results. Linear Multilinear Algebra 38:109–116
de Finneti B (1931) Sul concetto di media. G 1st Ital Attuari 2:369–396
Gibilisco P (2022) About the jensen inequality for numerical \(n\)-means. Int J Modern Phys A. https://doi.org/10.1142/S0217751X22430102
Gibilisco P, Hensen S (2017) An inequality for expectation of means of positive random variables. Ann Funct Anal 8:142–151
Ibragimov IA, Hasminskii RZ (1981) Statistical estimation : asymptotic theory. Springer
Kimeldorf G, Sampson A (1973) A class of covariance inequalities. J Am Statist Assoc 68:228–230
Kolmogorov A (1930) Sur la notion de la moyenne. Atti Acad Naz Lincei, Rend, VI Ser 12:388–391
Mitra SK (1973) Statistical proofs of some propositions on nonnegative definite matrices. Bull Inter Statist Inst XLV, pp 206–211
Nagumo M (1931) On mean values. Tokyo Baturigakto-zassi 40:520–527
Petz D, Temesi R (2005) Means of positive numbers. SIAM J Matrix Anal Appl 27:712–720
Prakasa Rao BLS (1990) On a matrix version of a covariance inequality. Gujarat Stat Rev 17A:150–159
Prakasa Rao BLS (2000) Inequality for random matrices with an application to statistical inference. Student 3:198–202
Rao CR (1996) Seven inequalities in statistical estimation theory. Student 1:149–158
Rao CR (2000) Statistical proofs of some matrix inequalities. Linear Algebra Appl 321:307–320
Kubo F, Ando T (1979) Means of positive linear operators. Math Ann 246:205–224
Hardy GH, Littlewood J, Polya G (1952) Inequalities, 2nd ed. Cambridge University Press, Cambridge
Acknowledgements
This work was supported under the scheme “INSA Honorary scientist” at the CR Rao Advanced Institute of Mathematics, Statistics and Computer Science, Hyderabad, India.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Prakasa Rao, B.L.S. (2023). On Some Matrix Versions of Covariance, Harmonic Mean and Other Inequalities: An Overview. In: Bapat, R.B., Karantha, M.P., Kirkland, S.J., Neogy, S.K., Pati, S., Puntanen, S. (eds) Applied Linear Algebra, Probability and Statistics. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-99-2310-6_1
Download citation
DOI: https://doi.org/10.1007/978-981-99-2310-6_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-99-2309-0
Online ISBN: 978-981-99-2310-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)