On Some Inequalities Involving Harmonic Mean and Moments

Authors

  • Madhu gupta Department of Mathematics, Chitkara University, Solan, India
  • Rajesh Sharma Department of Mathematics, H.P.U., Shimla, India
  • S.R. Sharma Department of Mathematics, Chitkara University, Solan, India

DOI:

https://doi.org/10.15415/mjis.2017.61006

Keywords:

Harmonic mean, variance, positive definite matrix, spread

Abstract

We derive bounds on the second order moment of a random variable in terms of its arithmetic and harmonic means. Both discrete and continuous cases are considered and it is shown that the present bounds provide refinements of the bounds which exist in literature. As an application we obtain a lower bound for the spread of a positive definite matrix A in terms of traces of A, A-1 and A2. Our results compare favourably with those obtained by Wolkowicz and Styan (Bounds for eigenvalues using traces, Lin. Alg. Appl. 29, 471-506, 1980).

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References

Kantorovich, L.V., Functional analysis and applied mathematics [in Russian],Uspekki Matematicheskikh Nauk 3, 89–185, (1948).

Goldman, A. J., A generalization of Rennie’s inequality J. Res. Nat. Bur.Standards Sect. B, 68 B, 59–63 (1964).

Sharma, R., Some more inequalities for arithmetic mean, harmonic mean and variance, Math. Inequal. Appl., 2, 109–114, (2008).

Krasnosel’skii, M. A. and Kreïn, S. G., An iteration process with minimal residuals [in Russian], Mat. Sbornik N.S. 31 (73), 315–334, (1952).

Merikoski, J.K., On the Trace and Sum of elements of a Matrix, Linear Algebra and Its Applications, 60: 177–185, (1984).

Wolkowicz, H. and Styan, P.H., Bounds for Eigenvalues Using Traces, Linear Algebra and Its Applications, 29: 471–506, (1980).

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Published

2017-09-01

How to Cite

Madhu gupta, Rajesh Sharma, and S.R. Sharma. 2017. “On Some Inequalities Involving Harmonic Mean and Moments”. Mathematical Journal of Interdisciplinary Sciences 6 (1):54-65. https://doi.org/10.15415/mjis.2017.61006.

Issue

Vol. 6 No. 1 (2017)

Section

Articles

License

 

Articles in Mathematical Journal of Interdisciplinary Sciences (Math. J. Interdiscip. Sci.) by Chitkara University Publications are Open Access articles that are published with licensed under a Creative Commons Attribution- CC-BY 4.0 International License. Based on a work at https://mjis.chitkara.edu.in. This license permits one to use, remix, tweak and reproduction in any medium, even commercially provided one give credit for the original creation.

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Creative Commons License Mathematical Journal of Interdisciplinary Sciences by Chitkara University Publications is licensed under a Creative Commons Attribution 4.0 International License. Based on a work at https://mjis.chitkara.edu.in
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