Math. J. Interdiscip. Sci.

On Some Inequalities Involving Harmonic Mean and Moments

Madhu gupta, Rajesh Sharma and S.R. Sharma


Harmonic mean, variance, positive definite matrix, spread.

PUBLISHED DATE September 2017
PUBLISHER The Author(s) 2017. This article is published with open access at

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).

Page(s) 54–65
ISSN Print : 2278-9561, Online : 2278-957X
DOI 10.15415/mjis.2017.61006
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