Bhatia–Davis inequality

In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance σ2 of any bounded probability distribution on the real line.

Statement

Let m and M be the lower and upper bounds, respectively, for a set of real numbers a1, ..., an , with a particular probability distribution. Let μ be the expected value of this distribution.

Then the Bhatia–Davis inequality states:

Equality holds if and only if every aj in the set of values is equal either to M or to m[1].

Comparisons to Other Inequalities

The Bhatia–Davis inequality is stronger than Popoviciu's inequality on variances as can be seen from the conditions for equality. Additionally, Sharma[2] has made further refinements on the Bhatia–Davis inequality.

See also

References
  1. Bhatia, Rajendra; Davis, Chandler (2000). "A Better Bound on the Variance". The American Mathematical Monthly. 107 (4): 353–357. doi:10.1080/00029890.2000.12005203. ISSN 0002-9890.
  2. Sharma, Rajesh (2008). "Some more inequalities for arithmetic mean, harmonic mean and variance". Journal of Mathematical Inequalities (1): 109–114. doi:10.7153/jmi-02-11. ISSN 1846-579X.
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