User:J47211/Bhatia–Davis inequality

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:


 * $$ \sigma^2 \le (M - \mu)(\mu - m). \, $$

Equality holds if and only if every aj in the set of values is equal either to M or to m with probability one.

Extensions of the Bhatia–Davis inequality
If $$\Phi$$ is a positive and unital linear mapping of a C* -algebra $$\mathcal{A}$$ into a C* -algebra $$\mathcal{B}$$, and A is a self-adjoint element of $$\mathcal{A}$$ satisfying m $$\leq$$ A $$\leq$$ M, then:

$$\Phi (A^2)-(\Phi A)^2\leq (M-\Phi A)(\Phi A - M)$$.

If $$\mathit{X}$$ is a discrete random variable such that

$$P (X=x_i)=p_i,$$ where $$i = 1, ..., n$$, then:

$$s_p^2=\sum_{1}^n p_ix_i^2-(\sum_{1}^n p_ix_i)^2\leq(M-\sum_{1}^n p_ix_i)(\sum_{1}^n p_ix_i-m)$$,

where $$0\leq p_i \leq1$$ and $$\sum_{1}^n p_i=1$$.

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. Equality holds in Popoviciu's inequality if and only if half of the aj are equal to the upper bounds and half of the aj are are equal to the lower bounds. Additionally, Sharma has made further refinements on the Bhatia–Davis inequality.