Bhatia–Davis inequality

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

Then the Bhatia–Davis inequality states:

${\displaystyle \sigma ^{2}\leq (M-\mu )(\mu -m).\,}$

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

Since ${\displaystyle m\leq A\leq M}$,

${\displaystyle 0\leq \mathbb {E} [(M-A)(A-m)]=-\mathbb {E} [A^{2}]-mM+(m+M)\mu }$.

Thus,

${\displaystyle \sigma ^{2}=\mathbb {E} [A^{2}]-\mu ^{2}\leq -mM+(m+M)\mu -\mu ^{2}=(M-\mu )(\mu -m)}$.

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

${\displaystyle \Phi (A^{2})-(\Phi A)^{2}\leq (M-\Phi A)(\Phi A-m)}$.

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

${\displaystyle P(X=x_{i})=p_{i},}$ where ${\displaystyle i=1,...,n}$, then:

${\displaystyle s_{p}^{2}=\sum _{1}^{n}p_{i}x_{i}^{2}-(\sum _{1}^{n}p_{i}x_{i})^{2}\leq (M-\sum _{1}^{n}p_{i}x_{i})(\sum _{1}^{n}p_{i}x_{i}-m)}$,

where ${\displaystyle 0\leq p_{i}\leq 1}$ and ${\displaystyle \sum _{1}^{n}p_{i}=1}$.

The Bhatia–Davis inequality is stronger than Popoviciu's inequality on variances (note, however, that Popoviciu's inequality does not require knowledge of the expectation or mean), as can be seen from the conditions for equality. Equality holds in Popoviciu's inequality if and only if half of the a_j are equal to the upper bounds and half of the a_j are equal to the lower bounds. Additionally, Sharma[2] has made further refinements on the Bhatia–Davis inequality.

• Cramér–Rao bound
• Chapman–Robbins bound
• Popoviciu's inequality on variances