Popoviciu's inequality on variances

In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance of any bounded probability distribution. Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution. Then Popoviciu's inequality states:

$\text{variance} \le \frac14 (M - m)^2.$

Sharma et al. have proved an improvement of the Popoviciu's inequality that says that:

${\text {variance} + ( \frac \text {Third central moment} \text{2 variance} )^2} \le \frac14 (M - m)^2.$

Equality holds precisely when half of the probability is concentrated at each of the two bounds.

Popoviciu's inequality is weaker than the Bhatia–Davis inequality.