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:^[1]

${\displaystyle {\text{variance}}\leq {\frac {1}{4}}(M-m)^{2}.}$

Sharma et al. have proved an improvement of the Popoviciu's inequality that says that:^[2]

${\displaystyle {{\text{variance}}+\left({\frac {\text{Third central moment}}{\text{2 variance}}}\right)^{2}}\leq {\frac {1}{4}}(M-m)^{2}.}$

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.

References

1. Popoviciu, T. (1935). "Sur les équations algébriques ayant toutes leurs racines réelles". Mathematica (Cluj). 9: 129–145.
2. Sharma, R., Gupta, M., Kapoor, G. (2010). "Some better bound on variance with applications". Journal of mathematical inequalities. 4: 355–363.