Bhatia—Davis inequality: first recorded occurrence?

Question

The Bhatia–Davis inequality states that, for any random variable $X$ such that $m \leq X \leq M$ a.s., $$ \operatorname{Var}[X] \leq (M-\mathbb{E}[X])(\mathbb{E}[X]-m) $$ This is a strenghtening of another inequality, attributed to Popovicius (1935). However, Bhatia and Davis only published their paper in 2000. Was there no earlier recorded occurrence of this inequality? It seems a little strange for it to have waited 65 years...

Answer 1

As I said in a comment, it is likely that this inequality appeared much earlier than 1999-2000, possibly hidden in proofs of other inequalities like Popoviciu's. I myself obtained it when proving Popoviciu's inequality by naturally expanding $\mathbb{E}[(M-X)(X-m)]$. As a side note, in the following, the first identity yields Bhatia—Davis's inequality, and the second yields Popoviciu's: \begin{align} \color{blue}{\operatorname{Var}[X] + \mathbb{E}[(M-X)(X-m)]} &\color{blue}{= (M-\mathbb{E}[X])(\mathbb{E}[X]-m)} \\ &\color{blue}{= \frac{(M-m)^2}{4} - \left(\mathbb{E}[X] - \frac{M+m}{2}\right)^2.} \end{align}

After a quick search in the literature, I can confirm my above hypothesis. Indeed, Bhatia—Davis's inequality appeared (though not in the exact form) in a paper published by H. E. Guterman in 1962 ("An Upper Bound for the Sample Standard Deviation"), highlighted in the red box below for your convenience:

[Update for clarification: Using the same notation in the question, the inequality in the red box reads $$\mathbb{E}\left[\left(X - \frac{M+m}{2}\right)^2\right] \le \frac{(M-m)^2}{4},$$ which is the inequality in question because \begin{align} \mathrm{LHS} - \mathrm{RHS} &= \mathbb{E}[X^2] - (M+m)E[X] + \frac{(M+m)^2}{4} - \frac{(M-m)^2}{4}\\ &= \mathbb{E}[X^2] - (M+m)E[X] + Mm \\ &= \operatorname{Var}[X] - (M-\mathbb{E}[X])(\mathbb{E}[X] - m). \end{align} ]

As you can see, it is hidden inside a proof of another inequality (which is also Popoviciu's if one looks more closely at the right-hand side in the red box, so my initial hypothesis is perfectly confirmed). Note that it is possible (or even likely) that this inequality was used even before 1962 by other researchers. If I can think of using $\mathbb{E}[(M-X)(X-m)]\ge 0$, so could our giants.