Let $X\sim beta(\alpha,\beta)$ be a random variable and let $\tau\in(0,1)$.


Are there any known closed-form bounds (I'm specifically interested in lower bounds) on
$$
\mathbb E[X\ | X\le \tau]?
$$



Are there known bounds on the expectation of the truncated Beta distribution?
Mathematica 12.0 is your friend, answering it by



```
Mean[TruncatedDistribution[{-Infinity,\[Tau]},BetaDistribution[\[Alpha], \[Beta]]]] 

```

Unfortunately the $\LaTeX$ form of the result
$$\frac{
\begin{array}{cc}
 \{ & 
\begin{array}{cc}
 \frac{\alpha }{\alpha +\beta } & \tau \geq 1 \\
 \frac{B\_{\tau }(\alpha +1,\beta )}{B(\alpha ,\beta )} & 0<\tau <1 \\
\end{array}
 \\
\end{array}
}{
\begin{array}{cc}
 \{ & 
\begin{array}{cc}
 I\_{\tau }(\alpha ,\beta ) & 0<\tau <1 \\
 1 & \tau \geq 1 \\
\end{array}
 \\
\end{array}
} ,$$
 where $B(\alpha ,\beta )$ and $ B\_\tau (\alpha ,\beta ) $ and $I\_\tau ( \alpha ,\beta ) $ are [the beta function and incomplete beta function and regularized incomplete beta function](https://en.wikipedia.org/wiki/Beta_function),
 leaves much to be desired. 
Next, the command



```
Normal[Series[%, {\[Tau], 0, 2}, Assumptions -> \[Tau] > 0]]

```

$$ \frac{\alpha \tau }{\alpha +1}-\frac{\alpha (\beta -1) \tau ^2}{(\alpha +1)^2 (\alpha +2)}$$
performs its asymptotics as $\tau\to 0,\, \tau >0$.


Many thanks from me to @Matt F for his valuable comment.


