  We show that an estimate by de la Pe\~na, Ibragimov and Jordan for
$\mathbb{E}(X-c)^+$, with $c$ a constant and $X$ a random variable of which the
mean, the variance, and $\mathbb{P}(X \leq c)$ are known, implies an estimate
by Scarf on the infimum of $\mathbb{E}(X \wedge c)$ over the set of positive
random variables $X$ with fixed mean and variance. This also shows, as a
consequence, that the former estimate implies an estimate by Lo on European
option prices.
