  The convex transform order is one way to make precise comparison between the
skewness of probability distributions on the real line. We establish a simple
and complete characterisation of when one Beta distribution is smaller than
another according to the convex transform order. As an application, we derive
monotonicity properties for the probability of Beta distributed random
variables exceeding the mean or mode of their distribution. Moreover, we obtain
a simple alternative proof of the mode-median-mean inequality for unimodal
distributions that are skewed in a sense made precise by the convex transform
order. This new proof also gives an analogous inequality for the anti-mode of
distributions that have a unique anti-mode. Such inequalities for Beta
distributions follow as special cases. Finally, some consequences for the
values of distribution functions of Binomial distributions near to their means
are mentioned.
