  We study classical weighted $L^p\to L^q$ inequalities for the fractional
maximal operators on $\R^d$, proved originally by Muckenhoupt and Wheeden in
the 70's. We establish a slightly stronger version of this inequality with the
use of a novel extension of Bellman function method.
  More precisely, the estimate is deduced from the existence of a certain
special function which enjoys appropriate majorization and concavity. From this
result and an explicit version of the ``$A_{p-\varepsilon}$ theorem," derived
also with Bellman functions, we obtain the sharp inequality of Lacey, Moen,
P\'erez and Torres.
