  We consider the dynamics of expanding semigroups generated by finitely many
rational maps on the Riemann sphere. We show that for an analytic family of
such semigroups, the Bowen parameter function is real-analytic and
plurisubharmonic. Combining this with a result obtained by the first author, we
show that if for each semigroup of such an analytic family of expanding
semigroups satisfies the open set condition, then the function of the Hausdorff
dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we
provide an extensive collection of classes of examples of analytic families of
semigroups satisfying all the above conditions and we analyze in detail the
corresponding Bowen's parameters and Hausdorff dimension function.
