  We consider the dynamics of rational semigroups (semigroups of rational maps)
on the Riemann sphere. We estimate the Bowen parameters (zeros of the pressure
functions) and the Hausdorff dimensions of the Julia sets of expanding finitely
generated rational semigroups. We show that the Bowen parameter is larger than
or equal to the ratio of the entropy of the skew product map $F$ and the
Lyapunov exponent of $F$ with respect to the maximal entropy measure for $F$.
Moreover, we show that the equality holds if and only if the generators $f_{j}$
are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a linear
fractional transformation. Furthermore, we show that there are plenty of
expanding finitely generated rational semigroups such that the Bowen parameter
is strictly larger than two.
