  We show that if $\beta>1$ is a rational number and the Julia set $J$ of the
holomorphic correspondence $z^{\beta}+c$ is a locally eventually onto
hyperbolic repeller, then the Hausdorff dimension of $J$ is bounded from above
by the zero of the associated pressure function. As a consequence, we conclude
that the Julia set of the correspondence has zero Lebesgue measure for
parameters close to zero, whenever $q^2<p$ and $\beta=p/q$ in lowest terms.
