  We prove that unicritical polynomials $f(z)=z^d+c$ which are semihyperbolic,
i.e., for which the critical point $0$ is a non-recurrent point in the Julia
set, are uniformly expanding on the Julia set with respect to the metric
$\rho(z) |dz|$, where $\rho(z) = 1+\frac{1}{\textrm{dist}(z,P(f))^{1-1/d}}$,
and where $P(f)$ is the postcritical set of $f$. We also show that this metric
is H\"older equivalent to the usual Euclidean metric.
