  We study long-range Bernoulli percolation on $\mathbb{Z}^d$ in which each two
vertices $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta
\|x-y\|^{-d-\alpha})$. It is a theorem of Noam Berger (CMP, 2002) that if
$0<\alpha<d$ then there is no infinite cluster at the critical parameter
$\beta_c$. We give a new, quantitative proof of this theorem establishing the
power-law upper bound \[ \mathbf{P}_{\beta_c}\bigl(|K|\geq n\bigr) \leq C
n^{-(d-\alpha)/(2d+\alpha)} \] for every $n\geq 1$, where $K$ is the cluster of
the origin. We believe that this is the first rigorous power-law upper bound
for a Bernoulli percolation model that is neither planar nor expected to
exhibit mean-field critical behaviour.
  As part of the proof, we establish a universal inequality implying that the
maximum size of a cluster in percolation on any finite graph is of the same
order as its mean with high probability. We apply this inequality to derive a
new rigorous hyperscaling inequality $(2-\eta)(\delta+1)\leq d(\delta-1)$
relating the cluster-volume exponent $\delta$ and two-point function exponent
$\eta$.
