  We study the transverse momentum of the leading jet in the limit where the
jet radius is small, $R\ll 1$. We introduce the leading-jet function to
calculate this cross section for an inclusive jet sample, and the
subleading-jet function when a loose veto on additional jets is imposed, i.e.
$p_{T,J} \gtrsim p_T^{\rm veto}$. These jet functions are calculated at
next-to-leading order in QCD and the resummation of jet radius logarithms is
explored. We present phenomenological results for Higgs + 1 jet production, for
both the jet and Higgs transverse momentum distribution. We find that, while
the $R \ll 1$ limit of the cross section provides a good description of the
full NLO result, even for values as large as $R=0.8$, simply retaining the
leading logarithm at this order does not. Indeed, the NLO contribution to the
hard function and, to a lesser extent, non-logarithmic corrections to the jet
function are sizable and must be included to obtain the correct cross section.
In the inclusive cross section we find that the $\alpha_s^2 \ln^2R$ corrections
are several precent, while in exclusive cross sections at large $p_{T,J}$ and
small $R$ they can reach 20%. However, it is not clear how important the
resummation of these logarithms is, given the presence of other large
corrections at NNLO.
