We study the transverse momentum of the leading jet in the limit where the jet radius is small, R 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 p veto T . 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 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 α 2 s ln 2 R 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. 1 Factorization formulae where a different number of jet functions appear at different orders in perturbation theory have also been obtained in refs. [23,24].2 Indeed, we extract dσpp→Hij by expanding eq. (2.2) to NLO, and subtracting the contribution involving the NLO jet function. This is discussed in detail in section 4.1.