  Within the framework of the dipole factorisation, we use a recent
collinearly-improved version of the Balitsky-Kovchegov equation to fit the HERA
data for inclusive deep inelastic scattering at small Bjorken $x$. The equation
includes an all-order resummation of double and single transverse logarithms
and running coupling corrections. Compared to similar equations previously
proposed in the literature, this work makes a direct use of Bjorken $x$ as the
rapidity scale for the evolution variable. We obtain excellent fits for
reasonable values for the four fit parameters. We find that the fit quality
improves when including resummation effects and a physically-motivated initial
condition. In particular, the resummation of the DGLAP-like single transverse
logarithms has a sizeable impact and allows one to extend the fit up to
relatively large photon virtuality $Q^2$.
