  In a previous publication, we have established a collinearly-improved version
of the Balitsky-Kovchegov (BK) equation, which resums to all orders the
radiative corrections enhanced by large double transverse logarithms. Here, we
study the relevance of this equation as a tool for phenomenology, by
confronting it to the HERA data. To that aim, we first improve the perturbative
accuracy of our resummation, by including two classes of single-logarithmic
corrections: those generated by the first non-singular terms in the DGLAP
splitting functions and those expressing the one-loop running of the QCD
coupling. The equation thus obtained includes all the next-to-leading order
corrections to the BK equation which are enhanced by (single or double)
collinear logarithms. We then use numerical solutions to this equation to fit
the HERA data for the electron-proton reduced cross-section at small Bjorken x.
We obtain good quality fits for physically acceptable initial conditions. Our
best fit, which shows a good stability up to virtualities as large as Q^2=400
GeV^2 for the exchanged photon, uses as an initial condition the
running-coupling version of the McLerran-Venugopalan model, with the QCD
coupling running according to the smallest dipole prescription.
