  We show that the forward-jet measurements performed at HERA allow for a
detailed study of corrections due to next-to-leading logarithms (NLL) in the
Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach. While the description of the
d\sigma/dx data shows small sensitivity to NLL-BFKL corrections, these can be
tested by the triple differential cross section d\sigma/dxdk_T^2dQ^2 recently
measured. These data can be successfully described using a
renormalization-group improved NLL kernel while the standard
next-to-leading-order QCD or leading-logarithm BFKL approaches fail to describe
the same data in the whole kinematic range. We present a detailed analysis of
the NLL scheme and renormalization-scale dependences and also discuss the
photon impact factors.
