Multiwavelength variability of blazars offers indirect insight into their powerful engines and on the mechanisms through which energy is propagated from the centre down the jet. The BL Lac object Mkn 421 is a TeV emitter, a bright blazar at all wavelengths, and therefore an excellent target for variability studies. Mkn 421 was observed by INTEGRAL and Fermi-LAT in an active state on 16-21 April 2013. Well sampled optical, soft, and hard X-ray light curves show the presence of two flares. The average flux in the 20-100 keV range is 9.1e-11 erg/s/cm2 (~4.5 mCrab) and the nuclear average apparent magnitude, corrected for Galactic extinction, is V ~12.2. In the time-resolved X-ray spectra (3.5-60 keV), which are described by broken power laws and, marginally better, by log-parabolic laws, we see a hardening that correlates with flux increase, as expected in refreshed energy injections in a population of electrons that later cool via synchrotron radiation. The hardness ratios between the JEM-X fluxes in two different bands and between the JEM-X and IBIS/ISGRI fluxes confirm this trend. During the observation, the variability level increases monotonically from the optical to the hard X-rays, while the large LAT errors do not allow a significant assessment of the MeV-GeV variability. The cross-correlation analysis during the onset of the most prominent flare suggests a monotonically increasing delay of the lower frequency emission with respect to that at higher frequency, with a maximum time-lag of about 70 minutes, that is however not well constrained. The spectral energy distributions from the optical to the TeV domain are satisfactorily described by homogeneous models of blazar emission based on synchrotron radiation and synchrotron self-Compton scattering, except in the state corresponding to the LAT softest spectrum and highest flux.
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