Synchrotron radiation mechanism, when electrons are accelerated in a relativistic shock, is known to have serious problems to explain the observed gamma-ray spectrum below the peak for most Gamma-Ray Bursts (GRBs); the synchrotron spectrum below the
peak is much softer than observed spectra. Recently, the possibility that electrons responsible for the radiation cool via Inverse Compton, but in the Klein-Nishina regime, has been proposed as a solution to this problem. We provide an analytical study of this effect and show that it leads to a hardening of the low energy spectrum but not by enough to make it consistent with the observed spectra for most GRBs (this is assuming that electrons are injected continuously over a time scale comparable to the dynamical time scale, as is expected for internal shocks of GRBs). In particular, we find that it is not possible to obtain a spectrum with alpha>-0.1 (f_{ u} propto u^{alpha}) whereas the typical observed value is alphasim0. Moreover, extreme values for a number of parameters are required in order that alphasim-0.1: the energy fraction in magnetic field needs to be less than about 10^{-4}, the thermal Lorentz factor of electrons should be larger than 10^6, and the radius where gamma-rays are produced should be not too far away from the deceleration radius. These difficulties suggest that the synchrotron radiation mechanism in internal shocks does not provide a self-consistent solution when alpha>-0.2.
Photons of energy larger than 100 MeV from long-GRBs arrive a few seconds after <10 MeV photons do. We show that this delay is a natural consequence of a magnetic dominated relativistic jet. The much slower acceleration of a magnetic jet with radius
(compared with a hot baryonic outflow) results in high energy gamma-ray photons to be converted to electron-positron pairs out to a larger radius whereas lower energy gamma-rays of energy less than ~10 MeV can escape when the jet crosses the Thomson-photosphere. The resulting delay for the arrival of high energy photons is found to be similar to the value observed by the Fermi satellite for a number of GRBs. A prediction of this model is that the delay should increase with photon energy (E) as E^{0.17} for E>100 MeV. The delay depends almost linearly on burst redshift, and on the distance from the central compact object where the jet is launched (R_0). Therefore, the delay in arrival of >10^2 MeV photons can be used to estimate burst redshift if the magnetic jet model for gamma-ray generation is correct and R_0 is roughly the same for long-GRBs.
