  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_{\nu} \propto \nu^{\alpha}) whereas the typical observed value
is \alpha\sim0. Moreover, extreme values for a number of parameters are
required in order that \alpha\sim-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.
