  Radiative energy losses are very important in regulating the cosmic ray
electron and/or positron (CRE) spectrum during their propagation in the Milky
Way. Particularly, the Klein-Nishina (KN) effect of the inverse Compton
scattering (ICS) results in less efficient energy losses of high-energy
electrons, which is expected to leave imprints on the propagated electron
spectrum. It has been proposed that the hardening of CRE spectra around 50 GeV
observed by Fermi-LAT, AMS-02, and DAMPE could be due to the KN effect. We show
in this work that the transition from the Thomson regime to the KN regime of
the ICS is actually quite smooth compared with the approximate treatment
adopted in some previous works. As a result, the observed spectral hardening of
CREs cannot be explained by the KN effect. It means that an additional
hardening of the primary electrons spectrum is needed. We also provide a
parameterized form for the accurate calculation of the ICS energy-loss rate in
a wide energy range.
