  Models of inflationary magnetogenesis with a coupling to the electromagnetic
action of the form $f^2 F_{\mu\nu}F^{\mu\nu}$, are known to suffer from several
problems. These include the strong coupling problem, the back reaction problem
and also strong constraints due to Schwinger effect. We propose a model which
resolves all these issues. In our model, the coupling function, $f$, grows
during inflation and transits to a decaying phase post inflation. This
evolutionary behaviour is chosen so as to avoid the problem of strong coupling.
By assuming a suitable power law form of the coupling function, we can also
neglect back reaction effects during inflation. To avoid back reaction
post-inflation, we find that the reheating temperature is restricted to be
below $ \approx 1.7 \times 10^4$ GeV. The magnetic energy spectrum is predicted
to be non-helical and generically blue. The estimated present day magnetic
field strength and the corresponding coherence length taking reheating at the
QCD epoch(150 MeV) are $ 1.4 \times 10^{-12}$ G and $6.1 \times 10^{-4}$ Mpc,
respectively. This is obtained after taking account of nonlinear processing
over and above the flux freezing evolution after reheating. If we consider also
the possibility of a non-helical inverse cascade, as indicated in direct
numerical simulations, the coherence length and the magnetic field strength are
even larger. In all cases mentioned above, the magnetic fields generated in our
models satisfy the $\gamma$-ray bound below a certain reheating temperature.
