  We consider the random-field O($N$) spin model with long-range exchange
interactions which decay with distance $r$ between spins as $r^{-d-\sigma}$
and/or random fields which correlate with distance $r$ as $r^{-d+\rho}$, and
reexamine the critical phenomena near the lower critical dimension by use of
the perturbative functional renormalization group. We compute the analytic
fixed points in the one-loop beta functions, and study their stability. We also
calculate the critical exponents at the analytical fixed points. We show that
the analytic fixed point which governs the phase transition in the system with
the long-range correlations of random fields can be destabilized by the
nonanalytic perturbation in both cases where the exchange interactions between
spins are short ranged and long ranged. For the system with the long-range
exchange interactions and uncorrelated random fields, we show that the $d\to
d-\sigma$ dimensional reduction at the leading order of the $d-2\sigma$
expansion holds only for $N>2(4+3{\sqrt{3}})\simeq 18.3923\cdots$. Our
investigation into the system with the long-range exchange interactions and
uncorrelated random fields also gives the value of the boundary between
critical behaviors in systems with long-range and short-range exchange
interactions, which is identical to that predicted by Sak [Phys. Rev. B
{\bf{8}}, 281 (1973)]. For the system with the long-range exchange interactions
and the long-range correlated random fields, we show that the $d\to
d-\sigma-\rho$ dimensional reduction does not hold within the present
framework, as far as $N$ is finite.
