  For a log Fano manifold (X, D) with D\neq 0 and with the log Fano pseudoindex
\geq 2, we prove that the restriction homomorphism Pic(X)\to Pic(D_1) of Picard
groups is injective for any irreducible component D_1\subset D.The strategy of
our proof is to run a certain minimal model program and is similar to the
argument of Casagrande's one. As a corollary, we prove that the Mukai
conjecture (resp. the generalized Mukai conjecture) implies the log Mukai
conjecture (resp. the log generalized Mukai conjecture).
