$n\geq 4$ , then $\{aaaa,aaaaa,aaaaaa,aaaaaaa,.....\} = aaaa\{ \epsilon,a,aa,aaa,.....\} =aaaaa^*$
$m\leq 3$, then $\{\epsilon,b,bb,bbb\}$
$L= \{a^nb^m \;|\;n\geq 4,m\leq 3\}$
Regular expression will be $aaaaa^*(\epsilon +b+bb+bbb)$
DFA for above regular expression will be
DFA for complement of L , i.e, L' will be
will have regular expression
$(\epsilon+a+aa+aaa)(\epsilon+b(a+b)^*) +aaaaa^*(b+bb)a(a+b)^*+aaaaa^*bbb(a+b)(a+b)^*$
or $(\epsilon+a+aa+aaa)(\epsilon+b(a+b)^*) +aaaaa^*(b+bb+bbb)a(a+b)^*+aaaaa^*bbbb(a+b)^*$