Hostname redirect to Subdirectory URL

Yes you can do it with iRules using HTTP::redirect For https connections, you will need to off load the SSL certificate to F5 so that the F5 can read the decrypted traffic

e.g.

 config

root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm virtual bar80
ltm virtual bar80 {
    destination 172.28.24.10:80
    ip-protocol tcp
    mask 255.255.255.255
    pool foo
    profiles {
        http { }
        tcp { }
    }
    rules {
        myrule
    }
    source 0.0.0.0/0
    source-address-translation {
        type automap
    }
    vs-index 12
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm virtual bar443
ltm virtual bar443 {
    destination 172.28.24.10:443
    ip-protocol tcp
    mask 255.255.255.255
    pool foo
    profiles {
        clientssl {
            context clientside
        }
        http { }
        tcp { }
    }
    rules {
        myrule
    }
    source 0.0.0.0/0
    source-address-translation {
        type automap
    }
    vs-index 13
}
root@(ve11a)(cfg-sync In Sync)(Active)(/Common)(tmos) list ltm rule myrule
ltm rule myrule {
    when HTTP_REQUEST {
  if { [HTTP::host] eq "example.com" and [HTTP::uri] eq "/" } {
    HTTP::redirect "https://[HTTP::host]/welcome/welcome.htm"
  }
}
}

 test

[root@ve11a:Active:In Sync] config  curl -I http://172.28.24.10/ -H "Host: example.com"
HTTP/1.0 302 Found
Location: https://example.com/welcome/welcome.htm
Server: BigIP
Connection: Keep-Alive
Content-Length: 0

[root@ve11a:Active:In Sync] config  curl -Ik https://172.28.24.10/ -H "Host: example.com"
HTTP/1.0 302 Found
Location: https://example.com/welcome/welcome.htm
Server: BigIP
Connection: Keep-Alive
Content-Length: 0