use Fcntl ':flock';
    ....
    # Code done in every child.
    open my $fh => ">> /path/to/log/file" or die "Failed to open: $!";
    flock $fh => LOCK_EX or die "Failed to lock: $!";
    print $fh $your_text;
    close $fh or die "Failed to close: $!";