%hash = map
  +(++$i % 2 ? lc($_) : $_),
  (foo => 'BAR', This => 'thAT', THESE => 'Those');

for (keys %hash) { print "$_ => $hash{$_}\n" }

__END__
this => thAT
these => Those
foo => BAR

##</code><code>##

%hash = map
  # the + before the parenthesis was a way of
  # disambiguating to Perl that the (...) is not the list
  # of arguments I'm passing to map().  but I don't need it

  ++$i % 2 ?  # first increment $i...
    lc($_) :  # if $i % 2 is 1, then lowercase the string
    $_        # otherwise, keep it as it is
  ,

  # then our hash initialization list as normal...
  (
    foo => 'BAR',
    This => 'thAT',
    THESE => 'Those',
  );