I'd build a "reverse hash", but since the values in the original hash will be repeated, I'd use arrays for the values in the new hash:

use Data::Dumper;

my %hash = (
    a => 'f',
    b => 'g',
    c => 'f',
    d => 'h',
    e => 'f',
);

my %dups;
while (my ($e, $g) = each %hash) {
    push @{$dups{$g}}, $e
}

while (my ($g, $e) = each %dups) {
    say "$g is duplicated in @$e"
      if @$e > 1;
}

print Dumper \%dups;
[download]

You seemed unsure as whether you'd produced a hash in your OP. Data::Dumper¹ is an excellent tool for inspecting arbitary data structures. Used as above it shows:

$VAR1 = {
          'h' => [
                   'd'
                 ],
          'g' => [
                   'b'
                 ],
          'f' => [
                   'e',
                   'c',
                   'a'
                 ]
        };
[download]

[1] Actuallly, I prefer Data::Dump but it isn't a core module.