For ismember, have you considered using a hash instead of an array? Then you could do

my $ismember = exists $hash{string};
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If you're going to do a bunch of lookups, you could convert the array to a hash temporarily.

my %hash = map { $_ => $_ } @array;
print($hash{string1}, "\n");
print($hash{string2}, "\n");
print($hash{string3}, "\n");
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The following returns the index or undefined, but create a list as big as the array in memory.

use List::Util qw( first );
my $idx = first { $array[$_] eq "string" } 0..$#array;
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A memory efficient version of the above.

sub ismember {
   my ($array, $search) = @_;
   for my $i (0..$#$array) {
      return $i if $array->[$i] eq $search;
   }
   return ();
}

my $idx = ismember(\@array, "string");
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Update: Applied Not_a_Number's fix. Thanks.