Ok, but what are you inheriting from string? Nothing. Use composition (that is, make a std::string a member of your class rather than inheriting anything).
The simplest solution to the problem your code is having now is to transform the entire string to lowercase and test whether that is a palindrome. Make a private "to lower" function for your class (a couple of lines, really), and call it from your constructor. Otherwise, you're creating a problem for yourself because 'h' is very different from 'H'.