Integral Hardy inequalities, their generalizations and related inequalities

Hardy inequalities have numerous applications in mathematical physics and spectral theory of unbounded operators. In this paper we describe direct generalizations of integral Hardy inequalities, their improvements and analogues. We systemize the relations between various interpretations of these inequalities and describe new one-dimensional integral inequalities. We show that these known and new inequalities are valid also for complex-valued functions.
We consider in details integral inequalities of Hardy, Rellich and Birman type for functions defined on bounded intervals. In particular, we provide the proofs for the generalizations and improvements of Birman integral inequalities for higher derivatives. We briefly discuss multidimensional analogues involving integrals of the powers of the modulus of the gradient of a function or of a polyharmonic operator.