  The principal aim of this paper is to extend Birman's sequence of integral
inequalities originally obtained in 1961, and containing Hardy's and Rellich's
inequality as special cases, to a sequence of inequalities that incorporates
power weights on either side and logarithmic refinements on the right-hand side
of the inequality as well. Our new technique of proof for this sequence of
inequalities relies on a combination of transforms originally due to Hartman
and M\"uller-Pfeiffer. The results obtained considerably improve on prior
results in the literature.
