The principal aim of this paper is to extend Birmans sequence of integral inequalities originally obtained in 1961, and containing Hardys and Rellichs 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 Muller-Pfeiffer. The results obtained considerably improve on prior results in the literature.
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