Passing from regular variation of a function to regular variation of its integral transform of Mellin-convolution form with kernel is an Abelian problem; its converse, under suitable Tauberian conditions, is a Tauberian one. In either case, one has a comparison statement that the ratio of and tends to a constant at infinity. Passing from a comparison statement to a regular-variation statement is a Mercerian problem. The prototype results here are the Drasin-Shea theorem (for non-negative ) and Jordan's theorem (for which may change sign). We free Jordan's theorem from its non-essential technical conditions which reduce its applicability. Our proof is simpler than the counter-parts of the previous results and does not even use the Pólya Peak Theorem which has been so essential before. The usefulness of the extension is highlighted by an application to Hankel transforms.
Nicholas H. BINGHAM. Akihiko INOUE. "Extension of the Drasin-Shea-Jordan theorem." J. Math. Soc. Japan 52 (3) 545 - 559, July, 2000. https://doi.org/10.2969/jmsj/05230545