TY - JOUR
AU - Eggleston, H. G.
AB - By H. G. EGGLESTON [Received 10 October 1949.—Read 17 November 1949] Introduction TH E lemma from which this paper takes its title is stated as Theorem 1. I t is the complex-variable analogue of the well-known real-variable result, </(*) = o(l) and x*f"(x) = 0(1) imply xf'(x) = 0(1)', where the variable x tends to infinity.* This lemma has also been considered by W. W. Rogosinski (11) and has been generalized by him in a number of ways. The generalizations con- sidered here are of a different type although the principal tool (Theorem 1) is very similar to Rogosinski's Theorem 2 (op. cit., p. 101). The main object of this paper is to show that several well-known results can be derived from Theorem 1 and to obtain refinements of these results by using generalizations of Theorem 1. This theorem is stated with respect to a general domain of regularity and boundedness of/(z); special cases of the theorem that arise in specific domains are stated as Theorems 2, 3, and 4. In these particular domains it is possible to establish modified forms of the lemma, similar to the real- variable lemma used previously by the author (5). These are stated 
TI - A Tauberian Lemma
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s3-1.1.28
DA - 1951-01-01
UR - https://www.deepdyve.com/lp/wiley/a-tauberian-lemma-amrsIPKQFX
SP - 28
EP - 45
VL - s3-1
IS - 1
DP - DeepDyve
ER -