ABSTRACT
Comments (i) (a), known as Abel’s lemma, is a simple consequence of Abel’s transformation or summation formula, an “integration by parts formula” for sequences:
n∑
wiai =Wnan +
n−1∑
Wi∆ai. (2)
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(ii) If a is non-negative and decreasing then we have the following result, simpler than (a), known as Abel’s inequality:
a1 min 1≤i≤n
Wi ≤ n∑ i=1
wiai ≤ a1 max 1≤i≤n
Wi. (3)
Extensions (a) [Bromwich] If w, a are n-tuples with a decreasing, and if
W k = max{Wi, 1 ≤ i ≤ k − 1}, W ′k = max{Wi, k ≤ i ≤ n}, rW k = min{Wi, 1 ≤ i ≤ k − 1}, W ′k = min{Wi, k ≤ i ≤ n},
for 1 ≤ k ≤ n then
W k(a1 − ak) +W ′kak ≤ n∑ i=1
wiai ≤W k(a1 − ak) +W ′kak. (4)
(b) [Redheffer] (i) Under the hypotheses of (b) above
An A1
≤ ( 1 +
n− 1 n∑ i=2
ai Ai−1
)n−1 ,
with equality if and only if for some λ > 0 Ai = λ i−1A1, 1 ≤ i ≤ n.
