ABSTRACT
Cˇakalov’s Inequality If a is an increasing positive non-constant ntuple, w a positive n-tuple, n > 2, then
λn { An(a;w)−Gn(a;w)
} ≥ λn−1{An−1(a;w)−Gn−1(a;w)}, where
λn = W 2n
(Wn−1 − w1) .
Comments (i) The proof, due to Cˇakalov, is the same as that of the more general result in the reference. (ii) Since it is easily seen that λ˜n = λn−1/λn > Wn−1/Wn, this inequal-
ity generalizes Rado’s Geometric-Arithmetic Mean Inequality Extension for this class of sequences; there is no analogous extension of Popoviciu’s Geometric-Arithmetic Mean Inequality Extension. (iii) In general λ˜n is an unattained lower bound, unlike the lower bound
Wn/Wn−1 in Rado’s geometric-arithmetic mean inequality extension. However λ˜n is best possible in that for any λ˜
′ n > λ˜n, n = 1, 2, . . . there are sequences a
for which the inequality would fail. (iv) The geometric mean can be replaced by a large class of quasi-arithmetic
M-means; those for which M−1 is increasing, convex, and 3-convex; for the definition of these terms see Quasi-arithmetic Mean Inequalities, n-Convex Sequence Inequalities.
