The harmonic, geometric, and arithmetic means are the special cases t=-1, 0, +1, respectively. We have 9)_t(a) = 1/19t(1/a), and, for positive constants k, 9t(ka) =kWxt(a). As is well known, 9St(a) is a continuous strictly increasing function of t -oo 1, the sign of equality holding for + oo > t> 1 if and only if (a) and (b) are proportional; the inequality sign in (2)is reversed for t < 1.
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E. F. Beckenbach (1950) studied this question.