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A monotonicity property of ratios of symmetric homogeneous means. (English) Zbl 1026.26025
The author continues his interesting work on strong inequalities and relative metrics [P. A. Hästö, J. Math. Anal. Appl. 274, No. 1, 38–58 (2002; Zbl 1019.54011)]. Given two symmetric homogeneous increasing means M(x,y),N(x,y) then MN when the function ρ(x)=M(x,1)/N(x,1),x1, is increasing. This is a strong inequality between the means and implies the inequalities NMCN where C=lim x ρ(x); both inequalities being sharp. In particular the author proves that for the extended means of E. B. Leach and M. C. Sholander [Am. Math. Mon. 85, 84–90 (1978; Zbl 0379.26012); ibid. 656 (1978; Zbl 0389.26008); J. Math. Anal. Appl. 92, 207–223 (1983; Zbl 0517.26007)], E s,t E p,q if and only if s+tp+q and min{s,t}min{p,q}; this is a strong version of a result of A. O. Pittenger [Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. fiz. 678–715, 15–18 (1980; Zbl 0469.26009)]. From this the following inequalities can be deduced: E s,t E p,q (q/p) 1/(p-q) (s/t) 1/(s-t) E s,t under the conditions: s>t, p>q, p+qs+t, tq. Strong inequalities are also found for the Gini means, arithmetic, geometric and logarithmic means, and a certain mean due to Seiffert. These results are used to introduce several new relative metrics.
MSC:
26E60Means
26D15Inequalities for sums, series and integrals of real functions