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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 [{\it 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 $M\succeq N$ when the function $ \rho(x)=M(x,1)/N(x,1), x\ge 1, $ is increasing. This is a strong inequality between the means and implies the inequalities $N\le M\le CN$ where $C= \lim_{x\to \infty} \rho(x)$; both inequalities being sharp. In particular the author proves that for the extended means of {\it E. B. Leach} and {\it 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)], $E_{s,t}\succeq E_{p,q}$ if and only if $s+t\ge p+q$ and $\min\{s,t\}\ge \min\{p,q\}$; this is a strong version of a result of {\it 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: $E_{s,t}\le E_{p,q}\le (q/p)^{1/(p-q)}(s/t)^{1/(s-t)}E_{s,t}$ under the conditions: $s>t$, $p>q$, $p+q\ge s+t$, $t\ge q$. 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.

26D15Inequalities for sums, series and integrals of real functions
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