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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 \(M\succeq N\) when the function \( \rho(x)=M(x,1)/N(x,1), x\geq 1, \) is increasing. This is a strong inequality between the means and implies the inequalities \(N\leq M\leq CN\) where \(C= \lim_{x\to \infty} \rho(x)\); 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)], \(E_{s,t}\succeq E_{p,q}\) if and only if \(s+t\geq p+q\) and \(\min\{s,t\}\geq \min\{p,q\}\); 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: \(E_{s,t}\leq E_{p,q}\leq (q/p)^{1/(p-q)}(s/t)^{1/(s-t)}E_{s,t}\) under the conditions: \(s>t\), \(p>q\), \(p+q\geq s+t\), \(t\geq q\). 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.

26E60 Means
26D15 Inequalities for sums, series and integrals
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