# zbMATH — the first resource for mathematics

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.

##### MSC:
 26E60 Means 26D15 Inequalities for sums, series and integrals
Full Text: