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26 D. M˘arghidanu

p 1 ,p 2 ,...,p n
The last two inequalities from enounce, those regarding the expression M , are
a 1 ,a 2 ,...,a n
1
obtained from the first two inequalities by the substitutions a k → , k = 1, n.
a k
As a direct consequence of Proposition 2, we derive the following result.

Corollary 1. Both expressions M p 1 ,p 2 ,...,p n and M p 1 ,p 2 ,...,p n can be expressed as weighted means
a 1 ,a 2 ,...,a n a 1 ,a 2 ,...,a n
of the numbers a 1 , a 2 , . . . , a n > 0.

References

[1] P.S. Bullen, D.S. Mitrinovi´c, P.M. Vasi´c, Means and Their Inequalities, D. Reidel Publishing
Company, Dordrecht/Boston, 1988.

[2] D. M˘arghidanu, Two Trigonometrical Means that Produce Continuous Refinement of the
Inequality of the Classic Means, OCTOGON Mathematical Magazine, Vol. 12 (2004), no.
2.A., pp. 664–667.

[3] D. M˘arghidanu, Proposed problem, Mathematical Inequalities, https://www.facebook.com
/photo/?fbid=8414222551970112&set=gm.3795219314099435&idorvanity=1486244404
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