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New means and its properties. (English) Zbl 1218.26034
Summary: The object of this paper is to introduce three new means on the basis of proportions and their dual forms, and to study certain properties, monotonicities and new inequalities involving them. Further, we define three new weighted means and their dual forms. Their properties are stated and there are deduced the ten neo-Pythagorean means, weighted oscillatory mean, weighted $$r$$th oscillatory mean, and also some familiar means and various other means.
##### MSC:
 26E60 Means 26D10 Inequalities involving derivatives and differential and integral operators 26D15 Inequalities for sums, series and integrals