Monotonic Refinements of a Ky Fan Inequality

Abstract

It is well-known that inequalities between means play a very important role in many branches of mathematics. Please refer to [1, 3, 7], etc. The main aims of the present article are: (i) to show that there are monotonic and continuous functions H(t), K(t), P (t) and Q(t) on [0, 1] such that for all t ∈ [0, 1], Hn ≤ H(t) ≤ Gn ≤ K(t) ≤ An and Hn/(1−Hn) ≤ P (t) ≤ Gn/Gn ≤ Q(t) ≤ An/An, where An, Gn and Hn are respectively the weighted arithmetic, geometric and harmonic means of the positive numbers x1, x2, ..., xn in (0, 1/2],with positive weightsα1, α2, ..., αn; while An and G ′ n are respectively the weighted arithmetic and geometric means of the numbers 1− x1, 1− x2, ..., 1− xn with the same positive weights α1, α2, ..., αn; (ii) to present more general monotonic refinements for the Ky Fan inequality as well as some inequalities involving means; and (iii) to present some generalized and new inequalities in this connection.

Cite this paper

@inproceedings{Qi2001MonotonicRO, title={Monotonic Refinements of a Ky Fan Inequality}, author={Feng Qi and K. K. Chong}, year={2001} }