On Some Inequalities Involving Three or More Means

Abstract

We investigate some results about mean-inequalities involving a large number of bivariate means. As application, we derive a lot of inequalities between four or more means among the standard means known in the literature.

A note on some inequalities involving the two Raissouli means

“…The previous means are mutually comparable, except U and A which are not comparable, see [7]. The two chains of inequalities H < G < L < P < A < M < T and L < V < P < U < M are well-known in the literature, see also [5,7,6]. A large number of papers concerning recent developments about the means P, T, M, U and V and their optimal bounds in terms of the other (power) standard means can be found in the literature, see for instance [13,14,15] and the related references cited therein.…”

“…As pointed out in [5], (2.4) is strict even if (2.3) is an equality, provided that m 1 = m 2 . A generalization of this result to three or more means is retrieved in [7].…”

“…Two different means m 1 and m 2 are called comparable if m 1 < m 2 or m 2 < m 1 holds, where we write m 1 < m 2 for meaning that m 1 (a, b) < m 2 (a, b) for all a, b > 0 with a = b. The previous means are mutually comparable, except U and A which are not comparable, see [7]. The two chains of inequalities H < G < L < P < A < M < T and L < V < P < U < M are well-known in the literature, see also [5,7,6].…”

A Theoretical Consideration on the Estimation of Interphase Poisson’s Ratio for Fibrous Polymeric Composites

“…Apparently, such a consideration is rather unrealistic. However, a reasonable conjecture for the calculation of coefficient is to focus on (30) or (35) which yields the Poisson's ratio in a closed form and try to search for the critical points = by means of other types of means as discussed in [23] since it has been proved that meantheory is very useful both from theoretical point of view and for practical (engineering) purposes. Yet, such an approach cannot include the case when < r f or > r and then one may examine the interval [0, r m ].…”

On a metric topology on the set of bivariate means

“…Another example of classical means is the Heron mean He t defined by, [8], He t = tG+(1−t)A with t ∈ [0, 1]. Clearly, He 0 = A and He 1 = G. For further examples of means, we refer the reader to [4,3,9] for instance and the related references cited therein. Otherwise, it is not hard to check that the following relationship…”