Abstract
The concept of strictly super-stabilizability for bivariate means has been defined recently by Raisoulli and Sandor (J. Inequal. Appl. 2014:28, 2014). We answer into affirmative to an open question posed in that paper, namely: Prove or disprove that the first Seiffert mean P is strictly -super-stabilizable. We use series expansions of the functions involved and reduce the main inequality to three auxiliary ones. The computations are performed with the aid of the computer algebra systems Maple and Maxima. The method is general and can be adapted to other problems related to sub- or super-stabilizability. MSC:26E60.
Highlights
A bivariate mean is a map m : (, ∞) → R satisfying the following statement:∀a, b >, min(a, b) ≤ m(a, b) ≤ max(a, b).Obviously m(a, a) = a for each a >
A mean m is symmetric if m(a, b) = m(b, a) for all a, b >, and monotone if (a, b) → m(a, b) is increasing in a and in b, that is, if a ≤ a m(a, b) ≤ m(a, b) (respectively m(a, b ) ≤ m(a, b ))
Two means m and m are comparable if m ≤ m or m ≤ m, and we say that m is between two comparable means m and m if min(m, m ) ≤ m ≤ max(m, m )
Summary
Some standard examples of means are given in the following (see [ ] and the references therein): a+b A := A(a, b) = ; The above means are strictly comparable, namely min < H < G < L < P < I < A < max . We have used Maple and Maxima, which already offered good results in proving inequalities for means (see, for example, [ ]). Definition Let m , m , and m be three given symmetric means.
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