The first Seiffert mean is strictly -super-stabilizable
© Anisiu and Anisiu; licensee Springer. 2014
Received: 6 March 2014
Accepted: 29 April 2014
Published: 13 May 2014
The concept of strictly super-stabilizability for bivariate means has been defined recently by Raïsoulli and Sándor (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.
Obviously for each . The maps and are means, and they are called the trivial means.
A mean m is symmetric if for all , and monotone if is increasing in a and in b, that is, if (respectively ) then (respectively ). For more details as regards monotone means, see .
For two means and we write if and only if for every , and if and only if for all with . Two means and are comparable if or , and we say that m is between two comparable means and if . If the above inequalities are strict then we say that m is strictly between and .
and are called the arithmetic, geometric, harmonic, logarithmic, identric means, respectively, the first Seiffert mean.
The next section presents some definitions and preliminary results, and the last section contains the main result. Its proof is based on some heavy computations, and a computer algebra system may be very helpful.
We have used Maple and Maxima, which already offered good results in proving inequalities for means (see, for example, ). Note that all the symbolic computations are exact, because only polynomials with rational coefficients are involved. We would like to point out that the method used in this paper is easily adaptable to other ‘stiff’ inequalities involving real analytic functions, if they contain subexpressions with algebraic derivatives.
both making use of exact (rational) arithmetic.
Definitions and preliminary results
At first we define the resultant mean-map of three means as in , where the properties of the resultant mean-map are studied.
stable if ;
stabilizable if there exist two nontrivial stable means and satisfying the relation . We then say that m is -stabilizable.
A study about the stability and stabilizability of the standard means was presented in . For example, the arithmetic, geometric, and harmonic means A, G, and H are stable. The logarithmic mean L is -stabilizable and -stabilizable, and the identric mean I is -stabilizable.
The next definitions were formulated in .
-sub-stabilizable if and m is between and ;
-super-stabilizable if and m is between and .
This definition extends that of stabilizability, in the sense that a mean m is -stabilizable if and only if (a) and (b) hold.
strictly -sub-stabilizable if and m is strictly between and ;
strictly -super-stabilizable if and m is strictly between and .
Example 6 
The geometric mean G is -super-stabilizable (but not strictly), and A is -sub-stabilizable.
The logarithmic mean L is strictly -super-stabilizable and strictly -sub-stabilizable. The identric mean I is strictly -sub-stabilizable.
In  it was proved that the first Seiffert mean P is strictly -sub-stabilizable. An open problem was proposed there, namely: prove or disprove that the first Seiffert mean P is strictly -super-stabilizable.
In what follows we shall prove that indeed the first Seiffert mean P is strictly -super-stabilizable.
for all .
Theorem 7 The first Seiffert mean P is strictly -super-stabilizable.
Note that a term was added to , because it can be seen that for sufficiently small. The coefficient was found using some estimations which are omitted because they are not essential for the proof.
It follows that , because has positive coefficients. Since , we have on and (i) is proved.
We proceed similarly for , .
hence and (ii) is proved.
Using the Sturm sequence as before we find that the polynomial has no roots in and .
(see , p.61).
It follows that , hence (iii) holds.
From (i)-(iii) it follows that (5) holds also on and the proof is complete. □
Remark 8 We have chosen the form of the polynomials , and dealing with three parameters: the degree of the polynomials m (), () and the coefficient c of the supplementary term in (). The value for r has been determined so that (6) is true, and the degree of the polynomials not too high (which would happened if we let ). The parameter r being fixed, c was related only to m, in such a way that (ii) to be true. Finally, the choice of the three parameters must make (iii) to be fulfilled.
The tests we performed showed that the degree of the polynomials cannot be less than 19, maybe this can be achieved by modifying slightly r, but it seems that the degree cannot be much smaller.
The authors would like to express their gratitude to the referees for the helpful suggestions.
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