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# The first Seiffert mean is strictly $(G,A)$-super-stabilizable

- Mira C Anisiu
^{1}and - Valeriu Anisiu
^{2}Email author

**2014**:185

https://doi.org/10.1186/1029-242X-2014-185

© Anisiu and Anisiu; licensee Springer. 2014

**Received:**6 March 2014**Accepted:**29 April 2014**Published:**13 May 2014

## Abstract

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 $(G,A)$-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.

## Keywords

- means
- stable means
- strictly super-stabilizable means

## Introduction

*bivariate mean*is a map $m:{(0,\mathrm{\infty})}^{2}\to \mathbb{R}$ satisfying the following statement:

Obviously $m(a,a)=a$ for each $a>0$. The maps $(a,b)\mapsto min(a,b)$ and $(a,b)\mapsto max$ $(a,b)$ are means, and they are called the *trivial means*.

A mean *m* is *symmetric* if $m(a,b)=m(b,a)$ for all $a,b>0$, and *monotone* if $(a,b)\mapsto m(a,b)$ is increasing in *a* and in *b*, that is, if ${a}_{1}\le {a}_{2}$ (respectively ${b}_{1}\le {b}_{2}$) then $m({a}_{1},b)\le m({a}_{2},b)$ (respectively $m(a,{b}_{1})\le m(a,{b}_{2})$). For more details as regards monotone means, see [1].

For two means ${m}_{1}$ and ${m}_{2}$ we write ${m}_{1}\le {m}_{2}$ if and only if ${m}_{1}(a,b)\le {m}_{2}(a,b)$ for every $a,b>0$, and ${m}_{1}<{m}_{2}$ if and only if ${m}_{1}(a,b)<{m}_{2}(a,b)$ for all $a,b>0$ with $a\ne b$. Two means ${m}_{1}$ and ${m}_{2}$ are *comparable* if ${m}_{1}\le {m}_{2}$ or ${m}_{2}\le {m}_{1}$, and we say that *m* is between two comparable means ${m}_{1}$ and ${m}_{2}$ if $min({m}_{1},{m}_{2})\le m\le max({m}_{1},{m}_{2})$. If the above inequalities are strict then we say that *m* is *strictly between* ${m}_{1}$ and ${m}_{2}$.

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, [3]). 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.

*p*, in order to find the number of roots in intervals $(c,d]$. This can be obtained in

*Maple*by

*Maxima*by

both making use of exact (rational) arithmetic.

## Definitions and preliminary results

At first we define the resultant mean-map of three means as in [4], where the properties of the resultant mean-map are studied.

**Definition 1**Let ${m}_{1}$, ${m}_{2}$, and ${m}_{3}$ be three given symmetric means. For all $a,b>0$, define the

*resultant mean-map*of ${m}_{1}$, ${m}_{2}$, and ${m}_{3}$ as

**Example 2**For ${m}_{1}=G$, ${m}_{3}=A$ and ${m}_{2}=P$ we get

**Definition 3**A symmetric mean

*m*is said to be

- (a)
*stable*if $\mathcal{R}(m,m,m)=m$; - (b)
*stabilizable*if there exist two nontrivial stable means ${m}_{1}$ and ${m}_{2}$ satisfying the relation $\mathcal{R}({m}_{1},m,{m}_{2})=m$. We then say that*m*is $({m}_{1},{m}_{2})$-stabilizable.

A study about the stability and stabilizability of the standard means was presented in [4]. For example, the arithmetic, geometric, and harmonic means *A*, *G*, and *H* are stable. The logarithmic mean *L* is $(H,A)$-stabilizable and $(A,G)$-stabilizable, and the identric mean *I* is $(G,A)$-stabilizable.

The next definitions were formulated in [5].

**Definition 4**Let ${m}_{1}$, ${m}_{2}$ be two nontrivial stable comparable means. A mean

*m*is called:

- (a)
$({m}_{1},{m}_{2})$-

*sub-stabilizable*if $\mathcal{R}({m}_{1},m,{m}_{2})\le m$ and*m*is between ${m}_{1}$ and ${m}_{2}$; - (b)
$({m}_{1},{m}_{2})$-

*super-stabilizable*if $m\le \mathcal{R}({m}_{1},m,{m}_{2})$ and*m*is between ${m}_{1}$ and ${m}_{2}$.

This definition extends that of stabilizability, in the sense that a mean *m* is $({m}_{1},{m}_{2})$-stabilizable if and only if (a) and (b) hold.

**Definition 5**Let ${m}_{1}$, ${m}_{2}$ be two nontrivial stable comparable means. A mean

*m*is called:

- (a)
*strictly*$({m}_{1},{m}_{2})$-*sub-stabilizable*if $\mathcal{R}({m}_{1},m,{m}_{2})<m$ and*m*is strictly between ${m}_{1}$ and ${m}_{2}$; - (b)
*strictly*$({m}_{1},{m}_{2})$-*super-stabilizable*if $m<\mathcal{R}({m}_{1},m,{m}_{2})$ and*m*is strictly between ${m}_{1}$ and ${m}_{2}$.

**Example 6** [5]

The geometric mean *G* is $(G,A)$-super-stabilizable (but not strictly), and *A* is $(G,A)$-sub-stabilizable.

The logarithmic mean *L* is strictly $(G,A)$-super-stabilizable and strictly $(A,H)$-sub-stabilizable. The identric mean *I* is strictly $(A,G)$-sub-stabilizable.

It is not known if the first Seiffert mean *P* is stabilizable or not. Several inequalities related to Seiffert means can be found in [6–10] and the references therein.

In [5] it was proved that the first Seiffert mean *P* is strictly $(A,G)$-sub-stabilizable. An open problem was proposed there, namely: prove or disprove that the first Seiffert mean *P* is strictly $(G,A)$-super-stabilizable.

In what follows we shall prove that indeed the first Seiffert mean *P* is strictly $(G,A)$-super-stabilizable.

## Main result

*G*and

*A*are stable. We have to prove that $P<\mathcal{R}(G,P,A)$, which, using (1), is equivalent with

for all $0<t<1$.

**Theorem 7** *The first Seiffert mean* *P* *is strictly* $(G,A)$-*super*-*stabilizable*.

*Proof*We have to prove that (3) holds for $0<t<1$. To this aim we denote, for $0\le t\le 1$,

*α*,

*β*,

*γ*are all increasing and for $t\ge r$

Note that a term was added to ${p}_{\gamma}$, because it can be seen that $\gamma (t)>{p}_{\gamma}(t)$ for $t>0$ sufficiently small. The coefficient $1/6$ was found using some estimations which are omitted because they are not essential for the proof.

- (i)
$\beta (t)<{p}_{\beta}(t)$, $0<t<1$;

- (ii)
$\gamma (t)<{\tilde{p}}_{\gamma}(t)$, $0<t<r$;

- (iii)
$4{p}_{\beta}(t){\tilde{p}}_{\gamma}(t)<\alpha (t)$, $0<t<1$.

**The coefficients**
${\mathit{a}}_{\mathbf{0}}\mathbf{,}{\mathit{a}}_{\mathbf{1}}\mathbf{,}\mathbf{\dots}\mathbf{,}{\mathit{a}}_{\mathbf{20}}$

4,770,923,133,534,208 | 45,577,737,845,735,424 | 208,413,045,121,875,968 |

606,289,729,726,119,936 | 1,257,880,992,601,669,632 | 1,977,530,020,741,201,920 |

2,443,128,584,682,799,104 | 2,427,671,832,468,406,272 | 1,969,549,944,783,110,144 |

1,316,806,964,788,476,928 | 729,144,476,396,464,128 | 334,812,607,979,053,568 |

127,214,924,071,312,896 | 39,763,039,787,401,392 | 10,120,564,836,367,104 |

2,064,597,817,622,592 | 329,565,434,362,848 | 39,662,337,834,126 |

3,384,693,012,652 | 182,584,573,899 | 4,681,655,741 |

It follows that ${f}_{2}(t)>0$, because ${F}_{2}(s)$ has positive coefficients. Since ${f}_{1}(0)=0$, we have ${f}_{1}(t)>0$ on $(0,1)$ and (i) is proved.

We proceed similarly for ${g}_{1}(t)={\tilde{p}}_{\gamma}(t)-\gamma (t)$, ${g}_{2}(t)={g}_{1}^{\prime}(t)={\tilde{p}}_{\gamma}^{\prime}(t)-\frac{1}{(t-2)\sqrt{1-t}}$.

**The coefficients**
${\mathit{b}}_{\mathbf{0}}\mathbf{,}{\mathit{b}}_{\mathbf{1}}\mathbf{,}\mathbf{\dots}\mathbf{,}{\mathit{b}}_{\mathbf{19}}$

816,042,556,423 | −8,304,263,424,095 | 57,076,977,341,817 |

−261,518,771,366,961 | 849,290,412,683,676 | −2,026,905,939,920,124 |

3,601,042,197,273,780 | −4,689,184,725,197,076 | 4,130,366,508,898,578 |

−1,578,461,017,820,258 | −1,834,983,544,254,146 | 4,260,109,227,548,850 |

−4,661,988,848,655,060 | 3,484,106,242,692,852 | −1,908,076,678,782,012 |

773,661,301,050,972 | −227,689,188,034,785 | 46,209,023,561,673 |

−5,802,339,420,719 | 340,462,481,719 |

*Maple*and in

*Maxima*) return two roots in $(0,1]$. Since ${G}_{2}(1)=1$, we find that ${G}_{2}(s)$ has a unique root in $(0,1)$. It follows that ${g}_{2}(t)$ has also a unique root ${t}_{1}\in (0,1)$, hence ${g}_{2}(t)>0$ on $(0,{t}_{1})$, and ${g}_{2}(t)<0$ on $({t}_{1},1)$. Therefore ${g}_{1}(t)>min({g}_{1}(0),{g}_{1}(r))$ on $(0,r)$. But ${g}_{1}(0)=0$ and

hence ${g}_{1}(t)>min({g}_{1}(0),{g}_{1}(r))=0$ and (ii) is proved.

Using the Sturm sequence as before we find that the polynomial ${h}_{2}$ has no roots in $(0,1)$ and ${t}^{6}{h}_{2}(t)>0$.

(see [11], p.61).

It follows that ${h}_{1}(t)>0$, hence (iii) holds.

From (i)-(iii) it follows that (5) holds also on $(0,r)$ and the proof is complete. □

**Remark 8** We have chosen the form of the polynomials ${p}_{\beta}$, ${p}_{\gamma}$ and ${\tilde{p}}_{\gamma}$ dealing with three parameters: the degree of the polynomials *m* ($m=19$), $r\in (0,1)$ ($r=0.995$) and the coefficient *c* of the supplementary term in ${\tilde{p}}_{\gamma}$ ($c=1/6$). 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 $r=1$). 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.

## Declarations

### Acknowledgements

The authors would like to express their gratitude to the referees for the helpful suggestions.

## Authors’ Affiliations

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