The concept of strictly super-stabilizability for bivariate means has been defined recently by Ra\”{\i}soulli and S\'{a}ndor (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.
Authors
Mira Cristiana Anisiu T. Popoviciu Institute of Numerical Analysis, Romanian Academy, Fantanele 53, Cluj-Napoca, România
Valeriu Anisiu
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
[1] Raïssouli M, Sándor J:On a method of construction of new means with applications.J. Inequal. Appl.2013.,2013:Article ID 89 Google Scholar
[2] Bullen P:Handbook of Means and Their Inequalities. Kluwer Academic, Dordrecht; 2003.BookMATHGoogle Scholar
[3] Anisiu M-C, Anisiu V:Logarithmic mean and weighted sum of geometric and anti-harmonic means.Rev. Anal. Numér. Théor. Approx.2012,41(2):95–98.MathSciNetMATHGoogle Scholar
[4] Raïssouli M:Stability and stabilizability for means.Appl. Math. E-Notes2011,11:159–174.MathSciNetMATHGoogle Scholar
[5] Raïssouli M, Sándor J:Sub-stabilizability and super-stabilizability for bivariate means.J. Inequal. Appl.2014.,2014:Article ID 28 Google Scholar
[6] Chu Y-M, Wang M-K, Wang Z-K:An optimal double inequality between Seiffert and geometric means.J. Appl. Math.2011.,2011:Article ID 26123 Google Scholar
[7] Chu Y-M, Wang M-K, Wang Z-K:A best-possible inequality between Seiffert and harmonic means.J. Inequal. Appl.2011.,2011:Article ID 94 Google Scholar
[8] Gong W-M, Song Y-Q, Wang M-K, Chu Y-M:A sharp double inequality between Seiffert, arithmetic, and geometric means.Abstr. Appl. Anal.2012.,2012:Article ID 684384 Google Scholar
[9] Song Y-Q, Qian W-M, Jiang Y-L, Chu Y-M:Optimal lower generalized logarithmic mean bound for the Seiffert mean.J. Appl. Math.2013.,2013:Article ID 273653 Google Scholar
[10] He Z-Y, Qian W-M, Jiang Y-L, Song Y-Q, Chu Y-M:Bounds for the combinations of Neuman-Sándor, arithmetic, and second Seiffert means in terms of contraharmonic mean.Abstr. Appl. Anal.2013.,2013:Article ID 903982 Google Scholar
[11] Gradshteyn I, Ryzhik I:Table of Integrals, Series and Products. Academic Press, Amsterdam; 2007.
2014-Anisiu-Anisiu-Thefirst
The first Seiffert mean is strictly ( G,AG, A )-super-stabilizable
Mira C Anisiu ^(1){ }^{1} and Valeriu Anisiu ^(2**){ }^{2 *}
*Correspondence: anisiu@math.ubbcluj.ro ^(2){ }^{2} Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 Kogălniceanu, Cluj-Napoca, 400084, România Full list of author information is available at the end of the article
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 PP is strictly (G,A)(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
A bivariate mean is a map m:(0,oo)^(2)rarrRm:(0, \infty)^{2} \rightarrow \mathbb{R} satisfying the following statement:
AA a,b > 0,quad min(a,b) <= m(a,b) <= max(a,b).\forall a, b>0, \quad \min (a, b) \leq m(a, b) \leq \max (a, b) .
Obviously m(a,a)=am(a, a)=a for each a > 0a>0. The maps (a,b)|->min(a,b)(a, b) \mapsto \min (a, b) and (a,b)|->max(a,b)(a, b) \mapsto \max (a, b) are means, and they are called the trivial means.
A mean mm is symmetric if m(a,b)=m(b,a)m(a, b)=m(b, a) for all a,b > 0a, b>0, and monotone if (a,b)|->m(a,b)(a, b) \mapsto m(a, b) is increasing in aa and in bb, that is, if a_(1) <= a_(2)a_{1} \leq a_{2} (respectively b_(1) <= b_(2)b_{1} \leq b_{2} ) then m(a_(1),b) <= m(a_(2),b)m\left(a_{1}, b\right) \leq m\left(a_{2}, b\right) (respectively m(a,b_(1)) <= m(a,b_(2))m\left(a, b_{1}\right) \leq m\left(a, b_{2}\right) ). For more details as regards monotone means, see [1].
For two means m_(1)m_{1} and m_(2)m_{2} we write m_(1) <= m_(2)m_{1} \leq m_{2} if and only if m_(1)(a,b) <= m_(2)(a,b)m_{1}(a, b) \leq m_{2}(a, b) for every a,b > 0a, b>0, and m_(1) < m_(2)m_{1}<m_{2} if and only if m_(1)(a,b) < m_(2)(a,b)m_{1}(a, b)<m_{2}(a, b) for all a,b > 0a, b>0 with a!=ba \neq b. Two means m_(1)m_{1} and m_(2)m_{2} are comparable if m_(1) <= m_(2)m_{1} \leq m_{2} or m_(2) <= m_(1)m_{2} \leq m_{1}, and we say that mm is between two comparable means m_(1)m_{1} and m_(2)m_{2} if min(m_(1),m_(2)) <= m <= max(m_(1),m_(2))\min \left(m_{1}, m_{2}\right) \leq m \leq \max \left(m_{1}, m_{2}\right). If the above inequalities are strict then we say that mm is strictly between m_(1)m_{1} and m_(2)m_{2}.
Some standard examples of means are given in the following (see [2] and the references therein):
and are called the arithmetic, geometric, harmonic, logarithmic, identric means, respectively, the first Seiffert mean.
The above means are strictly comparable, namely
min < H < G < L < P < I < A < max.\min <H<G<L<P<I<A<\max .
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.
In particular, during the proof, we needed the Sturm sequence associated to a univariate polynomial, say pp, in order to find the number of roots in intervals ( c,dc, d ]. This can be obtained in Maple by
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)m_{1}, m_{2}, and m_(3)m_{3} be three given symmetric means. For all a,b > 0a, b>0, define the resultant mean-map of m_(1),m_(2)m_{1}, m_{2}, and m_(3)m_{3} as
Definition 3 A symmetric mean mm is said to be
(a) stable if R(m,m,m)=m\mathcal{R}(m, m, m)=m;
(b) stabilizable if there exist two nontrivial stable means m_(1)m_{1} and m_(2)m_{2} satisfying the relation R(m_(1),m,m_(2))=m\mathcal{R}\left(m_{1}, m, m_{2}\right)=m. We then say that mm is (m_(1),m_(2))\left(m_{1}, m_{2}\right)-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,GA, G, and HH are stable. The logarithmic mean LL is (H,A)(H, A)-stabilizable and (A,G)(A, G)-stabilizable, and the identric mean II is ( G,AG, A )-stabilizable.
The next definitions were formulated in [5].
Definition 4 Let m_(1),m_(2)m_{1}, m_{2} be two nontrivial stable comparable means. A mean mm is called:
(a) ( m_(1),m_(2)m_{1}, m_{2} )-sub-stabilizable if R(m_(1),m,m_(2)) <= m\mathcal{R}\left(m_{1}, m, m_{2}\right) \leq m and mm is between m_(1)m_{1} and m_(2)m_{2};
(b) ( m_(1),m_(2)m_{1}, m_{2} )-super-stabilizable if m <= R(m_(1),m,m_(2))m \leq \mathcal{R}\left(m_{1}, m, m_{2}\right) and mm is between m_(1)m_{1} and m_(2)m_{2}.
This definition extends that of stabilizability, in the sense that a mean mm is ( m_(1),m_(2)m_{1}, m_{2} )stabilizable if and only if (a) and (b) hold.
Definition 5 Let m_(1),m_(2)m_{1}, m_{2} be two nontrivial stable comparable means. A mean mm is called:
(a) strictly ( m_(1),m_(2)m_{1}, m_{2} )-sub-stabilizable if R(m_(1),m,m_(2)) < m\mathcal{R}\left(m_{1}, m, m_{2}\right)<m and mm is strictly between m_(1)m_{1} and m_(2)m_{2};
(b) strictly ( m_(1),m_(2)m_{1}, m_{2} )-super-stabilizable if m < R(m_(1),m,m_(2))m<\mathcal{R}\left(m_{1}, m, m_{2}\right) and mm is strictly between m_(1)m_{1} and m_(2)m_{2}.
Example 6 [5] The geometric mean GG is ( G,AG, A )-super-stabilizable (but not strictly), and AA is ( G,AG, A )-sub-stabilizable.
The logarithmic mean LL is strictly ( G,AG, A )-super-stabilizable and strictly ( A,HA, H )-substabilizable. The identric mean II is strictly (A,G)(A, G)-sub-stabilizable.
It is not known if the first Seiffert mean PP 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 PP is strictly ( A,GA, G )-sub-stabilizable. An open problem was proposed there, namely: prove or disprove that the first Seiffert mean PP is strictly ( G,AG, A )-super-stabilizable.
In what follows we shall prove that indeed the first Seiffert mean PP is strictly (G,A)(G, A) -super-stabilizable.
Main result
It is well known that G < P < AG<P<A and both GG and AA are stable. We have to prove that P < R(G,P,A)P< \mathcal{R}(G, P, A), which, using (1), is equivalent with
for all a,b > 0a, b>0 with a!=ba \neq b. Without restricting the generality, we may consider that b > ab>a and after the substitution t=(b-a)//(b+a)t=(b-a) /(b+a) we reduce the problem to
for all 0 < t < 10<t<1. For r=0.995r=0.995, the inequality (5) is true on [r,1)[r, 1) because the functions alpha,beta\alpha, \beta, gamma\gamma are all increasing and for t >= rt \geq r
Note that a term was added to p_(gamma)p_{\gamma}, because it can be seen that gamma(t) > p_(gamma)(t)\gamma(t)>p_{\gamma}(t) for t > 0t>0 sufficiently small. The coefficient 1//61 / 6 was found using some estimations which are omitted because they are not essential for the proof.
We shall prove that:
(i) beta(t) < p_(beta)(t),0 < t < 1\beta(t)<p_{\beta}(t), 0<t<1;
(ii) gamma(t) < tilde(p)_(gamma)(t),0 < t < r\gamma(t)<\tilde{p}_{\gamma}(t), 0<t<r;
(iii) 4p_(beta)(t) tilde(p)_(gamma)(t) < alpha(t),0 < t < 14 p_{\beta}(t) \tilde{p}_{\gamma}(t)<\alpha(t), 0<t<1.
We denote by f_(1)(t)=p_(beta)(t)-beta(t),f_(2)(t)=f_(1)^(')(t)=p_(beta)^(')(t)-(1)/((2+t)sqrt(1+t))f_{1}(t)=p_{\beta}(t)-\beta(t), f_{2}(t)=f_{1}^{\prime}(t)=p_{\beta}^{\prime}(t)-\frac{1}{(2+t) \sqrt{1+t}}. We substitute t=(1+s)^(2)-1,0 < s < sqrt2-1t=(1+ s)^{2}-1,0<s<\sqrt{2}-1, and we get f_(2)((1+s)^(2)-1)=(s^(19))/(34,359,738,368(2+2s+s^(2))(1+s))F_(2)(s)f_{2}\left((1+s)^{2}-1\right)=\frac{s^{19}}{34,359,738,368\left(2+2 s+s^{2}\right)(1+s)} F_{2}(s), where F_(2)(s)=sum_(n=0)^(20)a_(n)s^(n)F_{2}(s)= \sum_{n=0}^{20} a_{n} s^{n} is a polynomial of 20 th degree whose coefficients are given in Table 1, a_(0),a_(1),a_(2)a_{0}, a_{1}, a_{2} etc. in rows.
It follows that f_(2)(t) > 0f_{2}(t)>0, because F_(2)(s)F_{2}(s) has positive coefficients. Since f_(1)(0)=0f_{1}(0)=0, we have f_(1)(t) > 0f_{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)^(')(t)= tilde(p)_(gamma)^(')(t)-(1)/((t-2)sqrt(1-t))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}}.
We substitute t=1-s^(2),0 < s < 1t=1-s^{2}, 0<s<1, and get g_(2)(1-s^(2))=(1)/(103,079,215,104(1+s^(2))s)G_(2)(s)g_{2}\left(1-s^{2}\right)=\frac{1}{103,079,215,104\left(1+s^{2}\right) s} G_{2}(s), where G_(2)(s)=-103,079,215,104+sum_(n=0)^(19)b_(n)s^(2n+1)G_{2}(s)= -103,079,215,104+\sum_{n=0}^{19} b_{n} s^{2 n+1} is a polynomial of 39 th degree, the coefficients b_(n)b_{n} being given in Table 2, three in a row.
By using the Sturm sequence for the polynomial G_(2)(s)G_{2}(s) as stated at the end of the Introduction, both functions (in Maple and in Maxima) return two roots in ( 0,1 ]. Since G_(2)(1)=1G_{2}(1)=1, we find that G_(2)(s)G_{2}(s) has a unique root in ( 0,1 ). It follows that g_(2)(t)g_{2}(t) has also a unique root t_(1)in(0,1)t_{1} \in(0,1), hence g_(2)(t) > 0g_{2}(t)>0 on (0,t_(1))\left(0, t_{1}\right), and g_(2)(t) < 0g_{2}(t)<0 on (t_(1),1)\left(t_{1}, 1\right). Therefore g_(1)(t) > min(g_(1)(0),g_(1)(r))g_{1}(t)>\min \left(g_{1}(0), g_{1}(r)\right) on (0,r)(0, r). But g_(1)(0)=0g_{1}(0)=0 and
hence g_(1)(t) > min(g_(1)(0),g_(1)(r))=0g_{1}(t)>\min \left(g_{1}(0), g_{1}(r)\right)=0 and (ii) is proved.
We consider now the function h_(1)(t)=alpha(t)-4p_(beta)(t) tilde(p)_(gamma)(t)h_{1}(t)=\alpha(t)-4 p_{\beta}(t) \tilde{p}_{\gamma}(t) and its series expansion h_(1)(t)=t^(6)h_(2)(t)+O(t^(40))h_{1}(t)= t^{6} h_{2}(t)+O\left(t^{40}\right), where
Using the Sturm sequence as before we find that the polynomial h_(2)h_{2} has no roots in ( 0,1 ) and t^(6)h_(2)(t) > 0t^{6} h_{2}(t)>0.
Now we write h_(1)(t)=t^(6)h_(2)(t)+sum_(n=40)^(oo)q_(n)t^(n)h_{1}(t)=t^{6} h_{2}(t)+\sum_{n=40}^{\infty} q_{n} t^{n}. We notice that all the coefficients q_(n),n >= 40q_{n}, n \geq 40 are in fact the coefficients of the series expansion of alpha(t)=(arcsin t)^(2)\alpha(t)=(\arcsin t)^{2}, due to the fact that p_(beta) tilde(p)_(gamma)p_{\beta} \tilde{p}_{\gamma} is polynomial of degree less that 40 . It is known that these are all positive (those of odd order being null), namely
(see [11], p.61).
It follows that h_(1)(t) > 0h_{1}(t)>0, hence (iii) holds.
From (i)-(iii) it follows that (5) holds also on ( 0,r0, r ) and the proof is complete.
Remark 8 We have chosen the form of the polynomials p_(beta),p_(gamma)p_{\beta}, p_{\gamma} and tilde(p)_(gamma)\tilde{p}_{\gamma} dealing with three parameters: the degree of the polynomials m(m=19),r in(0,1)(r=0.995)m(m=19), r \in(0,1)(r=0.995) and the coefficient cc of the supplementary term in tilde(p)_(gamma)(c=1//6)\tilde{p}_{\gamma}(c=1 / 6). The value for rr has been determined so that (6) is true, and the degree of the polynomials not too high (which would happened if we let r=1r=1 ). The parameter rr being fixed, cc was related only to mm, 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 rr, but it seems that the degree cannot be much smaller.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both authors jointly worked, read and approved the final manuscript.
Author details
^(1){ }^{1} T. Popoviciu Institute of Numerical Analysis, Romanian Academy, Fantanele 53, Cluj-Napoca, România. ^(2){ }^{2} Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 Kogălniceanu, Cluj-Napoca, 400084, România.
Acknowledgements
The authors would like to express their gratitude to the referees for the helpful suggestions.
Received: 6 March 2014 Accepted: 29 April 2014 Published: 13 May 2014
References
Raïssouli, M, Sándor, J: On a method of construction of new means with applications. J. Inequal. Appl. 2013, 89 (2013)
Bullen, P: Handbook of Means and Their Inequalities. Kluwer Academic, Dordrecht (2003)
Anisiu, M-C, Anisiu, V: Logarithmic mean and weighted sum of geometric and anti-harmonic means. Rev. Anal. Numér. Théor. Approx. 41(2), 95-98 (2012)
Raïssouli, M: Stability and stabilizability for means. Appl. Math. E-Notes 11, 159-174 (2011)
Raïssouli, M, Sándor, J: Sub-stabilizability and super-stabilizability for bivariate means. J. Inequal. Appl. 2014, 28 (2014)
Chu, Y-M, Wang, M-K, Wang, Z-K: An optimal double inequality between Seiffert and geometric means. J. Appl. Math. 2011, 261237 (2011)
Chu, Y-M, Wang, M-K, Wang, Z-K: A best-possible inequality between Seiffert and harmonic means. J. Inequal. Appl. 2011, 94 (2011)
Gong, W-M, Song, Y-Q, Wang, M-K, Chu, Y-M: A sharp double inequality between Seiffert, arithmetic, and geometric means. Abstr. Appl. Anal. 2012, 684384 (2012)
Song, Y-Q, Qian, W-M, Jiang, Y-L, Chu, Y-M: Optimal lower generalized logarithmic mean bound for the Seiffert mean. J. Appl. Math. 2013, 273653 (2013)
He, Z-Y, Qian, W-M, Jiang, Y-L, Song, Y-Q, Chu, Y-M: Bounds for the combinations of Neuman-Sándor, arithmetic, and second Seiffert means in terms of contraharmonic mean. Abstr. Appl. Anal. 2013, 903982 (2013)
Gradshteyn, I, Ryzhik, I: Table of Integrals, Series and Products. Academic Press, Amsterdam (2007)
10.1186/1029-242X-2014-185
Cite this article as: Anisiu and Anisiu: The first Seiffert mean is strictly ( G,AG, A )-super-stabilizable. Journal of Inequalities and Applications 2014, 2014:185
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