Open Access

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

Journal of Inequalities and Applications20142014:185

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

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

A bivariate mean is a map m : ( 0 , ) 2 R satisfying the following statement:
a , b > 0 , min ( a , b ) m ( a , b ) max ( a , b ) .

Obviously m ( a , a ) = a for each a > 0 . The maps ( a , b ) min ( a , b ) and ( a , b ) 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 ) m ( a , b ) is increasing in a and in b, that is, if a 1 a 2 (respectively b 1 b 2 ) then m ( a 1 , b ) m ( a 2 , b ) (respectively m ( a , b 1 ) 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 m 2 if and only if m 1 ( a , b ) 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 b . Two means m 1 and m 2 are comparable if m 1 m 2 or m 2 m 1 , and we say that m is between two comparable means m 1 and m 2 if min ( m 1 , m 2 ) m max ( m 1 , m 2 ) . If the above inequalities are strict then we say that m is strictly between m 1 and m 2 .

Some standard examples of means are given in the following (see [2] and the references therein):
A : = A ( a , b ) = a + b 2 ; G : = G ( a , b ) = a b ; H : = H ( a , b ) = 2 a b a + b ; L : = L ( a , b ) = b a ln b ln a , L ( a , a ) = a ; I : = I ( a , b ) = e 1 ( b b a a ) 1 / ( b a ) , I ( a , a ) = a ; P : = P ( a , b ) = b a 4 arctan b / a π = b a 2 arcsin b a b + a , P ( a , a ) = a ,

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 .

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 p, in order to find the number of roots in intervals ( c , d ] . This can be obtained in Maple by
sturm ( sturmseq ( p , c , d ) )
or in Maxima by
nroot ( p , c , d ) ,

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
R ( m 1 , m 2 , m 3 ) ( a , b ) = m 1 ( m 2 ( a , m 3 ( a , b ) ) , m 2 ( m 3 ( a , b ) , b ) ) .
Example 2 For m 1 = G , m 3 = A and m 2 = P we get
R ( G , P , A ) ( a , b ) = | b a | 4 ( arcsin b a 3 a + b arcsin b a a + 3 b ) 1 / 2 .
(1)
Definition 3 A symmetric mean m is said to be
  1. (a)

    stable if R ( m , m , m ) = m ;

     
  2. (b)

    stabilizable if there exist two nontrivial stable means m 1 and m 2 satisfying the relation 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:
  1. (a)

    ( m 1 , m 2 ) -sub-stabilizable if R ( m 1 , m , m 2 ) m and m is between m 1 and m 2 ;

     
  2. (b)

    ( m 1 , m 2 ) -super-stabilizable if m 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:
  1. (a)

    strictly ( m 1 , m 2 ) -sub-stabilizable if R ( m 1 , m , m 2 ) < m and m is strictly between m 1 and m 2 ;

     
  2. (b)

    strictly ( m 1 , m 2 ) -super-stabilizable if m < 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 [610] 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

It is well known that G < P < A and both G and A are stable. We have to prove that P < R ( G , P , A ) , which, using (1), is equivalent with
b a 2 arcsin b a b + a < | b a | 4 ( arcsin b a 3 a + b arcsin b a a + 3 b ) 1 / 2 ,
or
4 arcsin b a 3 a + b arcsin b a a + 3 b < ( arcsin b a b + a ) 2 ,
(2)
for all a , b > 0 with a b . Without restricting the generality, we may consider that b > a and after the substitution t = ( b a ) / ( b + a ) we reduce the problem to
( arcsin t ) 2 > 4 arcsin t 2 t arcsin t 2 + t
(3)

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 t 1 ,
α ( t ) = ( arcsin t ) 2 , β ( t ) = arcsin t 2 + t , γ ( t ) = arcsin t 2 t ,
(4)
and we shall prove that
α ( t ) > 4 β ( t ) γ ( t )
(5)
for all 0 < t < 1 . For r = 0.995 , the inequality (5) is true on [ r , 1 ) because the functions α, β, γ are all increasing and for t r
α ( t ) 4 β ( t ) γ ( t ) α ( r ) 4 β ( 1 ) γ ( 1 ) = 0.027 > 0 .
(6)
In order to prove that (5) is true also on ( 0 , r ) we shall use series expansions up to 20th degree. We obtain
β ( t ) = p β ( t ) + O ( t 20 ) , γ ( t ) = p γ ( t ) + O ( t 20 ) ,
where
p β ( t ) = 1 2 t 1 4 t 2 + 7 48 t 3 3 32 t 4 + 83 1 , 280 t 5 73 1 , 536 t 6 + 523 14 , 336 t 7 119 4 , 096 t 8 + 14 , 051 589 , 824 t 9 13 , 103 655 , 360 t 10 + 98 , 601 5 , 767 , 168 t 11 15 , 565 1 , 048 , 576 t 12 + 1 , 423 , 159 109 , 051 , 904 t 13 1 , 361 , 617 117 , 440 , 512 t 14 + 10 , 461 , 043 1 , 006 , 632 , 960 t 15 1 , 259 , 743 134 , 217 , 728 t 16 + 623 , 034 , 403 73 , 014 , 444 , 032 t 17 603 , 217 , 979 77 , 309 , 411 , 328 t 18 + 4 , 681 , 655 , 741 652 , 835 , 028 , 992 t 19 ; p γ ( t ) = 1 2 t + 1 4 t 2 + 7 48 t 3 + 3 32 t 4 + 83 1 , 280 t 5 + 73 1 , 536 t 6 + 523 14 , 336 t 7 + 119 4 , 096 t 8 + 14 , 051 589 , 824 t 9 + 13 , 103 655 , 360 t 10 + 98 , 601 5 , 767 , 168 t 11 + 15 , 565 1 , 048 , 576 t 12 + 1 , 423 , 159 109 , 051 , 904 t 13 + 1 , 361 , 617 117 , 440 , 512 t 14 + 10 , 461 , 043 1 , 006 , 632 , 960 t 15 + 1 , 259 , 743 134 , 217 , 728 t 16 + 623 , 034 , 403 73 , 014 , 444 , 032 t 17 + 603 , 217 , 979 77 , 309 , 411 , 328 t 18 + 4 , 681 , 655 , 741 652 , 835 , 028 , 992 t 19 .
We shall use a slightly modified polynomial p ˜ γ given by
p ˜ γ ( t ) = p γ ( t ) + ( 1 / 6 ) t 19 .

Note that a term was added to p γ , because it can be seen that γ ( t ) > p γ ( 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.

We shall prove that:
  1. (i)

    β ( t ) < p β ( t ) , 0 < t < 1 ;

     
  2. (ii)

    γ ( t ) < p ˜ γ ( t ) , 0 < t < r ;

     
  3. (iii)

    4 p β ( t ) p ˜ γ ( t ) < α ( t ) , 0 < t < 1 .

     
We denote by f 1 ( t ) = p β ( t ) β ( t ) , f 2 ( t ) = f 1 ( t ) = p β ( t ) 1 ( 2 + t ) 1 + t . We substitute t = ( 1 + s ) 2 1 , 0 < s < 2 1 , and we get f 2 ( ( 1 + s ) 2 1 ) = s 19 34 , 359 , 738 , 368 ( 2 + 2 s + s 2 ) ( 1 + s ) F 2 ( s ) , where F 2 ( s ) = n = 0 20 a n s n is a polynomial of 20th degree whose coefficients are given in Table 1, a 0 , a 1 , a 2 etc. in rows.
Table 1

The coefficients a 0 , a 1 , , a 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 ) = p ˜ γ ( t ) γ ( t ) , g 2 ( t ) = g 1 ( t ) = p ˜ γ ( t ) 1 ( t 2 ) 1 t .

We substitute t = 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 ) , where G 2 ( s ) = 103 , 079 , 215 , 104 + n = 0 19 b n s 2 n + 1 is a polynomial of 39th degree, the coefficients b n being given in Table 2, three in a row.
Table 2

The coefficients b 0 , b 1 , , b 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

 
By using the Sturm sequence for the polynomial 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 ) = 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 ( 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
g 1 ( r ) = 7 , 545 , 035 , 064 , 001 , 924 , 896 , 095 , 274 , 314 , 681 , 659 , 544 , 598 , 682 , 938 , 990 , 892 , 623 , 399 , 131 5 , 242 , 022 , 432 , 353 , 119 , 161 , 548 , 800 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 arcsin ( 199 201 ) = 0.00972 > 0 ,

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

We consider now the function h 1 ( t ) = α ( t ) 4 p β ( t ) p ˜ γ ( t ) and its series expansion h 1 ( t ) = t 6 h 2 ( t ) + O ( t 40 ) , where
h 2 ( t ) = 1 / 48 + 11 480 t 2 + 547 26 , 880 t 4 + 199 11 , 520 t 6 + 207 , 401 14 , 192 , 640 t 8 + 920 , 021 73 , 801 , 728 t 10 + 95 , 240 , 443 8 , 856 , 207 , 360 t 12 8 , 128 , 804 , 483 25 , 092 , 587 , 520 t 14 + 1 / 6 t 15 155 , 266 , 321 , 334 , 306 , 053 1 , 999 , 672 , 863 , 904 , 235 , 520 t 16 + 1 / 16 t 17 2 , 624 , 741 , 122 , 147 , 393 , 885 110 , 381 , 942 , 087 , 513 , 800 , 704 t 18 + 73 2 , 304 t 19 374 , 493 , 942 , 658 , 190 , 094 , 451 58 , 870 , 369 , 113 , 340 , 693 , 708 , 800 t 20 + 119 6 , 144 t 21 + 55 , 315 , 568 , 609 , 924 , 639 , 053 117 , 740 , 738 , 226 , 681 , 387 , 417 , 600 t 22 + 13 , 103 983 , 040 t 23 + 34 , 201 , 104 , 415 , 289 , 943 , 432 , 424 , 777 9 , 833 , 706 , 456 , 692 , 429 , 477 , 117 , 952 , 000 t 24 + 15 , 565 1 , 572 , 864 t 25 + 38 , 113 , 578 , 427 , 551 , 231 , 317 , 881 7 , 820 , 295 , 659 , 001 , 835 , 851 , 612 , 160 t 26 + 1 , 361 , 617 176 , 160 , 768 t 27 + 50 , 378 , 210 , 487 , 327 , 721 , 746 , 099 , 089 9 , 147 , 580 , 300 , 695 , 808 , 976 , 458 , 088 , 448 t 28 + 1 , 259 , 743 201 , 326 , 592 t 29 + 82 , 855 , 982 , 945 , 562 , 549 , 731 , 871 , 465 , 739 14 , 407 , 438 , 973 , 595 , 899 , 137 , 921 , 489 , 305 , 600 t 30 + 603 , 217 , 979 115 , 964 , 116 , 992 t 31 + 47 , 775 , 326 , 983 , 451 , 755 , 017 , 721 , 677 , 497 8 , 258 , 730 , 450 , 127 , 109 , 454 , 998 , 326 , 476 , 800 t 32 .

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 .

Now we write h 1 ( t ) = t 6 h 2 ( t ) + n = 40 q n t n . We notice that all the coefficients q n , n 40 are in fact the coefficients of the series expansion of α ( t ) = ( arcsin t ) 2 , due to the fact that p β p ˜ γ is polynomial of degree less that 40. It is known that these are all positive (those of odd order being null), namely
q 2 k = 2 2 k 2 ( ( k 1 ) ! ) 2 ( 2 k 1 ) ! k

(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 β , p γ and p ˜ γ dealing with three parameters: the degree of the polynomials m ( m = 19 ), r ( 0 , 1 ) ( r = 0.995 ) and the coefficient c of the supplementary term in p ˜ γ ( 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

(1)
T. Popoviciu Institute of Numerical Analysis, Romanian Academy
(2)
Faculty of Mathematics and Computer Science, Babeş-Bolyai University

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© Anisiu and Anisiu; licensee Springer. 2014

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