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**Kathuria L ^{*} and Raka M**

Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, India

- *Corresponding Author:
- Kathuria L

Centre for Advanced Study in Mathematics

Panjab University, Chandigarh-160014

India

**Tel:**08754216121

**E-mail:**[email protected]

**Received Date:** February 10, 2015; **Accepted**** Date:** March 23, 2015; **Published Date:** April 15, 2015

**Citation:** Kathuria L, Raka M (2015) Refined Estimates on Conjectures of Woods and Minkowski-I. J Appl Computat Math 4:209. doi: 10.4172/2168-9679.1000209

**Copyright:** © 2015 Kathuria L, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

Lattice; Covering; Non-homogeneous; Product of linear forms; Critical determinant; Korkine and Zolotare reduction; Hermite's constant; Centre density

Let ,be n real linear forms in n variables x1; : : : ; xn and having determinant The following conjecture is attributed to H. Minkowski:

Conjecture I: For any given real numbers c1; : : : ; cn, there exists integers x1; : : : ; xn such that

(1.1)

Equality is necessary if and only if after a suitable unimodular transformation the linear forms L_{i} have the form* 2c _{i}x_{l}*for

This result is known to be true for For a detailed history and the related results,

Minkowski's Conjecture is equivalent to saying that [1]

where is given by

Chebotarev proved the weaker inequality

(1.2)

Since then several authors have tried to improve upon this estimate. The bounds have been obtained in the form

(1.3)

where Clearly by considering the linear forms Li=xi and for During 1949-1986, many authors such as Davenport, Woods, Bombieri, Gruber, Skubenko, Andrijasjan, Il'in and Malyshev obtained V_{n} for large n. obtained for all n [2-4] improved Mordell's estimates for Hans-Gill et al. [12,14] got improvements on the results of [5-8] for Since recently V_{n} 9=2^{9/2} has been established by the authors [9], we study V_{n} for in a series of three papers.

In this paper we obtain improved estimates on Minkowski's Conjecture for n=10; 11 and 12. In next papers [10-12], we shall derive improved estimates on Minkowski's Conjecture for n=13; 14; 15 and for respectively [13-16]. For sake of comparison, we give results by our improved Vn in **Table 1**.

Estimates by Mordell | Estimates by Il’in | Estimates by Hans-Gill et al | Our improved Estimates | |
---|---|---|---|---|

n | V_{n} |
V_{n} |
V_{n} |
V_{n} |

10 | 2.899061 | 3.47989 | 24.3627506 | 27.60348 |

11 | 2.973102 | 3.52291 | 29.2801145 | 33.47272 |

12 | 3.040525 | 3.55024 | 32.2801213 | 39.59199 |

13 | 3.102356 | 3.57856 | 34.8475153 | 45.40041 |

14 | 3.159373 | 3.60209 | 37.8038391 | 51.26239 |

15 | 3.21218 | 3.61116 | 40.905198 | 57.00375 |

16 | 3.261252 | 3.61908 | 44.3414913 | 57.4702 |

17 | 3.306972 | 3.63924 | 47.2339309 | 57.67598 |

18 | 3.349652 | 3.66176 | 46.7645724 | 57.38876 |

19 | 3.389556 | 3.66734 | 47.2575897 | 60.09339 |

20 | 3.426907 | 3.67236 | 46.8640155 | 58.48592 |

21 | 3.461897 | 3.67692 | 46.0522028 | 56.42571 |

22 | 3.494699 | 3.68408 | 43.6612034 | 53.94142 |

23 | 3.525464 | 3.68633 | 37.8802374 | 50.98842 |

24 | 3.55433 | 3.68978 | 32.5852958 | 47.74632 |

25 | 3.581421 | 3.69295 | 27.8149432 | 42.39088 |

26 | 3.606852 | 3.69589 | 23.0801951 | 38.8657 |

27 | 3.630729 | 3.70012 | 17.3895105 | 31.93316 |

28 | 3.653149 | 3.70263 | 12.9938763 | 26.10663 |

29 | 3.674203 | 3.70497 | 9.5796191 | 19.96254 |

30 | 3.693976 | 3.70867 | 6.7664335 | 16.06884 |

31 | 3.712547 | 3.72558 | 4.745972 | 11.23872 |

32 | 3.729989 | 8.325879 | ||

33 | 3.746371 | 5.411488 |

**Table 1:** The weaker result.

We shall follow the Remak-Davenport approach. For the sake of convenience of the reader we give some basic results of this approach. Minkowski's Conjecture can be restated in the terminology of lattices as : Any lattice ^ of determinant d(^) in Rn is a covering lattice for the set

The weaker result (1.3) is equivalent to saying that any lattice ^ of determinant d(^) in Rn is a covering lattice for the set

Define the homogeneous minimum of ^ as

**Proposition 1. **Suppose that Minkowski Conjecture has been proved for dimensions 1, 2,…., n - 1: Then it holds for all lattices ^ in Rn for which MH(^)=0.

**Proposition 2**. If ^ is a lattice in Rn for with with MH(^) ≠ 0 then there exists an ellipsoid having n linearly independent points of ^ on its boundary and no point of ^ other than O in its interior.

It is well known that using these results, Minkowski's Conjecture would follow from

Conjecture II. If ^ is a lattice in Rn of determinant 1 and there is a sphere |X| <R which contains no point of ^ other than O in its interior and has n linearly independent points of ^ on its boundary then ^ is a covering lattice for the closed sphere of radius Equivalently, every closed sphere of radius lying in Rn contains a point of ^.

They formulated a conjecture from which Conjecture-II follows immediately. To state Woods' conjecture, we need to introduce some terminology [17,18].

Let L be a lattice in Rn. By the reduction theory of quadratic forms introduced by a cartesian co-ordinate system may be chosen in Rn in such a way that L has a basis of the form [19-22],

where A1;A2; : : : ;An are all positive and further for each i=1; 2; : : : ; n any two points of the lattice in R^{n-i+1} with basis

are at a distance atleast Ai apart. Such a basis of L is called a reduced basis [23].

**Conjecture III (Woods)**: If A_{1}A_{2}…A_{n}=1 and for each i then any closed sphere in Rn of radius contains a point of L.

Woods [10] proved this conjecture for Hans-Gill et al. [12] gave a unified proof of Woods' Conjecture for Hans-Gill et al. [12,14] proved Woods' Conjecture for n=7 and n=8 and thus completed the proof of Minkowski's Conjecture for n=7 and 8 Woods [10,24] proved Conjecture and hence Minkowski's Conjecture for n=9. With the assumptions as in Conjecture III, a weaker result would be that

If any closed sphere in Rn of radius contains a point of L [25,26].

Hans-Gill et al. [12,14] obtained the estimates w_{n} on Woods' Conjecture for As w_{9}=9 has been established by the authors [17] recently, we study w_{n} for in a series of three papers. In this paper we obtain improved estimates wn on Woods' Conjecture for n=10; 11 and 12. In next papers [18,19], we shall derive improved estimates w_{n}on Woods' Conjecture for n=13; 14; 15 and for respectively. Together with the following result of Hans-Gill et al. [12], we get improvements of w_{n} for

**Proposition 3.** Let L be a lattice in R_{n} with A_{1}A_{2}…A_{n}=1 and for each i. Let where l_{n} and m_{n} are real numbers. Then L is a covering lattice for the sphere where Wn is defined inductively by

Here we prove

Theorem 1. Let n=10; 11; 12. If d(L)=A1 : : :An=1 and for i=2;….; n, then any closed sphere in Rn of radius contains a point of L, where

The earlier best known values were w_{10}=10:5605061, w_{11}=11:9061976 and w_{12}=13:4499927.

To deduce the results on the estimates of Minkowski's Conjecture we also need the following generalization of Proposition 1

**Proposition 4. **Suppose that we know

Let v_{n}<min V_{k1 }V_{k2}…V_{ks }, where the minimum is taken over all (k_{1}; k_{2}; ; k_{s}) such that n=k_{1}+k_{2}+ : : :+k_{s}, k_{i} positive integers for all i and . Then for all lattices in Rn with homogeneous minimum MH(<)=0, the estimate V_{n} holds for Minkowski's Conjecture.

Since by arithmetic-geometric inequality the sphere is a subset of Propositions 2 and 4 immediately imply

**Theorem 2:** The values of Vn for the estimates of Minkowski's Conjecture can be taken as

For these values are listed in **Table 1**. In Section 2 we state some preliminary results and in Sections 3-5 we prove Theorem 1 for n=10; 11 and 12.

Let L be a lattice in Rn reduced in the sense of Korkine and Zolotare. Let (Sn) denotes the critical determinant of the unit sphere D Sn with centre O in R^{n} i.e.

has no point other than O in the interior of S_{n}}

Let be the Hermite's constant i.e. is the smallest real number such that for any positive de nite quadratic form Q in n variables of determinant D, there exist integers u_{1}; u_{2};…; u_{n} not all zero satisfying

It is well known that We write A^{2}B_{i}=Bi.

We state below some preliminary lemmas. Lemmas 1 and 2 are due to Woods [25], Lemma 3 is due to Korkine and Zolotare [21] and Lemma 4 is due to Pendavingh and Van Zwam [24]. In Lemma 5, the cases n=2 and 3 are classical results of Lagrange and Gauss; n=4 and 5 are due to Korkine and Zolotare [21] while n=6; 7 and 8 are due to Blichfeldt [3].

**Lemma 1**. If then any closed sphere of radius

in R^{n} contains a point of L.

**Lemma 2.** For a Fixed integer i with denote by L_{1} the lattice in Ri with reduced basis

and denote by L2 the lattice in R^{n-i} with reduced basis

If any closed sphere in R_{i} of radius r1 contains a point of L_{1} and if any closed sphere in R_{n-i }of radius r_{2} contains a point of L_{2} then any closed sphere in R_{n} of radius contains a point of L:

**Lemma 3.** For all relevant i,

(2.1)

**Lemma 4.** For all relevant i,

(2.2)

**Throughout the paper we shall denote 0.46873 by ε .**

**Lemma 5.** for n=2; 3; 4; 5; 6; 7 and 8 respectively:

**Lemma 6. **For any integer s;

and (2.4)

This is Lemma 4 of Hans-Gill et al. [12].

**Lemma 7.**

(2.5)

This is Lemma 6 of Hans-Gill et al. [14].

**Remark 1.** Let

=the best centre density of packings of unit spheres in R_{n};

=the best centre density of lattice packings of unit spheres in Rn:

Then it is known that

(2.6)

and hence is known for Also =4 has been proved by Cohn and Kumar [6]. For using the bounds on given by Cohn and Elkies [5] and inequality (2.6) we find that , ,,

We assume that Theorem 1 is false and derive a contradiction. Let L be a lattice satisfying the hypothesis of the conjecture. Suppose that there exists a closed sphere of radius in R^{n} that contains no point of L in R^{n}.

Since B_{i}=A^{2}_{ i }and d(L)=1; we have B_{1}B_{2} : : :B_{n}=1:

We give some examples of inequalities that arise. Let L1 be a lattice in R4 with basis (A_{1}; 0; 0; 0), (a_{2;1};A_{2}; 0; 0); (a_{3;1}; a_{3;2};A_{3}; _{0}); (a_{4;1;} a_{4;2;} a_{4;3};A_{4}); and Li for be lattices in R1 with basis (Ai+3). Applying Lemma 2 repeatedly and using Lemma 1 we see that if then any closed sphere of radius

contains a point of L: By the initial hypothesis this radius exceeds Since and this results in the conditional inequality : if then

(2.7)

We call this inequality (4; 1;…; 1); since it corresponds to the ordered partition (4; 1;…; 1) of n for the purpose of applying Lemma 2. Similarly the conditional inequality (1;…; 1; 2; 1;…; 1) corresponding to the ordered partition (1;…; 1; 2; 1;…; 1) is : if then

(2.8)

Since (2.8) gives

One may remark here that the condition is necessary only if we want to use inequality (2.8), but it is not necessary if we want to use the weaker inequality (2.9). This is so because if using the partition (1; 1) in place of (2) for the relevant part, we get the upper bound which is clearly less than 2B_{i+1}. We shall call inequalities of type (2.9) as weak inequalities and denote it by (1;…; 1; 2; 1;….; 1)_{w}.

If is an ordered partition of n, then the conditional inequality arising from it, by using Lemmas 1 and 2, is also denoted by If the conditions in an inequality are satisfied then we say that holds. Sometimes, instead of Lemma 2, we are able to use induction. The use of this is indicated by putting (*) on the corresponding part of the partition. For example, if for n=10, B_{5} is larger than each of B_{6};B_{7};….;B_{10}, and if the inequality (4; 6*) gives

(2.10)

In particular the inequality ((n-1)*; 1) always holds. This can be written as

(2.11)

Also we have because if B_{1}<1, then for each I contradicting B_{1}B_{2}:::Bn=1.

Using the upper bounds on and the inequality (2.5), we obtain numerical lower and upper bounds on Bn, which we denote by ln and mn respectively. We use the approach of Hans-Gill et al. [14], but our method of dealing with

Is somewhat different. In Sections 3-5 we give proof of Theorem 1 for n=10; 11 and 12 respectively. The proof of these cases is based on the truncation of the interval [ln;mn] from both the sides.

In this paper we need to maximize or minimize frequently functions of several variables. When we say that a given function of several variables in x; y; is an increasing/decreasing function of x; y;…., it means that the concerned property holds when function is considered as a function of one variable at a time, all other variables being fixed.

**Proof of Theorem 1 for n=10**

Here we have W_{10}=10:3,B_{1}<_{10}<2:2636302. Using (2.5), we have l10=0:4007<B_{10}<1:9770808=m10.

The inequality (9*; 1) gives 9(B_{10}) ^{-1/9} + B_{10}<10:3. But for 0:4398 B_{10} 1:9378, this inequality is not true. Hence we must have either B_{10}<0:4398 or B_{10}>1:9378. We will deal with the two cases 0:4007< B_{10}<0:4398 and 1:9378<B_{10}<1:9770808 separately:

**0:4007<B _{10}<0:4398**

Using the Lemmas 3 & 4 we have:

**Claim(i) **B_{2}>1:7046

The inequality (2; 2; 2; 2; 2)w gives 2B_{2} +2B_{4} +2B_{6} +2B_{8} +2B_{10}>10:3. Using (3.1), we find that this inequality is not true for B_{2} ≤ 1:7046. Hence we must have B_{2}>1:7046.

**Claim(ii) **

Suppose then using (3.1) and that we find that and So the inequality (1,4,4,1) holds, i.e.B_{1}+4B_{2}- Applying AM-GM inequality we get Now since , and we find that the left side is a decreasing function of B_{10} and B_{6}. So replacing B_{10} by and we get Now the left side is a decreasing function of B_{2}, so replacing B_{2} by 1.8815 we find that for 1<B_{1}<2:2636302, a contradiction. Hence we must have B_{2}<1:8815.

**Claim (iii)** B_{3}<1:5652

Suppose From (3.1) we have B_{4}B_{5}B_{6}<1:6524 and B_{8}B_{9}B_{10} <0:1702, so we find that and for B_{3}>1:49.

Applying AM-GM to inequality (2,4,4) we get Since and we find that left side is a decreasing function of B_{1} and B_{7}. So we replace B_{1} by B_{2}, B_{7} by and get that

But left side is a decreasing function of B_{3}, so replacing B_{3} by 1.5652 we find that for 1:7046<B_{2}<1:8815, a contradiction. Hence we must have B_{3}<1:5652.

**Claim (iv)** B_{1}>1:9378

Suppose Using (3.1) and that B_{3}<1:5652, B_{2}>1:7046, we find that B_{2} is larger than each of B_{3}; B_{4};…;B_{10}. So the inequality (1; 9,*) holds. This gives which is not true for So we must have B_{1}>1:9378.

**Claim (v) **B_{3}<1:5485

Suppose We proceed as in Claim(iii) and replace B_{1} by 1.9378 and B_{7} by to get that

One easily checks that for 1.5485≤ B_{3}<1:5652 and 1:7046< B_{2}<1:8815. Hence we have B_{3}<1:5485.

**Claim (vi) **B_{1}<2:0187

Suppose Using (3.1) and Claims (ii), (v) we have B_{2}B_{3}B_{4}<4:11. Therefore As we see using (3.1) that B_{5} is larger than each of B_{6};B_{7},… ;B_{10}. Hence the inequality (4; 6,*) holds. This gives Left side is an increasing function of B_{2}B_{3}B_{4} and decreasing function of B_{1}. So we can replace B_{2}B_{3}B_{4} by 4:11 and B_{1} by 2.0187 to find a contradiction. Hence we have B_{1}<2:0187.

**Claim (vii)** B_{4}<1:337

Suppose then using (3.1) we get Applying AMGM to inequality (1,2,4,2,1) we have

Since B_{2}>1:7046, and we find that left side is a decreasing function of B_{2}, B_{8} and B_{10}. So we can replace B_{2} by 1.7046; B_{8} by and B_{10} by to get

Now left side is a decreasing function of B_{4}, replacing B_{4} by 1:337, we find that for 1<B_{1}<2:0187 and 1<B_{3}<1:5485, a contradiction. Hence we have B_{4}<1:337.

**Claim (viii) **B_{5}<1:1492

Suppose Using (3.1), we get B_{6}B_{7}B_{8}<0:5445: Therefore Also using Lemma 3 & 4, 2 .So the inequality (4*; 4; 2) holds, i.e.4 Now left side is a decreasing function of B_{5} and B_{9}. So we replace B_{5} by 1.1492 and B_{9} by and get that where x=B_{6}B_{7}B_{8}. Using Lemma 3 & 4 we have x==B_{6}B_{7}B_{8} and It can be verified that for and giving thereby a contradiction. Hence we must have B_{5}<1:1492.

**Claim (ix) **B_{2}<1:766.

Suppose We have B_{3}B_{4}B_{5}<2:3793. So Also Therefore B_{6} is larger than each of Hence the inequality (1; 4; 5,*) holds. This gives Left side is an increasing function of B_{3}B_{4}B_{5}, a decreasing function of B_{2} and an increasing function of B_{1}. One easily checks that this inequality is not true for B_{1}<2:0187;

and B_{3}B_{4}B_{5}<2:3793: Hence we have B_{2}<1:766.

Final contradiction

As 2(B_{2}+B_{4}+B_{6}+B_{8}+B_{10})<2(1:766+1:337+0:9383+0:6597+0:4398)<10:3, the weak inequality (2; 2; 2; 2; 2)w gives a contradiction.

**9378<B _{10}<1:9770808**

Here and B_{2}=(B_{1}B_{3}…B_{10})^{-1 Which implies i.e. B2<1:75076.}

Similarly

These respectively give B_{3}<1:46138, B_{4}<1:22883, B_{6}<0:896058 and B_{8}<0:721763. So we have Also and Applying AMGM to inequality (4,2,2,1,1) we have 4B_{1} + 4B_{5} + 4B_{7} + B_{9} +B_{10} We find that left side is a decreasing function of B_{7} and B_{5}, so can replace B_{7} by and B_{5} by then it is a decreasing function of B_{1}, so replacing B_{1} by B_{10} we have which is not true for and 1:9378<B_{10}<1:9770808. Hence we get a contradiction.

Here we have w_{11}=11.62, Using (2.5), we have l_{11}=0:3673<B_{11}<2:1016019=m_{11}.

The inequality (10*; 1) gives 10:3 But for this inequality is not true. So we must have either or

**Claim (i)** B_{10}<0:4692

Suppose then 2B_{10}>B_{11}, so (9*; 2) holds, i.e. 9 As left side is a decreasing function of B_{10}, we can replace B_{10} by 0.4692 and find that it is not true for 0:3673<B_{11}<0:4409.

Hence we must have B_{10}<0:4692.

Using Lemmas 3 and 4 we have:

(4.1)

**Claim (ii) B _{2}>1:913**

The inequality (2; 2; 2; 2; 2; 1) w gives 2B_{2}+2B_{4}+2B_{6}+2B_{8}+2B_{10}+B_{11} > 11:62. Using (4.1) we find that this inequality is not true for B ≤1.913 so we must have B_{2}>1:913.

**Claim(iii)** B_{3}<1:761

Suppose then we have and

2. Applying AM-GM to the inequality (2,4,4,1) we get One easily finds that it is not true for and Hence we must have B_{3}<1:761:

**Claim (iv) **B_{1}<2:2436

Suppose As B_{2}B_{3}B_{4}<2:13557×1:761×1:50151<5:6468, we have
Also so B_{5} is larger than each of B_{6};B_{7}…;B_{11}. Hence the inequality (4; 7,*) holds. This gives Left side is an increasing function of B_{2}B_{3}B_{4} and decreasing function of B_{1}. One easily checks that the inequality is not true for B_{2}B_{3}B_{4}<5:6468 and B_{1} ≥ 2:2436. Hence we have B_{1}<2:2436.

**Claim (v) B _{4}<1.4465 and B_{2}>1:9686**

Suppose B_{4} ≥ 1.4465 We have B_{5}B_{6}B_{7}<1:2569 and B_{9}B_{10}B_{11}<0:1295. Therefore for B_{4}>1:36, we have and So the inequality (1,2,4,4) holds. Applying AM-GM to inequality(1,2,4,4), we get A simple calculation shows that this is not true for ,, and Hence we have B_{4}<1:4465.

Further if then 2B_{2}+2B_{4}+2B_{6}+2B_{8}+2B_{10}+B_{11}<11:62. So the inequality (2; 2; 2; 2; 2; 1)_{w} gives a contradiction.

**Claim (vi) B _{4}<1:4265 and B_{2}>1:9888**

Suppose We proceed as in Claim (v) and get a contradiction with improved bounds on B_{2} and B_{4}.

**Claim (vii) B _{1}<2:2056**

Suppose As B_{3}B_{4}B_{5}<1:761 × 1:4265 × 1:3347<3:3529, we have Also so B_{6} is larger than each of B_{7};B_{8},…,B_{11}. Hence the inequality (1; 4; 6*) holds, i.e. B_{1} + 4B_{2} -

**Claim (ix) B _{1}<2:1669**

Suppose We proceed as in Claim(iv) and get a contradiction with improved bounds on B_{1}, B_{2} and B_{4}.

**Claim (x) B _{4}<1:403 and B_{2}>2:012**

Suppose We proceed as in Claim(v) and get a contradiction with improved bounds on B_{2} and B_{4}.

**Final Contradiction:**

As now B_{3}B_{4}B_{5}<1:761×1:403 1:3347<3:2977, we have for B_{2}>2:012. Also each of B_{7}; B_{8}; B_{11}. Hence the inequality (1; 4; 6) holds. Proceeding as in Claim (viii) we find that this inequality is not true for B_{1}<2:1669; B_{2}>2:012 and B_{3}B_{4}B_{5}<3:2977; giving thereby a contradiction.

Here Therefore using Lemmas 3 & 4 we have B_{10}=(B_{1} B_{9}B_{11})^{-1}

Similarly

which gives B_{4}<1:37661.

**Claim (i) B _{10}<0:4402**

The inequality (9*; 1; 1) gives But this inequality is not true for and 2:018<B_{11}<2:1016019. Hence we must have B_{10}<0:4402.

Now we have < 0:58694, and

**Claim (ii) B _{7}<0:768**

Suppose Then so (6*; 4; 1) holds. This gives where x=B_{1}B_{2} : : :B_{6}. The function has its maximum value at Therefore which is less than 11:62 for 2:018<B_{11}<2:1016019. This gives a contradiction.

Now and

**Claim (iii) B _{2}<1:795**

Suppose then and Applying AMGM to the inequality (1,4,4,1,1) p we get B_{1} + 4B_{2} + 4B_{6} + B_{10} + B_{11} - We find that left side is a decreasing function of B_{6}, so we first replace B_{6} by εB_{2} then it is a decreasing function of B_{2}, so we replace B_{2} by 1.795 and get that

Now so which can be verified to be at most 11.62 for and 2:018<B_{1}<2:393347, giving thereby a contradiction.

**Claim (iv) B _{5}<0:98392**

Suppose We have and Also Applying AM-GM to the inequality (4; 4; 2; 1) we get One can easily check that left side is a decreasing function of B_{9} and B_{1} so we can replace B_{9} by εB and B_{1} by B_{11} to get Now the left side is a decreasing function of B_{5}, so replacing B_{5} by 0.98392 we see that and 2:018<B_{11} < 2:1016019, a contradiction.

**Final Contradiction:**

As in Claim(iv), we have Also each of B_{6};B_{7},…,B_{10}. Therefore the inequality (4; 6*; 1) holds, i.e. Left side is an increasing function of B_{2}B_{3}B_{4} and B_{11} and decreasing function of B_{1}. Using B_{5}<0:98392, we have and One easily checks that for B_{2}B_{3}B_{4}<1:795 × 1:47588 × 1:311894, B_{11}<2:1016019 and Hence we have a contradiction.

Here we have w_{12}=13, Using (2.5), we have l12 =0:3376<B_{12}<2:2254706=m12 and using (2.3) we have i.e

The inequality (11*; 1) gives 11:62(B_{12})-1/11 +B_{12}>13. But this is not true for So we must have either B_{12}<0:4165 or B_{12}>2:17.

Claim (i) B_{11}<0:459

Suppose then and 2B_{11}>B_{12}, so (10*; 2) holds, i.e, Left side is a decreasing function of B_{11}, so we can replace B_{11} by .459 to find that for 0:34425<B_{12}<0:4165, a contradiction. Hence we have B_{11}<0:459.

**Claim (ii) B _{10}<0:5432**

Suppose From Lemma 3, and B_{12}.Therefore and so the inequality (9*; 3) holds, i.e. One easily checks that it is not true noting that left side is a decreasing function of B_{10}. Hence we must have B_{10}<0:5432.

**Claim (iii) B _{9}<0:6655**

Suppose then So the inequality (8*; 4) holds. This gives where x=B_{1}B_{2} … B_{8}. The function has its maximum value at so for where x=B_{1}B_{2}… B_{8}. The function has its maximum value at so for 0:6655 This gives a contradiction.

Using Lemmas 3 & 4 we have:

Claim (iv) B_{2}>1:828, B_{4}>1:426, B_{6}>1:019 and B_{8}>0:715

Suppose Then 2(B_{2}+B_{4}+B_{6}+B_{8}+B_{10}+B_{12})<2(1:828+ 1:7384+1:1589+0:8148+0:5432+0:4165)<13, giving thereby a contradiction to the weak inequality (2; 2; 2; 2; 2; 2) w.

Similarly we obtain lower bounds on B_{4};B_{6} and B_{8} using (2; 2; 2; 2; 2; 2)w.

**Claim(v)** B_{2}>2:0299

Suppose Consider following two cases:

**Case (i) B _{3}>B_{4}**

We have B_{3}>B_{4}>1:426>each of B_{5},….,B_{12}. So the inequality (2; 10*) holds, i.e. The left side is a decreasing function of B_{1}, so replacing B_{1} by B_{2} we get 2B_{2} + 10:3 which is not true for

**Case (ii) **

As B_{4}>1:426>each of B_{5},….,B_{12}, the inequality (3; 9*) holds, i.e. where X=B_{2}B_{3}<min say. Now is an increasing function of X for and so which can be seen to be less than 13. Hence we have B_{2}>2:0299.

**Claim (vi) **B_{1}>2:17 and B_{3}<1:9517

Using (2.3) we have Therefore B_{2}>2:0299>each of B_{3},…,B_{12}. So the inequality (1; 11*) holds, i.e. B_{1} + 11:62 But this is not true for So we must have B_{1}>2:17: Again using (2.3) we have

**Claim (vii) B _{4}<1:646**

Suppose From (5.1) and Claims (i)-(iii), we have and Applying AM-GM to the inequality (1,2,4,4,1) we get We find that left side is a decreasing function of B_{2}, B_{8} and B_{12}. So we can replace B_{2} by 2:0299, B_{8} by "B_{4} and B_{12} by . Then it turns a decreasing function of e 2B_{4}, so can replace B_{4} by 1.646 to find that ,a contradiction. Hence we have B_{4}<1:646.

**Claim (viii) B _{1}<2:4273**

Suppose B_{1}≥2:4273. Consider following two cases:

Case (i) B_{5}>B_{6}

Here B_{5}>each of B_{6},…,B_{12} as B_{5} ≥ B_{1}>1.137 >each of B_{7},…, B_{12}. Also B_{2}B_{3}B_{4}<2:2254706×1:9517×1:646<7:15. So Hence the inequality (4; 8*) holds. This gives

Left side is an increasing function of B_{2}B_{3}B_{4} and decreasing function of B_{1}. So we can replace B_{2}B_{3}B_{4} by 7.15 and B_{1} by 2.4273 to get a contradiction.

**Case (ii) **

Using (5.1) we have <1:1589 and so Therefore as B_{2}>2:0299 and B_{3}<1:9517. Also from Claim (iv), B_{6}>1:019>each of B_{7},….,B_{12}. Hence the inequality (1; 4; 7*) holds. This gives 7(B_{1}B_{2}B_{3}B_{4}B_{5})-1/7>13: Left side is an increasing function of B_{3}B_{4}B_{5} and B_{1} and a decreasing function of B_{2}. One can check that inequality is not true for B_{3}B_{4}B_{5}<1:9517×1:5452 ×1:1589, B_{1}<2:5217871 and for B_{2}>2:0299: Hence we must have B_{1}<2:4273:

**Claim (ix) B _{5}<1:396**

Suppose From (5.1), B_{6}B_{7}B_{8}<0:925 and B_{10}B_{11}B_{12}<0:104, so we have and Applying AMGM to the inequality (1,2,1,4,4) we get + We find that left side is a decreasing function of B_{2} and B_{9}. So we replace B_{2} by 2:0299 and B_{9} by ε B_{5}. Now it becomes a decreasing function of B_{5} and an increasing function of B_{1} so replacing

B_{5} by 1.396 and B_{1} by 2.4273, we find that above inequality is not true for 1:522<B_{3}<1:9517 and 1:426<B_{4}<1:646, giving thereby a contradiction. Hence we must have B_{5}<1:396.

**Claim (x) B _{3}>1:7855**

Suppose We have B_{4}>1:426>each of B_{5};B_{6},…,B_{12}, hence the inequality (1; 2; 9*) holds. It gives It is easy to check that left side of above inequality is a decreasing function of B_{2} and an increasing function of B_{1} and B_{3}. So replacing B_{1} by 2.4273, B_{3} by 1.7855 and B_{2} by 2.0299 we get -15<13; a contradiction. Hence we have B_{3}>1:7855.

**Claim (xi)**

Suppose We have B_{3}>1:7855>each of B_{4};B_{5},…,B_{12}, hence the inequality (2; 10*) holds. It gives The left side is a decreasing function of B_{1} and an increasing function of B_{2}, so replacing B_{1} by 2:17 and B_{2} by 2.0733 we get a contradiction.

**Claim (xii) B _{7}<0:92 and B_{5}<1:38**

Suppose Here we have B_{4}B_{5}B_{6}<2:67 and B_{8}B_{9}B_{10}<0:295, so and Also Applying AM-GM to the inequality (2,4,4,2) we get We find that left side is a decreasing function of B_{1} and B_{11}. So we can replace B_{1} by 2:17 and B_{11} by ε B_{7}. Then left side becomes a decreasing function of B_{7} and an increasing function of B_{2}, so can replace B_{7} by 0.92 and B_{2} by 2.2254706 to see that for 1:7855<B_{3}<1:9517 and 0:3376<B_{12}<0:4156, a contradiction. Hence B_{7}<0:92. Further gives B_{5}<1:38.

**Claim (xiii) B _{6}<1:097**

Suppose Here we have B_{3}B_{4}B_{5}<4:44 and B_{7}B_{8}B_{9}<0:5, so and Also Applying AM-GM to the inequality (1,4,4,2,1) we get We find that left side is a decreasing function of B_{10}, B_{12} and B_{11}. So we can replace B_{10} by εB_{6} and B_{12} by 0.3376 and B_{11} by B_{6}. Then left side becomes a decreasing function of B_{6}, so we can replace B_{6} by 1.097 to find that Hence we must have B_{6}<1:097.

Claim (xiv) B_{5}>B_{6} and

First suppose B_{5} ≤ B_{6}, then B_{4}B_{5}B_{6}<1:646 × 1:0972<1:981 and Also each of B_{8},…,B_{12}. Hence the inequality (2; 4; 6*) holds, i.e. Now the left side is a decreasing function of B_{1} and B_{3} as well; also it is an increasing function of B_{2} and B_{4}B_{5}B_{6}. But one can check that this inequality is not true for B_{1}>2:17, B_{3}>1:7855, B_{2}<2:2254706 and B_{4}B_{5}B_{6}<1:981, giving thereby a contradiction. Further suppose then as B_{5}>B_{6}>1:019>each of B_{7},…, B_{12}, the inequality (4; 8*) holds. Now working as in Case (i) of Claim (viii) we get contradiction for B_{1}>2:17 and B_{2}B_{3}B_{4}<2:2254706 × 1:9517 × 1:646<7:14934.

**Claim (xv) **B_{3}<1:9 and B_{1}<2:4056

Suppose ,then for B_{4}B_{5}B_{6}<1:646×1:38×1:097<2:492, Also B_{7}≥ε B_{3}>0:89>each of B_{8},…,B_{12}. Hence the inequality (2; 4; 6*) holds. Now working as in Claim (xiv) we get contradiction for B_{1}>2:17, B_{2}<2:2254706, B_{3}>1:9 and B_{4}B_{5}B_{6}<2:492. So B_{3}<1:9. Further if B_{1}≥ 2:4056, then contradicting Claim (xiv).

**Claim (xvi)** B_{4}<1:58 and B_{1}<2:373

Suppose then for B_{5}B_{6}B<1:38 × 1:097 × 0:92<1:393, Also B_{8} B_{4}>0:74>each of B_{9},…,B_{12}. Hence the inequality (1; 2; 4; 5*) holds, i.e. -19=B_{1} + 4B_{2} - Left side is a decreasing function of B_{2} and B_{4}.

So we replace B_{2} by 2.0733 and B_{4} by 1.58. Then it becomes an increasing function of B_{1}, B_{3} and B_{5}B_{6}B_{7}. So we replace B_{1} by 2.4056, B_{3} by 1.9 and B_{5}B_{6}B_{7} by 1.393 to find that -19<13, a contradiction. Further if B_{1} ≥ 2:373, then contradicting Claim (xiv).

We have B_{3}B_{4}B<1:9×1:58×1:38<4:15. Therefore >2 Also B_{6}>1:019>each of B_{7},…,B_{12}. Hence the inequality (1; 4; 7*) holds. Now we get contradiction working as in Case (ii) of Claim (viii).

5.2 2:17<B_{12}<2:2254706

Here 2.17Using Lemma 3 and 4, we have

B^{11}=(B_{1}B_{2}…B_{10}B_{12})^{-1}<

Claim (i) Either B_{11}<0:4307 or B_{11}>1:818

Suppose 0:4307 ≤B_{11}≤1.818 The inequality (10*; 1; 1) gives 10:3 which is not true for 0:4307 ≤B_{11}≤1.818 and 2:17<B_{12}<2:2254706. So we must have either B_{11}<0:4307 or B_{11}>1:818

**Claim (ii) B _{11}<0:4307**

Suppose B_{11} ≥ 0.4307 then using Claim(i) we have B_{11}>1:818. Now we have using Lemmas 3 & 4,

This gives B_{2}<1:777.

This gives B_{3}<1:487

This gives B_{4}<1:213.

This gives B_{6}<0:826.

This gives B_{7}<0:697.

This gives B_{8}<0:559.

This gives B_{9}<0:478.

Therefore we have and each of B_{6},…,B_{10}. So the inequality (4; 6*; 1; 1) holds, i.e.4B_{1} (B_{1}B_{2}B_{3}B_{4}B_{11}B_{12})^{-1/6} +B_{11}+B_{12}>13 Now the left side is an increasing function of B_{2}B_{3}B_{4}, B_{11} and of B_{12} as well. Also it is a decreasing function of B_{1}. So we replace B_{2}B_{3}B_{4} by 1:777 × 1:487 × 1:213, B_{11} by 1.8223, B_{12} by 2.2254706 and B_{1} by 2.17 to arrive at a contradiction. Hence we must have B_{11}<0:4307.

Claim (iii) B_{10}<0:445

Suppose then 2B_{10}>B_{11}. So the inequality (9*; 2; 1) holds, i.e. B_{10} and B_{12}>2:2254706, the left side is an increasing function of B_{12} and a decreasing function of B_{10}, so replacing B_{12} by 2.2254706 and B_{10} by 0.445 we find that ,for 3 4(0:445)<B_{11}<0:4307, a contradiction. Hence we must have B_{10}<0:445.

Using Lemmas 3 and 4 we have:

Claim (iv) B_{3}<1:62

Suppose From (5.2), we have B_{4}B_{5}B_{6}<1:712 and B_{8}B_{9}B_{10}<0:178, so and Applying AM-GM to the inequality (2,4,4,1,1) we get We find that left side is a decreasing function of B_{1}, B_{7} and B_{11}. So we can replace B_{1} by B_{12}, B_{7} by ε B_{3} and B_{11} by e 2B_{3}. Then it becomes a decreasing function of B_{3}, so replacing B_{3} by 1.62 we find that φ 21<13; for 1:6275<B_{2}<2:0255 and 2:17<B_{12}<2:2254706, a contradiction. Hence we must have B_{3}<1:62.

Claim (v) B_{12}>2:196

Suppose From (5.2), we have B_{2}B_{3}B_{4}<4:674 and Also each of B_{6},…, B_{11}. Therefore the inequality (4; 7*; 1) holds, i.e. Left side is an increasing function of B_{2}B_{3}B_{4} and of B_{12} as well. Also it is a decreasing function of B_{1}. So we can replace B_{2}B_{3}B_{4} by 4.674, B_{12} by 2.196 and B_{1} by 2.17 to get Φ 22<13, a contradiction. Hence we must have B_{12}>2:196.

**Final Contradiction**

Now we have B_{1} ≥B_{12}>2:196. We proceed as in Claim(v) and use (4; 7*; 1). Here we replace B_{2}B_{3}B_{4} by 4.674, B_{12} by 2.2254706 and B_{1} by 2.196 to get Φ22<13, a contradiction.

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