Proving Inequalities with Difference Substitution Yu-Dong Wu, Zhi-Hua Zhang
and Yu-Rui Zhang vol. 8, iss. 3, art. 81, 2007
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PROVING INEQUALITIES IN ACUTE TRIANGLE WITH DIFFERENCE SUBSTITUTION
YU-DONG WU ZHI-HUA ZHANG
Xinchang High School Zixing Educational Research Section
Xinchang City, Zhejiang Province 312500 Chenzhou City, Hunan Province 423400
P.R. China. P. R. China.
EMail:zjxcwyd@tom.com EMail:zxzh1234@163.com
YU-RUI ZHANG
Xinchang High School
Xinchang City, Zhejiang Province 312500 P. R. China.
EMail:xczxzyr@163.com
Received: 09 October, 2006
Accepted: 04 April, 2007
Communicated by: B. Yang 2000 AMS Sub. Class.: 26D15.
Key words: Inequalities, Acute Triangle, Difference Substitution, Linear transformation.
Abstract: In this paper, we prove several inequalities in the acute triangle by means of so- called Difference Substitution. As generalization of the method, we also consider an example that the greatest interior angle is less than or equal to120◦in the triangle.
Acknowledgements: The authors would like to thank Professor Lu Yang for his enthusiastic help.
Proving Inequalities with Difference Substitution Yu-Dong Wu, Zhi-Hua Zhang
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Contents
1 Introduction 3
2 Some Problems and their Proofs 8
2.1 The Problems . . . 8
2.2 The Proof of Inequality (2.1) . . . 8
2.3 The Proof of Inequality (2.2) . . . 11
2.4 Remarks . . . 16
3 Generalization of the Method 17
Proving Inequalities with Difference Substitution Yu-Dong Wu, Zhi-Hua Zhang
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1. Introduction
In [1, 2], L. Yang suggested the use of Difference Substitution to prove asymmetric polynomial inequalities, as it had been used previously to deal with symmetric ones.
Ifx1 ≤x2 ≤x3 ≤ · · · ≤xnwithn∈N∗, then we set
(1.1)
x1 =t1, x2 =t1+t2, x3 =t1+t2+t3,
· · · ·
xn=t1+t2+t3+· · ·+tn, whereti ≥0for2≤i≤nandi∈N∗.
The expansion (1.1) is so-called a “splitting” transformation, and{t1, t2, . . . , tn} is simply the difference sequence of{x1, x2, . . . , xn}.
In general, for then-variant polynomials, there aren!different orders of{x1, x2, . . . , xn}, sorting by size. In the instance ofn= 3, we letx≤y ≤z, and take
(1.2)
x=u, y=u+v, z =u+v+w, wherev ≥0,w≥0.
Analogously, ify≤x≤z, then its “splitting” transformation is
(1.3)
y=u, x=u+v, z =u+v+w,
Proving Inequalities with Difference Substitution Yu-Dong Wu, Zhi-Hua Zhang
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wherev ≥0,w≥0.
Sequentially, for y ≤ z ≤ xorz ≤ x ≤ yorz ≤ y ≤ xorx ≤ z ≤ y, we set four similar linear transformations.
For a 3-variant polynomial F(x, y, z), by using the six linear transformations above, we obtain 6 membersPi(u, v, w)with1≤i≤6, and call the set{P1, P2, . . . , P6} the Difference Substitution of F(x, y, z) and denote this byDS(F). If all the co- efficients of these membersDS(F)are nonnegative, then F ≥ 0whenever x, y, z all are nonnegative. In other words, F is positive semi-definite on R3+. Difference substitution is a very valid method for proving inequalities. For more information on Difference Substitution, please refer to [3] and [4].
In this paper, by using Difference Substitution, the authors prove several inequal- ities in acute triangles.
Throughout the paper we denoteA, B, C as the interior angles,a, b, cas the side- lengths, S as the area, sas the semi-perimeter, R as the circumradius, r as the in- radius, ha, hb, hc as the altitudes, ma, mb, mc as the medians, and ra, rb, rc as the radii of the described circles of triangleABC respectively. Moreover, we will cus- tomarily use the cyclic sum symbol, that is: P
f(a) = f(a) +f(b) +f(c), and Pf(a, b) =f(a, b) +f(b, c) +f(c, a), etc.
Let us begin with the well-known Walker’s inequality [5]. In the acute triangle, show that
(1.4) s2 ≥2R2+ 8Rr+ 3r2,
or
(1.5) −2a3b3+a4b2−a4bc+a5b+ab5+b5c+b4c2
−2b3c3+b2c4 −2c3a3+c4a2+c5a+c5b+c2a4
+a5c−ab4c+a2b4−b6−c6−a6−abc4 ≥0.
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Let
(1.6)
x= b+c−a2 >0, y = c+a−b2 >0, z = a+b−c2 >0.
Then inequality (1.4) or (1.5) is equivalent to
(1.7) F(x, y, z) = 6xyz4+ 2xy2z3+ 2xy3z2 + 6xy4z+ 2x2yz3+ 2x2y3z + 2x3yz2+ 2x3y2z+ 6x4yz−x4y2−x2z4 −2x3z3−x4z2
−2x3y3−y4z2−y4x2−2y3z3−y2z4−18x2y2z2 ≥0.
There is no harm in supposingx ≤ y ≤ z since inequality (1.7) is symmetric for x, y, z. Then, by using (1.2), for the acute triangle, it follows that
b2+c2−a2 = (z+x)2+ (x+y)2−(y+z)2 = 2[x2+ (y+z)x−yz]
(1.8)
= 2{u2+ [(u+v) + (u+v+w)]u−(u+v)(u+v+w)}
= 2(2u2−v2 −vw)>0, andF(x, y, z)in (1.7) is transformed into
F(x, y, z) (1.9)
=P(u, v, w)
= 2u2−v2−vw 4v2+ 4w2+ 4vw u2 + 8v3+ 20vw2+ 12v2w+ 8w3
u+ 4v4+ 8v3w+ 2w4 +18vw3+ 22v2w2
+ 24v3w2+ 36v2w3+ 12vw4 u + 34v3w3+ 19v2w4+ 2vw5+ 17v4w2.
Proving Inequalities with Difference Substitution Yu-Dong Wu, Zhi-Hua Zhang
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Obviously F(x, y, z) = P(u, v, w) ≥ 0 from (1.8) and u > 0, v ≥ 0, w ≥ 0, i.e., inequality (1.4) or (1.5) is true.
Now, let us consider another semi-symmetric inequality [6] in the acute triangle
(1.10) cos(B −C)≤ ha
ma. It is equivalent to
(1.11) −a4+ 3b2+ 3c2
a2−2 (b−c)2(b+c)2 ≥0, and from (1.6), this equals
(1.12) F(x, y, z)
= −y2−z2+ 14yz
x2−(y+z) z2−14yz+y2
x+yz(y+z)2 ≥0.
Calculating DS(F), it consists of 3 polynomials with u > 0, v ≥ 0, w ≥ 0 as follows
(1.13) P1(u, v, w)
= 40u4+ 112u3v + 108u2v2+ 56u3w+ 14u2w2 + 40uv3+ 20uvw2 + 60uv2w+ 108u2vw+ 8v3w+ 5v2w2+vw3+ 4v4,
(1.14) P2(u, v, w)
= 2u2−v2−vw
20u2+ (24w+ 52v)u+ 53v2+ 6w2+ 52vw + 72v3+ 36vw2+ 108v2w
u+ 57v2w2 + 51v4+ 6vw3+ 102v3w,
Proving Inequalities with Difference Substitution Yu-Dong Wu, Zhi-Hua Zhang
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and
(1.15) P3(u, v, w)
= 2u2−v2−vw
20u2+ (52v + 28w)u+ 53v2+ 54vw+ 7w2 + 72v3+ 36vw2+ 108v2w
u+ 57v2w2 + 51v4+ 6vw3+ 102v3w.
By (1.8), we immediately obtain Pi(u, v, w) ≥ 0for1 ≤ i ≤ 3. Hence, inequality (1.10) is proved.
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2. Some Problems and their Proofs
2.1. The Problems
In 2004-2005, J. Liu [7,8] posed the following conjectures for the inequality in the acute triangle.
Problem 1. Let4ABC be an acute triangle. Prove the following inequalities
(2.1) X
sin 2A sinB+ sinC
2
≤ 3 4, and
(2.2) sinA
2 ≤
√mbmc 2ma . 2.2. The Proof of Inequality (2.1)
Proof. Usingsin 2α= 2 sinαcosα, we find that inequality (2.1) is equivalent to 4a10b2 −10a5b5c2 −24b6c5a+ 16a9b3+ 4a8b4−5b4c6a2
(2.3)
−8a11b−5a4b6c2+ 8a9c2b+ 4a8b2c2−16a10cb+ 8a9cb2−5a6b4c2 + 32a8b3c+ 8a7b4c+ 8a7c4b−5a6c4b2+ 32a8c3b+ 4a10c2−8a11c
−4a12−4b12−4c12+ 4a8c4+ 16a9c3−8a7b5−8a6b6 −8a6c6
−8a7c5−8a5b7 + 4a4b8−24a6b5c−24a5b6c+ 2a5b3c4 + 6a4b4c4 + 4c10b2−26a6b3c3+ 2a5b4c3−24a5c6b−5a4c6b2−24a6c5b
−10a5c5b2+ 8a4b7c−8c11b+ 32b8a3c+ 8b9a2c+ 4b8a2c2 + 2b5c3a4−26b6c3a3+ 2b5c4a3−5b6c4a2−16b10ca+ 8b9c2a
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+ 32b8c3a+ 8b7c4a−10b5c5a2+ 16b9a3+ 4b10a2−8b11a
−8b11c+ 4b10c2+ 16b9c3+ 4b8c4−8b7c5−8b6c6+ 4a4c8
−8a5c7−24b5c6a+ 8a4c7b+ 2c5b4a3−26c6b3a3+ 2c5b3a4
+ 4c8b2a2+ 8c9a2b+ 32c8a3b+ 8b4c7a−8c11a+ 32c8b3a+ 8c9b2a
−16c10ab+ 4c10a2+ 16c9a3−8b5c7+ 4b4c8+ 16c9b3 ≥0.
From (1.6), inequality (2.3) equals F(x, y, z)
(2.4)
=−4576x7z5−5590x6z6−116x10z2−2453x4z8−2453x8z4
−4576x5z7−788x3z9−788x9z3−2453x8y4−4576y7z5
−2453y8z4−2453x4y8 −788x9y3−788x3y9−5590x6y6
−4576x7y5−4576x5y7 −2453y4z8−4576y5z7−5590y6z6
−116y10z2−788y9z3−116y10x2−788y3z9−116y2z10 + 13448x6y5z+ 8176x7yz4+ 13448x6yz5 + 13448x5yz6 + 8176x4yz7+ 6448x2y3z7+ 1220xy9z2+ 6448x2y7z3 + 6448x3y7z2+ 1220x9yz2+ 10862x4y6z2+ 14288x2y5z5 + 10862x2y4z6+ 14288x5y2z5+ 10862x6y2z4+ 6448x7y2z3
−28248x3y5z4−28248x4y5z3+ 14288x5y5z2−57474x4y4z4
−28248x3y4z5−8672x3y3z6+ 6448x3y2z7+ 10862x2y6z4
−8672x3y6z3−28248x5y4z3+ 10862x6y4z2−28248x4y3z5
−28248x5y3z4−8672x6y3z3+ 6448x7y3z2+ 10862x4y2z6
+ 3420x8yz3+ 3420x8y3z+ 1220x2yz9+ 280xy10z+ 4066x2y2z8
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+ 3420x3yz8+ 3420x3y8z+ 4066x8y2z2+ 4066x2y8z2 + 1220xy2z9+ 8176x7y4z+ 3420xy3z8+ 3420xy8z3
+ 1220x2y9z+ 280xyz10+ 280x10yz+ 1220x9y2z+ 8176xy4z7 + 13448xy5z6+ 13448xy6z5+ 8176x4y7z+ 8176xy7z4
+ 13448x5y6z−116x10y2 −116x2z10≥0.
Since (2.4) is symmetric for x, y, z, there is no harm in supposing that x ≤ y ≤ z.
Using the transformation (1.2), thenF(x, y, z)in (2.4) becomes F(x, y, z)
(2.5)
=P(u, v, w)
=(2u2 −v2−vw)[(180224w2+ 180224v2+ 180224vw)u8 + (1794048v2w+ 1810432vw2+ 606208w3+ 1196032v3)u7 + (4360192vw3+ 7030784v3w+ 771072w4+ 7875584v2w2 + 3515392v4)u6+ (6049280v5+ 520704w5+ 19394048v3w2 + 13967872v2w3+ 4689152vw4+ 15123200v4w)u5
+ (2838144vw5+ 6838400v6+ 12647648v2w4 + 30324704v4w2 + 210048w6+ 26457408v3w3+ 20515200v5w)u4 + (19291776v6w + 52480w7 + 1074176vw6+ 5511936v7+ 32787968v5w2
+ 33740480v4w3+ 20662912v3w4+ 6899776v2w5)u3
+ (32727200v5w3+ 7968w8+ 2528912w6v2+ 24395856v4w4 + 27385760v6w2+ 14122880v7w+ 268096w7v+ 3530720v8
+ 10723072v3w5)u2+ (9558576v8w+ 4185944v3w6+ 13383144v4w5 + 676240v2w7 + 2124128v9+ 672w9+ 24737624v5w4+ 45200vw8
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+ 20832112v7w2+ 28305704v6w3)u+ 15686836v5w5+ 6092840v4w6 + 1326664v3w7+ 139150v2w8+ 24651416v6w4+ 24921352v7w3 + 1378920v10+ 6894600v9w+ 16572238v8w2+ 5112vw9+ 24w10] + (27659640v9w2+ 10558592v10w+ 689380v3w8+ 4642800v4w7 + 36001700v6w5+ 45278940v8w3+ 16715660v5w6 + 720vw10 + 49540936v7w4+ 1919744v11+ 44048v2w9)u+ 5020v2w10 + 49008067v8w4+ 142314v3w9+ 23121662v10w2+ 8144784v11w + 1451049v4w8+ 40947790v9w3 + 24vw11+ 7353016v5w7
+ 1357464v12+ 39938152v7w5+ 21582818v6w6.
This impliesF(x, y, z) =P(u, v, w)≥0from (1.8). Hence, inequality (2.1) holds.
The proof is completed.
2.3. The Proof of Inequality (2.2) Proof. Inequality (2.2) is equivalent to
(2.6) sin4 A
2 ≤ m2bm2c 16m4a.
By using the formulacosα = 1−2 sin2α2, the law of cosines and the formulas of the medians, we find that (2.6) is simply the following inequality
(2.7) −a8−4b8 + 6a6c2−34b2c6+ 20b3a4c+ 12a2c6−32b5a2c−32c5a2b
−34b6c2−51b4c4−4c8−4a6bc+ 20c3a4b−26a4b2c2+ 54a2b4c2
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+ 54a2b2c4−13a4b4−13a4c4 + 12a2b6 + 6a6b2+ 16b7c+ 16c7b
−64b3c3a2+ 48b5c3+ 48b3c5 ≥0.
Considering (1.6), inequality (2.7) is transformed into F(x,y, z)
(2.8)
=x8+ (4z+ 4y)x7+ (2z2+ 40yz+ 2y2)x6 + (−8z3+ 84yz2 −8y3 + 84y2z)x5
+ (−20y2z2+ 76yz3+ 76y3z−7z4−7y4)x4
+ (4z5+ 48y4z−248y3z2−248y2z3+ 4y5 + 48yz4)x3
+ (4y6−234y4z2−234y2z4+ 28y5z+ 4z6 + 28yz5+ 32y3z3)x2 + (−84z5y2+ 8z6y−84y5z2+ 256y3z4+ 256y4z3+ 8y6z)x + 84z5y3−63z4y4−12y6z2−12z6y2+ 84y5z3 ≥0.
It it easy to see that inequality (2.8) is symmetric fory, z. Therefore, we only need to prove that inequality (2.8) holds whenx≤y≤z,y≤x≤zandy≤z ≤x.
Calculating DS(F), it consists of 3 polynomials with u > 0, v ≥ 0, w ≥ 0 as follows
P1(u, v, w) (2.9)
=(2u2−v2−vw)[(192w2+ 768v2+ 768vw)u4 + (256w3+ 2112vw2+ 4800v2w+ 3200v3)u3
+ (5808v4+ 80w4+ 7376v2w2+ 11616v3w+ 1568vw3)u2
+ (6336v5+ 15840v4w+ 13440v3w2+ 16w5+ 4320v2w3+ 416vw4)u + 5112v6+ 15336v5w+ 48vw5+ 16560v4w2+ 7560v3w3
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+ 1272v2w4] + (7344v7+ 432w5v2+ 25704v6w+ 33912v5w2 + 20520v4w3+ 5400v3w4)u+ 20772v7w+ 5193v8+ 36w6v2 + 1332w5v3+ 32418v6w2+ 24552v5w3+ 9009v4w4
forx≤y≤z, P2(u, v, w) (2.10)
=(2u2−v2−vw)[(−384vw+ 192v2+ 192w2)u4 + (−192vw2+ 896v3−960v2w+ 256w3)u3
+ (−976v2w2+ 1776v4+ 80w4+ 224vw3−288v3w)u2
+ (2032v5−480v3w2+ 16w5+ 1328v4w+ 128v2w3+ 240vw4)u + 1640v6+ 2128v5w+ 544v4w2+ 328v3w3+ 416v2w4+ 80vw5] + (2064v5w2+ 32w6v+ 4176v6w+ 776v4w3+ 2320v7
+ 416v2w5+ 968v3w4)u+ 1640v8+ 2708w2v6 + 817w4v4 + 524w5v3+ 956w3v5+ 84w6v2+ 3768wv7
fory≤x≤z, and P3(u,v, w) (2.11)
=384u6v2+ 11072w2u2v4+ 20992w2u3v3+ 19552w2u4v2 + 8832w2u5v+ 2008w4uv3+ 5296w4u2v2+ 5376w4u3v + 36w2v6+ 1536w2u6 + 2792w3uv4+ 10400w3u2v3 + 15744w3u3v2+ 10816w3u4v+ 2368w2uv5+ 840w5uv2 + 1344w5u2v+ 2816w3u5+ 132w3v5+ 1888w4u4+ 193w4v4
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+ 184w6uv+ 144w5v3+ 640w5u3+ 1200wuv6+ 13120wu3v4 + 6256wu2v5+ 13824wu4v3+ 58w6v2 + 128w6u2+ 7296wu5v2 + 3360u4v4+ 1792u5v3+ 288uv7+ 1504u2v6+ 3168u3v5 + 1536wu6v+ 12w7v+w8+ 16w7u
fory≤z ≤x.
It is not difficult to see that P1(u, v, w) ≥ 0 and P3(u, v, w) ≥ 0 because u >
0, v ≥0, w ≥0and2u2 −v2−vw >0.
In order to proveP2(u, v, w)≥0, we only need prove the following inequality p(u,v, w)
(2.12)
=(−384vw+ 192v2 + 192w2)u4
+ (−192vw2+ 896v3−960v2w+ 256w3)u3
+ (−976v2w2+ 1776v4+ 80w4+ 224vw3−288v3w)u2
+ (2032v5−480v3w2 + 16w5+ 1328v4w+ 128v2w3+ 240vw4)u + 1640v6+ 2128v5w+ 544v4w2+ 328v3w3+ 416v2w4+ 80vw5
≥0,
whereu >0,v ≥0andw≥0.
(i)Foru >0, v ≥w≥0, takingv =w+twitht ≥0, then we have
p(u, v, w) =192t2u4+ (576tw2+ 1728t2w+ 896t3)u3+ (816w4+ 4512w3t + 8816w2t2+ 6816wt3+ 1776t4)u2+ (2032t5+ 11488wt4 + 3264w5+ 14528w4t+ 26976w3t2+ 25152w2t3)u+ 50544w4t2
+ 56584w3t3+ 5136w6+ 24552w5t+ 1640t6+ 35784w2t4+ 11968wt5. It obviously follows thatp(u, v, w)≥0, i.e., inequality (2.12) holds.
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(ii)Whenu >0, w ≥v ≥0, settingw=v+tfort≥0, we get p(u, v, w) =(2u+ 10v)t5 + (10u2+ 40uv+ 102v2)t4
+ (156uv2+ 32u3+ 349v3+ 68u2v)t3
+ (24u4 + 188uv3+ 22u2v2+ 72u3v + 603v4)t2 +v2(783v3−72u3+ 224uv2−156u2v)t
+ 6v4(17u2 + 68uv+ 107v2)
=p1(u, v, t) +p2(u, v, t), where
p1(u, v, t) = (2u+ 10v)t5+ (10u2+ 40uv+ 102v2)t4
+ (156uv2+ 32u3+ 349v3+ 68u2v)t3 ≥0, and
(2.13) p2(u, v, t) = (24u4+ 188uv3+ 22u2v2 + 72u3v+ 603v4)t2 +v2(783v3−72u3+ 224uv2−156u2v)t
+ 6v4(17u2+ 68uv+ 107v2).
It is easy to see that 24u4 + 188uv3 + 22u2v2 + 72u3v + 603v4 > 0, and the discriminant of the quadratic function (2.13) with respect totis
(2.14) ∆(u, v) =−v4(935415v6+ 480144u3v3+ 1116096uv5
+ 803456u2v4+ 4608u6+ 196032u4v2+ 46080u5v)≤0.
This is to say thatp2(u, v, t)≥0.
Hence,P2(u, v, w)≥0. From the proof above, the required result (2.6) is proved.
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2.4. Remarks
Remark 1. By the same argument as above, we also prove the following inequalities conjectures [9,10,11] in the acute triangle
(2.15) X
m2ah2a ≥X m2ara2,
(2.16) X
sin8A≥X
cos8 A 2,
(2.17) X
(b−c)2 ≥X a b+c
2
(rb−rc)2, and
(2.18) X
(hb+hc −ha)3 ≥3mambmc.
Remark 2. The operations in this paper were performed using mathematical software Maple 9.0.
Proving Inequalities with Difference Substitution Yu-Dong Wu, Zhi-Hua Zhang
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3. Generalization of the Method
In fact, Difference Substitution can go even further. Now, we consider the following inequality [12]. In4ABC, ifmax (A, B, C)≤ 2π3 , then
(3.1) s2 ≥R2+ 10Rr+ 3r2.
Utilizing the known formulasR = abc4S, r = Ss andS = p
s(s−a)(s−b)(s−c), from (1.6), inequality (3.1) is equivalent to
(3.2) 3c2a2b2−a2bc3−a3bc2−a3b2c−a2b3c−ab2c3−ab3c2 −2b3c3
−2a3c3+c4b2−2a3b3+c5b+a4c2+b4c2+b5a+a5c+c5a
−a6+a4b2+a5b−b6−c6+a2b4+b5c+a2c4 ≥0, or
(3.3) F(x, y, z)
=−42x2y2z2+ 14y4zx+ 14xyz4+ 2xy2z3 + 2x2y3z+ 2xy3z2 + 14x4yz+ 2x3yz2+ 2x2yz3+ 2x3y2z−x4y2−x2z4
−2x3z3−x4z2−2x3y3−y4z2 −y4x2−2y3z3−y2z4 ≥0, wherex >0, y >0, z >0.
Since inequality (3.3) is symmetric with x, y, z, there is no harm in supposing thatx≤y≤z. From (1.2),F(x, y, z)in (3.3) is transformed into
F(x, y, z) =P(u, v, w) (3.4)
=(8u2+ 4uv+ 2uw−v2−vw)(8uvw2+ 12uv2w+ 4u2vw + 4v4+ 4u2v2+ 2uw3+ 8uv3+ 8v3w+ 7v2w2+ 3vw3) + 2w2(v+ 2u)2(v+ 2u+w)2 ≥0,
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and formax (A, B, C)≤ 2π3 andy = cosxdecreasing inx∈(0, π), we have b2+c2+bc−a2 =b2+c2− 1
2bccos2π 3 −a2 (3.5)
= 3x2+ 3(y+z)x−yz
= 8u2+ 4uv+ 2uw−v2−vw
≥b2+c2−1
2bccosA−a2 = 0.
SinceF(x, y, z) = P(u, v, w) ≥ 0foru > 0, v ≥ 0andw ≥ 0, inequality (3.1) is obtained.
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References
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[11] B.-Q. LIU, Private Communication, 2003.
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