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Transcript of Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities?...
![Page 1: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/1.jpg)
InequalitiesBasic tools and general techniques
Peng Shi
Department of MathematicsDuke University
September 30, 2009
![Page 2: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/2.jpg)
Inequalities?
Involves proving A ≥ B. Not only in math contests but widely usedin mathematical sciences.
Many tools available (see formula sheet)
Fundamental problem solving ideas:
1 Smoothing
2 Substitution
3 Clever manipulation
4 Bash (not recommended)
Due to the nature of the topic, there will be quite a mess of expressions,so don’t feel that you have to follow every line.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 2/18
![Page 3: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/3.jpg)
Inequalities?
Involves proving A ≥ B. Not only in math contests but widely usedin mathematical sciences.
Many tools available (see formula sheet)
Fundamental problem solving ideas:
1 Smoothing
2 Substitution
3 Clever manipulation
4 Bash (not recommended)
Due to the nature of the topic, there will be quite a mess of expressions,so don’t feel that you have to follow every line.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 2/18
![Page 4: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/4.jpg)
Inequalities?
Involves proving A ≥ B. Not only in math contests but widely usedin mathematical sciences.
Many tools available (see formula sheet)
Fundamental problem solving ideas:
1 Smoothing
2 Substitution
3 Clever manipulation
4 Bash (not recommended)
Due to the nature of the topic, there will be quite a mess of expressions,so don’t feel that you have to follow every line.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 2/18
![Page 5: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/5.jpg)
Inequalities?
Involves proving A ≥ B. Not only in math contests but widely usedin mathematical sciences.
Many tools available (see formula sheet)
Fundamental problem solving ideas:
1 Smoothing
2 Substitution
3 Clever manipulation
4 Bash (not recommended)
Due to the nature of the topic, there will be quite a mess of expressions,so don’t feel that you have to follow every line.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 2/18
![Page 6: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/6.jpg)
Inequalities?
Involves proving A ≥ B. Not only in math contests but widely usedin mathematical sciences.
Many tools available (see formula sheet)
Fundamental problem solving ideas:
1 Smoothing
2 Substitution
3 Clever manipulation
4 Bash (not recommended)
Due to the nature of the topic, there will be quite a mess of expressions,so don’t feel that you have to follow every line.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 2/18
![Page 7: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/7.jpg)
Inequalities?
Involves proving A ≥ B. Not only in math contests but widely usedin mathematical sciences.
Many tools available (see formula sheet)
Fundamental problem solving ideas:
1 Smoothing
2 Substitution
3 Clever manipulation
4 Bash (not recommended)
Due to the nature of the topic, there will be quite a mess of expressions,so don’t feel that you have to follow every line.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 2/18
![Page 8: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/8.jpg)
Inequalities?
Involves proving A ≥ B. Not only in math contests but widely usedin mathematical sciences.
Many tools available (see formula sheet)
Fundamental problem solving ideas:
1 Smoothing
2 Substitution
3 Clever manipulation
4 Bash (not recommended)
Due to the nature of the topic, there will be quite a mess of expressions,so don’t feel that you have to follow every line.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 2/18
![Page 9: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/9.jpg)
Inequalities?
Involves proving A ≥ B. Not only in math contests but widely usedin mathematical sciences.
Many tools available (see formula sheet)
Fundamental problem solving ideas:
1 Smoothing
2 Substitution
3 Clever manipulation
4 Bash (not recommended)
Due to the nature of the topic, there will be quite a mess of expressions,so don’t feel that you have to follow every line.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 2/18
![Page 10: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/10.jpg)
Inequalities?
Involves proving A ≥ B. Not only in math contests but widely usedin mathematical sciences.
Many tools available (see formula sheet)
Fundamental problem solving ideas:
1 Smoothing
2 Substitution
3 Clever manipulation
4 Bash (not recommended)
Due to the nature of the topic, there will be quite a mess of expressions,so don’t feel that you have to follow every line.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 2/18
![Page 11: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/11.jpg)
Basic tools
Theoremx2 ≥ 0 with equality iff x = 0.
Examplex2 + y 2 + z2 ≥ xy + yz + xz because (x − y)2 + (y − z)2 + (z − x)2 ≥ 0.
TheoremAM-GM If x1, · · · , xn are positive real numbers, then
x1 + · · ·+ xn
n≥ n√
x1 · · · xn
with equality iff x1 = x2 = · · · = xn.
Examplex3 + y 3 + z3 ≥ 3xyz
Peng Shi, Duke University Inequalities, Basic tools and general techniques 3/18
![Page 12: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/12.jpg)
Basic tools
Theoremx2 ≥ 0 with equality iff x = 0.
Examplex2 + y 2 + z2 ≥ xy + yz + xz because (x − y)2 + (y − z)2 + (z − x)2 ≥ 0.
TheoremAM-GM If x1, · · · , xn are positive real numbers, then
x1 + · · ·+ xn
n≥ n√
x1 · · · xn
with equality iff x1 = x2 = · · · = xn.
Examplex3 + y 3 + z3 ≥ 3xyz
Peng Shi, Duke University Inequalities, Basic tools and general techniques 3/18
![Page 13: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/13.jpg)
Basic tools
Theoremx2 ≥ 0 with equality iff x = 0.
Examplex2 + y 2 + z2 ≥ xy + yz + xz because (x − y)2 + (y − z)2 + (z − x)2 ≥ 0.
TheoremAM-GM If x1, · · · , xn are positive real numbers, then
x1 + · · ·+ xn
n≥ n√
x1 · · · xn
with equality iff x1 = x2 = · · · = xn.
Examplex3 + y 3 + z3 ≥ 3xyz
Peng Shi, Duke University Inequalities, Basic tools and general techniques 3/18
![Page 14: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/14.jpg)
Basic tools
Theoremx2 ≥ 0 with equality iff x = 0.
Examplex2 + y 2 + z2 ≥ xy + yz + xz because (x − y)2 + (y − z)2 + (z − x)2 ≥ 0.
TheoremAM-GM If x1, · · · , xn are positive real numbers, then
x1 + · · ·+ xn
n≥ n√
x1 · · · xn
with equality iff x1 = x2 = · · · = xn.
Examplex3 + y 3 + z3 ≥ 3xyz
Peng Shi, Duke University Inequalities, Basic tools and general techniques 3/18
![Page 15: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/15.jpg)
Basic tools
Theorem(Cauchy-Schwarz) For any real numbers a1, · · · , an, b1, · · · , bn,
(a21 + · · ·+ a2
n)(b21 + · · ·+ b2
n) ≥ (a1b1 + · · ·+ anbn)2
with equality iff the two sequences are proportional. (‖~a‖‖~b‖ ≥ ~a · ~b.)
Example
1
a+
1
b+
4
c+
16
d≥ 64
a + b + c + d
because
(a + b + c + d)(1
a+
1
b+
22
c+
42
d) ≥ (1 + 1 + 2 + 4)2
Peng Shi, Duke University Inequalities, Basic tools and general techniques 4/18
![Page 16: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/16.jpg)
Basic tools
Theorem(Cauchy-Schwarz) For any real numbers a1, · · · , an, b1, · · · , bn,
(a21 + · · ·+ a2
n)(b21 + · · ·+ b2
n) ≥ (a1b1 + · · ·+ anbn)2
with equality iff the two sequences are proportional. (‖~a‖‖~b‖ ≥ ~a · ~b.)
Example
1
a+
1
b+
4
c+
16
d≥ 64
a + b + c + d
because
(a + b + c + d)(1
a+
1
b+
22
c+
42
d) ≥ (1 + 1 + 2 + 4)2
Peng Shi, Duke University Inequalities, Basic tools and general techniques 4/18
![Page 17: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/17.jpg)
Basic tools
Theorem(Jensen) Let f be a convex function. Then for any x1, · · · , xn ∈ I andany non-negative reals w1, · · · ,wn,
∑i wi = 1
w1f (x1) + · · ·+ wnf (xn) ≥ f (w1x1 + · · ·+ wnxn)
If f is concave, then the inequality is flipped.
ExampleOne proof of the AM-GM inequality uses the fact that f (x) = log(x) isconcave, so
1
b(log x1 + · · ·+ log xn) ≤ log
x1 + · · ·+ xn
n
from which AM-GM follows by taking exponents of both sides.
For other tools, see the formula sheet.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 5/18
![Page 18: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/18.jpg)
Basic tools
Theorem(Jensen) Let f be a convex function. Then for any x1, · · · , xn ∈ I andany non-negative reals w1, · · · ,wn,
∑i wi = 1
w1f (x1) + · · ·+ wnf (xn) ≥ f (w1x1 + · · ·+ wnxn)
If f is concave, then the inequality is flipped.
ExampleOne proof of the AM-GM inequality uses the fact that f (x) = log(x) isconcave, so
1
b(log x1 + · · ·+ log xn) ≤ log
x1 + · · ·+ xn
n
from which AM-GM follows by taking exponents of both sides.
For other tools, see the formula sheet.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 5/18
![Page 19: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/19.jpg)
Basic tools
Theorem(Jensen) Let f be a convex function. Then for any x1, · · · , xn ∈ I andany non-negative reals w1, · · · ,wn,
∑i wi = 1
w1f (x1) + · · ·+ wnf (xn) ≥ f (w1x1 + · · ·+ wnxn)
If f is concave, then the inequality is flipped.
ExampleOne proof of the AM-GM inequality uses the fact that f (x) = log(x) isconcave, so
1
b(log x1 + · · ·+ log xn) ≤ log
x1 + · · ·+ xn
n
from which AM-GM follows by taking exponents of both sides.
For other tools, see the formula sheet.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 5/18
![Page 20: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/20.jpg)
Idea 1: Smoothing
By altering terms and arguing what happens, we can sometimes reduceproving A ≥ B in general to checking a canonical case.
For example, while fixing B, say that we can decrease A by moving termscloser, then it suffices to check the case when all terms are equal.
ExampleProve the AM-GM inequality(∑
i xi
n
)n
≥∏
i
xi
Proof.Let x̄ =
Pi xi
n . Say that xi < x̄ and xj > x̄ . Consider replacing (xi , xj) by(x̄ , 2x̄). Note that xixj ≤ x̄(2x̄ − xi ). Hence, this fixes LHS but increasesRHS. So for fixed LHS, the RHS is maximized when xi = x̄ ∀i .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 6/18
![Page 21: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/21.jpg)
Idea 1: Smoothing
By altering terms and arguing what happens, we can sometimes reduceproving A ≥ B in general to checking a canonical case.
For example, while fixing B, say that we can decrease A by moving termscloser, then it suffices to check the case when all terms are equal.
ExampleProve the AM-GM inequality(∑
i xi
n
)n
≥∏
i
xi
Proof.Let x̄ =
Pi xi
n . Say that xi < x̄ and xj > x̄ . Consider replacing (xi , xj) by(x̄ , 2x̄). Note that xixj ≤ x̄(2x̄ − xi ). Hence, this fixes LHS but increasesRHS. So for fixed LHS, the RHS is maximized when xi = x̄ ∀i .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 6/18
![Page 22: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/22.jpg)
Idea 1: Smoothing
By altering terms and arguing what happens, we can sometimes reduceproving A ≥ B in general to checking a canonical case.
For example, while fixing B, say that we can decrease A by moving termscloser, then it suffices to check the case when all terms are equal.
ExampleProve the AM-GM inequality(∑
i xi
n
)n
≥∏
i
xi
Proof.Let x̄ =
Pi xi
n . Say that xi < x̄ and xj > x̄ . Consider replacing (xi , xj) by(x̄ , 2x̄). Note that xixj ≤ x̄(2x̄ − xi ). Hence, this fixes LHS but increasesRHS. So for fixed LHS, the RHS is maximized when xi = x̄ ∀i .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 6/18
![Page 23: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/23.jpg)
Idea 1: Smoothing
By altering terms and arguing what happens, we can sometimes reduceproving A ≥ B in general to checking a canonical case.
For example, while fixing B, say that we can decrease A by moving termscloser, then it suffices to check the case when all terms are equal.
ExampleProve the AM-GM inequality(∑
i xi
n
)n
≥∏
i
xi
Proof.Let x̄ =
Pi xi
n . Say that xi < x̄ and xj > x̄ . Consider replacing (xi , xj) by(x̄ , 2x̄). Note that xixj ≤ x̄(2x̄ − xi ). Hence, this fixes LHS but increasesRHS. So for fixed LHS, the RHS is maximized when xi = x̄ ∀i .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 6/18
![Page 24: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/24.jpg)
Connection to convexity
Usually smoothing is equivalent to arguing about a convex function: forfixed
∑i xi and f convex,
∑ni f (xi ) is minimized when all xi ’s are equal.
Sometimes the function is not convex, in which case the argument needsto be more intricate.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 7/18
![Page 25: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/25.jpg)
Connection to convexity
Usually smoothing is equivalent to arguing about a convex function: forfixed
∑i xi and f convex,
∑ni f (xi ) is minimized when all xi ’s are equal.
Sometimes the function is not convex, in which case the argument needsto be more intricate.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 7/18
![Page 26: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/26.jpg)
Another example
ExampleIf a, b, c, d, e are real numbers such that
a + b + c + d + e = 8 (1)a2 + b2 + c2 + d2 + e2 = 16 (2)
What is the largest possible value of e?
Solution.Relax the second constraint to a2 + b2 + c2 + d2 + e2 ≤ 16 (2*). Call a 5-tuple valid if it satisfies(1) and (2*). We seek the valid 5-tuple with the largest possible e. For any valid (a, b, c, d, e),
setting k = (a+b+c+d)4 , (k, k, k, k, e) is also valid (smoothing). So we just need to find the largest
e s.t. for some k4k + e = 8
4k2 + e ≤ 16
=⇒„
8− e
4
«2
= k2 ≤16− e2
4
=⇒ (5e − 16)e ≤ 0 ⇒ 0 ≤ e ≤16
5
Conversely, ( 65 ,
65 ,
65 ,
65 ,
165 ) satisfies the original equation, so the largest possible value is 16
5 .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 8/18
![Page 27: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/27.jpg)
Another example
ExampleIf a, b, c, d, e are real numbers such that
a + b + c + d + e = 8 (1)a2 + b2 + c2 + d2 + e2 = 16 (2)
What is the largest possible value of e?
Solution.Relax the second constraint to a2 + b2 + c2 + d2 + e2 ≤ 16 (2*). Call a 5-tuple valid if it satisfies(1) and (2*). We seek the valid 5-tuple with the largest possible e.
For any valid (a, b, c, d, e),
setting k = (a+b+c+d)4 , (k, k, k, k, e) is also valid (smoothing). So we just need to find the largest
e s.t. for some k4k + e = 8
4k2 + e ≤ 16
=⇒„
8− e
4
«2
= k2 ≤16− e2
4
=⇒ (5e − 16)e ≤ 0 ⇒ 0 ≤ e ≤16
5
Conversely, ( 65 ,
65 ,
65 ,
65 ,
165 ) satisfies the original equation, so the largest possible value is 16
5 .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 8/18
![Page 28: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/28.jpg)
Another example
ExampleIf a, b, c, d, e are real numbers such that
a + b + c + d + e = 8 (1)a2 + b2 + c2 + d2 + e2 = 16 (2)
What is the largest possible value of e?
Solution.Relax the second constraint to a2 + b2 + c2 + d2 + e2 ≤ 16 (2*). Call a 5-tuple valid if it satisfies(1) and (2*). We seek the valid 5-tuple with the largest possible e. For any valid (a, b, c, d, e),
setting k = (a+b+c+d)4 , (k, k, k, k, e) is also valid (smoothing). So we just need to find the largest
e s.t. for some k4k + e = 8
4k2 + e ≤ 16
=⇒„
8− e
4
«2
= k2 ≤16− e2
4
=⇒ (5e − 16)e ≤ 0 ⇒ 0 ≤ e ≤16
5
Conversely, ( 65 ,
65 ,
65 ,
65 ,
165 ) satisfies the original equation, so the largest possible value is 16
5 .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 8/18
![Page 29: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/29.jpg)
Another example
ExampleIf a, b, c, d, e are real numbers such that
a + b + c + d + e = 8 (1)a2 + b2 + c2 + d2 + e2 = 16 (2)
What is the largest possible value of e?
Solution.Relax the second constraint to a2 + b2 + c2 + d2 + e2 ≤ 16 (2*). Call a 5-tuple valid if it satisfies(1) and (2*). We seek the valid 5-tuple with the largest possible e. For any valid (a, b, c, d, e),
setting k = (a+b+c+d)4 , (k, k, k, k, e) is also valid (smoothing). So we just need to find the largest
e s.t. for some k4k + e = 8
4k2 + e ≤ 16
=⇒„
8− e
4
«2
= k2 ≤16− e2
4
=⇒ (5e − 16)e ≤ 0 ⇒ 0 ≤ e ≤16
5
Conversely, ( 65 ,
65 ,
65 ,
65 ,
165 ) satisfies the original equation, so the largest possible value is 16
5 .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 8/18
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Idea 2: Substitution
Use substitutions to transform the given inequality into a simpler or“nicer” form.
Commonly used subsitutions
Simple manipluations
Triangle related substitutions
Homogenization
Peng Shi, Duke University Inequalities, Basic tools and general techniques 9/18
![Page 31: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/31.jpg)
Idea 2: Substitution
Use substitutions to transform the given inequality into a simpler or“nicer” form.Commonly used subsitutions
Simple manipluations
Triangle related substitutions
Homogenization
Peng Shi, Duke University Inequalities, Basic tools and general techniques 9/18
![Page 32: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/32.jpg)
Idea 2: Substitution
Use substitutions to transform the given inequality into a simpler or“nicer” form.Commonly used subsitutions
Simple manipluations
Triangle related substitutions (cool)
Homogenization (will show later)
Peng Shi, Duke University Inequalities, Basic tools and general techniques 9/18
![Page 33: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/33.jpg)
Triangle related substitutions
If a, b, c are sides of a triangle, then let x = (b + c − a)/2,y = (a + c − b)/2, z = (a + b− c)/2, so that a = y + z , b = x + z ,c = x + y , and x , y , z are arbitrary positive real numbers.
If a2 + b2 = 1, let a = cos θ, b = sin θ.
If a + b + c = abc, a, b, c > 0, let a = tan A, b = tan B, c = tan C ,where A,B,C are angles in a triangle.
If a2 + b2 + c2 + 2abc = 1, let a = cos A, b = cos B, c = cos C ,where A,B,C are angles in a triangle.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 10/18
![Page 34: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/34.jpg)
Triangle related substitutions
If a, b, c are sides of a triangle, then let x = (b + c − a)/2,y = (a + c − b)/2, z = (a + b− c)/2, so that a = y + z , b = x + z ,c = x + y , and x , y , z are arbitrary positive real numbers.
If a2 + b2 = 1, let a = cos θ, b = sin θ.
If a + b + c = abc, a, b, c > 0, let a = tan A, b = tan B, c = tan C ,where A,B,C are angles in a triangle.
If a2 + b2 + c2 + 2abc = 1, let a = cos A, b = cos B, c = cos C ,where A,B,C are angles in a triangle.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 10/18
![Page 35: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/35.jpg)
Triangle related substitutions
If a, b, c are sides of a triangle, then let x = (b + c − a)/2,y = (a + c − b)/2, z = (a + b− c)/2, so that a = y + z , b = x + z ,c = x + y , and x , y , z are arbitrary positive real numbers.
If a2 + b2 = 1, let a = cos θ, b = sin θ.
If a + b + c = abc, a, b, c > 0, let a = tan A, b = tan B, c = tan C ,where A,B,C are angles in a triangle.
If a2 + b2 + c2 + 2abc = 1, let a = cos A, b = cos B, c = cos C ,where A,B,C are angles in a triangle.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 10/18
![Page 36: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/36.jpg)
Triangle related substitutions
If a, b, c are sides of a triangle, then let x = (b + c − a)/2,y = (a + c − b)/2, z = (a + b− c)/2, so that a = y + z , b = x + z ,c = x + y , and x , y , z are arbitrary positive real numbers.
If a2 + b2 = 1, let a = cos θ, b = sin θ.
If a + b + c = abc, a, b, c > 0, let a = tan A, b = tan B, c = tan C ,where A,B,C are angles in a triangle.
If a2 + b2 + c2 + 2abc = 1, let a = cos A, b = cos B, c = cos C ,where A,B,C are angles in a triangle.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 10/18
![Page 37: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/37.jpg)
Slick example
ExampleFor positive real numbers a, b, c with a + b + c = abc, show that
1√1 + a2
+1√
1 + b2+
1√1 + c2
≤ 3
2
Proof.WLOG, let a = tan A, b = tan B, c = tan C , where A,B,C are angles ina triangle. The inequality is equivalent to
cos A + cos B + cos C ≤ 3
2
But cos A + cos B + cos C = 1 + 4 sin A2 sin B
2 sin C2 , so inequality follows
from f (x) = log sin x being concave, where 0 ≤ x ≤ π2 .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 11/18
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Clever Manipulation
Manipulate the algebra in a possibly strange way and magically yield theresult.
Guidelines:
Be creative; try random things
Use equality case to help you
Envision what kind of expressions you need and try to manipulatethe given into a similar form
Can only learn this through experience.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 12/18
![Page 39: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/39.jpg)
Clever Manipulation
Manipulate the algebra in a possibly strange way and magically yield theresult.Guidelines:
Be creative; try random things
Use equality case to help you
Envision what kind of expressions you need and try to manipulatethe given into a similar form
Can only learn this through experience.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 12/18
![Page 40: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/40.jpg)
Clever Manipulation
Manipulate the algebra in a possibly strange way and magically yield theresult.Guidelines:
Be creative; try random things
Use equality case to help you
Envision what kind of expressions you need and try to manipulatethe given into a similar form
Can only learn this through experience.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 12/18
![Page 41: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/41.jpg)
Clever Manipulation
Manipulate the algebra in a possibly strange way and magically yield theresult.Guidelines:
Be creative; try random things
Use equality case to help you
Envision what kind of expressions you need and try to manipulatethe given into a similar form
Can only learn this through experience.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 12/18
![Page 42: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/42.jpg)
Clever Manipulation
Manipulate the algebra in a possibly strange way and magically yield theresult.Guidelines:
Be creative; try random things
Use equality case to help you
Envision what kind of expressions you need and try to manipulatethe given into a similar form
Can only learn this through experience.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 12/18
![Page 43: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/43.jpg)
Clever Manipulation
Manipulate the algebra in a possibly strange way and magically yield theresult.Guidelines:
Be creative; try random things
Use equality case to help you
Envision what kind of expressions you need and try to manipulatethe given into a similar form
Can only learn this through experience.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 12/18
![Page 44: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/44.jpg)
How did they come up with that?Example(IMO 2005 P3) Let x , y , z be three positive reals such that xyz ≥ 1. Prove that
x5 − x2
x5 + y 2 + z2+
y 5 − y 2
x2 + y 5 + z2+
z5 − z2
x2 + y 2 + z5≥ 0
Solution.By Cauchy-Schwarz
(x5 + y 2 + z2)(1
x+ y 2 + z2) ≥ (x2 + y 2 + z2)2
So1x
+ y 2 + z2
x2 + y 2 + z2≥ x2 + y 2 + z2
x5 + y 2 + z2
=⇒ yz − x2
x2 + y 2 + z2+ 1 ≥ x2 − x5
x5 + y 2 + z2+ 1
=⇒ LHS ≥ x2 + y 2 + z2 − xy − yz − xz
x2 + y 2 + z2≥ 0
Peng Shi, Duke University Inequalities, Basic tools and general techniques 13/18
![Page 45: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/45.jpg)
How did they come up with that?Example(IMO 2005 P3) Let x , y , z be three positive reals such that xyz ≥ 1. Prove that
x5 − x2
x5 + y 2 + z2+
y 5 − y 2
x2 + y 5 + z2+
z5 − z2
x2 + y 2 + z5≥ 0
Solution.By Cauchy-Schwarz
(x5 + y 2 + z2)(1
x+ y 2 + z2) ≥ (x2 + y 2 + z2)2
So
1x
+ y 2 + z2
x2 + y 2 + z2≥ x2 + y 2 + z2
x5 + y 2 + z2
=⇒ yz − x2
x2 + y 2 + z2+ 1 ≥ x2 − x5
x5 + y 2 + z2+ 1
=⇒ LHS ≥ x2 + y 2 + z2 − xy − yz − xz
x2 + y 2 + z2≥ 0
Peng Shi, Duke University Inequalities, Basic tools and general techniques 13/18
![Page 46: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/46.jpg)
How did they come up with that?Example(IMO 2005 P3) Let x , y , z be three positive reals such that xyz ≥ 1. Prove that
x5 − x2
x5 + y 2 + z2+
y 5 − y 2
x2 + y 5 + z2+
z5 − z2
x2 + y 2 + z5≥ 0
Solution.By Cauchy-Schwarz
(x5 + y 2 + z2)(1
x+ y 2 + z2) ≥ (x2 + y 2 + z2)2
So1x
+ y 2 + z2
x2 + y 2 + z2≥ x2 + y 2 + z2
x5 + y 2 + z2
=⇒ yz − x2
x2 + y 2 + z2+ 1 ≥ x2 − x5
x5 + y 2 + z2+ 1
=⇒ LHS ≥ x2 + y 2 + z2 − xy − yz − xz
x2 + y 2 + z2≥ 0
Peng Shi, Duke University Inequalities, Basic tools and general techniques 13/18
![Page 47: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/47.jpg)
How did they come up with that?Example(IMO 2005 P3) Let x , y , z be three positive reals such that xyz ≥ 1. Prove that
x5 − x2
x5 + y 2 + z2+
y 5 − y 2
x2 + y 5 + z2+
z5 − z2
x2 + y 2 + z5≥ 0
Solution.By Cauchy-Schwarz
(x5 + y 2 + z2)(1
x+ y 2 + z2) ≥ (x2 + y 2 + z2)2
So1x
+ y 2 + z2
x2 + y 2 + z2≥ x2 + y 2 + z2
x5 + y 2 + z2
=⇒ yz − x2
x2 + y 2 + z2+ 1 ≥ x2 − x5
x5 + y 2 + z2+ 1
=⇒ LHS ≥ x2 + y 2 + z2 − xy − yz − xz
x2 + y 2 + z2≥ 0
Peng Shi, Duke University Inequalities, Basic tools and general techniques 13/18
![Page 48: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/48.jpg)
How did they come up with that?Example(IMO 2005 P3) Let x , y , z be three positive reals such that xyz ≥ 1. Prove that
x5 − x2
x5 + y 2 + z2+
y 5 − y 2
x2 + y 5 + z2+
z5 − z2
x2 + y 2 + z5≥ 0
Solution.By Cauchy-Schwarz
(x5 + y 2 + z2)(1
x+ y 2 + z2) ≥ (x2 + y 2 + z2)2
So1x
+ y 2 + z2
x2 + y 2 + z2≥ x2 + y 2 + z2
x5 + y 2 + z2
=⇒ yz − x2
x2 + y 2 + z2+ 1 ≥ x2 − x5
x5 + y 2 + z2+ 1
=⇒ LHS ≥ x2 + y 2 + z2 − xy − yz − xz
x2 + y 2 + z2≥ 0
Peng Shi, Duke University Inequalities, Basic tools and general techniques 13/18
![Page 49: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/49.jpg)
Idea 4: Bash (for symmetric inequalities)
Not really an idea, but what you do when you run out of ideas and feellike an algebraic workout.
Preliminaries:
Symmetric notation: i.e.
(2, 1, 1) =∑sym
x2yz = x2yz + x2zy + y 2xz + y 2zx + z2xy + z2yx
Homogenization: i.e. If xyz = 1, then
x5 − x2
x5 + y 2 + z2=
x5 − x3yz
x5 + xy 3z + xyz3
Equivalently one can subsitute x = bca , y = ac
b , z = abc .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 14/18
![Page 50: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/50.jpg)
Idea 4: Bash (for symmetric inequalities)
Not really an idea, but what you do when you run out of ideas and feellike an algebraic workout.
Preliminaries:
Symmetric notation: i.e.
(2, 1, 1) =∑sym
x2yz = x2yz + x2zy + y 2xz + y 2zx + z2xy + z2yx
Homogenization: i.e. If xyz = 1, then
x5 − x2
x5 + y 2 + z2=
x5 − x3yz
x5 + xy 3z + xyz3
Equivalently one can subsitute x = bca , y = ac
b , z = abc .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 14/18
![Page 51: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/51.jpg)
Idea 4: Bash (for symmetric inequalities)
Not really an idea, but what you do when you run out of ideas and feellike an algebraic workout.
Preliminaries:
Symmetric notation: i.e.
(2, 1, 1) =∑sym
x2yz = x2yz + x2zy + y 2xz + y 2zx + z2xy + z2yx
Homogenization: i.e. If xyz = 1, then
x5 − x2
x5 + y 2 + z2=
x5 − x3yz
x5 + xy 3z + xyz3
Equivalently one can subsitute x = bca , y = ac
b , z = abc .
Peng Shi, Duke University Inequalities, Basic tools and general techniques 14/18
![Page 52: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/52.jpg)
Ultimate Bash ToolboxDefinition(Majorization) Sequence x1, · · · , xn is said to majorize sequence y1, · · · , yn if
x1 ≥ y1
x1 + x2 ≥ y1 + y2
· · · · · ·Pn−1i=1 xi ≥
Pn−1i=1 yiPn
i=1 xi =Pn
i=1 yi
Theorem(Muirhead) Suppose the sequence a1, · · · , an majorizes the sequenceb1, · · · , bn. Then for any positive reals x1, · · · , xn,X
sym
xa11 xa2
2 · · · xann ≥
Xsym
xb11 xb2
2 · · · xbnn
where the sums are taken over all permutations of the n variables.
ExamplePsym x4y ≥
Psym x3yz (In simplified notation, (4, 1, 0) ≥ (3, 1, 1))
Peng Shi, Duke University Inequalities, Basic tools and general techniques 15/18
![Page 53: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/53.jpg)
Ultimate Bash ToolboxDefinition(Majorization) Sequence x1, · · · , xn is said to majorize sequence y1, · · · , yn if
x1 ≥ y1
x1 + x2 ≥ y1 + y2
· · · · · ·Pn−1i=1 xi ≥
Pn−1i=1 yiPn
i=1 xi =Pn
i=1 yi
Theorem(Muirhead) Suppose the sequence a1, · · · , an majorizes the sequenceb1, · · · , bn. Then for any positive reals x1, · · · , xn,X
sym
xa11 xa2
2 · · · xann ≥
Xsym
xb11 xb2
2 · · · xbnn
where the sums are taken over all permutations of the n variables.
ExamplePsym x4y ≥
Psym x3yz (In simplified notation, (4, 1, 0) ≥ (3, 1, 1))
Peng Shi, Duke University Inequalities, Basic tools and general techniques 15/18
![Page 54: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/54.jpg)
Ultimate Bash ToolboxDefinition(Majorization) Sequence x1, · · · , xn is said to majorize sequence y1, · · · , yn if
x1 ≥ y1
x1 + x2 ≥ y1 + y2
· · · · · ·Pn−1i=1 xi ≥
Pn−1i=1 yiPn
i=1 xi =Pn
i=1 yi
Theorem(Muirhead) Suppose the sequence a1, · · · , an majorizes the sequenceb1, · · · , bn. Then for any positive reals x1, · · · , xn,X
sym
xa11 xa2
2 · · · xann ≥
Xsym
xb11 xb2
2 · · · xbnn
where the sums are taken over all permutations of the n variables.
ExamplePsym x4y ≥
Psym x3yz (In simplified notation, (4, 1, 0) ≥ (3, 1, 1))
Peng Shi, Duke University Inequalities, Basic tools and general techniques 15/18
![Page 55: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/55.jpg)
Ultimate Bash Toolbox
Theorem(Schur) Let a, b, c be nonnegative reals and r > 0. Then
ar (a− b)(a− c) + br (b − c)(b − a) + c r (c − a)(c − b) ≥ 0
with equality iff a = b = c or some two of a, b, c are equal and the otheris 0. (
∑sym ar+2 +
∑sym ar bc ≥ 2
∑sym ar+1b)
Example ∑sym
x3 +∑sym
xyz ≥ 2∑sym
x2y
Equivalently
2(x3 + y 3 + z3 + 3xyz) ≥ 2(x2(y + z) + y 2(x + z) + z2(x + y))
Peng Shi, Duke University Inequalities, Basic tools and general techniques 16/18
![Page 56: Inequalities - Basic tools and general techniquespengshi/math149/inequalities.pdf · Inequalities? Involves proving A B. Not only in math contests but widely used in mathematical](https://reader031.fdocuments.net/reader031/viewer/2022030400/5a72c0a77f8b9ac0538df8dd/html5/thumbnails/56.jpg)
Ultimate Bash Toolbox
Theorem(Schur) Let a, b, c be nonnegative reals and r > 0. Then
ar (a− b)(a− c) + br (b − c)(b − a) + c r (c − a)(c − b) ≥ 0
with equality iff a = b = c or some two of a, b, c are equal and the otheris 0. (
∑sym ar+2 +
∑sym ar bc ≥ 2
∑sym ar+1b)
Example ∑sym
x3 +∑sym
xyz ≥ 2∑sym
x2y
Equivalently
2(x3 + y 3 + z3 + 3xyz) ≥ 2(x2(y + z) + y 2(x + z) + z2(x + y))
Peng Shi, Duke University Inequalities, Basic tools and general techniques 16/18
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The Idiot’s Guide to Symmetric Inequalities
1 Homogenize
2 Multiply out all denomiators, expand, and rewrite using symmetricnotation.
3 Apply AM-GM, Muirhead, and Schurs.
Peng Shi, Duke University Inequalities, Basic tools and general techniques 17/18
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Example of BashExample(IMO 2005 P3) Let x, y , z be three positive reals such that xyz ≥ 1. Prove that
x5 − x2
x5 + y2 + z2+
y5 − y2
x2 + y5 + z2+
z5 − z2
x2 + y2 + z5≥ 0
Solution.Homogenizing and rearranging, it suffices to show
3 ≥ (x3yz + xy3z + xyz3)
„1
x5 + xy3z + xyz3+
1
x3yz + y5 + xyz3+
1
x3yz + xy3z + z5
«Multiply out and using symmetric notation, this is equivalent to
Xsym
x10yz + 4Xsym
x7y5 +Xsym
x6y3z3 ≥ 2Xsym
x6y5z +Xsym
x8y2z2 +Xsym
x5y5z2 +Xsym
x6y4z2
Which follows from(10, 1, 1) + (6, 3, 3) ≥ 2(8, 2, 2)
(8, 2, 2) ≥ (6, 4, 2)2(7, 5, 0) ≥ 2(6, 5, 1)
(7, 5, 0) ≥ (5, 5, 2)(7, 5, 0) ≥ (6, 4, 2)
Peng Shi, Duke University Inequalities, Basic tools and general techniques 18/18
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Example of BashExample(IMO 2005 P3) Let x, y , z be three positive reals such that xyz ≥ 1. Prove that
x5 − x2
x5 + y2 + z2+
y5 − y2
x2 + y5 + z2+
z5 − z2
x2 + y2 + z5≥ 0
Solution.Homogenizing and rearranging, it suffices to show
3 ≥ (x3yz + xy3z + xyz3)
„1
x5 + xy3z + xyz3+
1
x3yz + y5 + xyz3+
1
x3yz + xy3z + z5
«Multiply out and using symmetric notation, this is equivalent to
Xsym
x10yz + 4Xsym
x7y5 +Xsym
x6y3z3 ≥ 2Xsym
x6y5z +Xsym
x8y2z2 +Xsym
x5y5z2 +Xsym
x6y4z2
Which follows from(10, 1, 1) + (6, 3, 3) ≥ 2(8, 2, 2)
(8, 2, 2) ≥ (6, 4, 2)2(7, 5, 0) ≥ 2(6, 5, 1)
(7, 5, 0) ≥ (5, 5, 2)(7, 5, 0) ≥ (6, 4, 2)
Peng Shi, Duke University Inequalities, Basic tools and general techniques 18/18