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inequality $(a+b+c+d)(a^3+b^3+c^3+d^3) > (a^2+b^2+c^2+d^2)^2$

1

$begingroup$

I am trying to prove the inequality:

$$(a+b+c+d)(a^3+b^3+c^3+d^3) > (a^2+b^2+c^2+d^2)^2$$ given $a,b,c,d$ are positive and unequal.

starting from LHS

since AM of mth power > mth power of AM

$$(a^3+b^3+c^3+d^3)/4 > ((a+b+c+d)/4)^3$$

multiplying both sides by $(a+b+c+d)$

$$(a+b+c+d)(a^3+b^3+c^3+d^3)/4 > (a+b+c+d)^4/(4^3)$$

$implies$
$$(a+b+c+d)(a^3+b^3+c^3+d^3) > (a+b+c+d)^4/16$$

now taking expression on the RHS and using AM of mth power > mth power of AM

$$(a^2+b^2+c^2+d^2)/4 > ((a+b+c+d)/4)^2$$

squaring both sides

$$(a^2+b^2+c^2+d^2)^2/16 > (a+b+c+d)^4/(4^4)$$

$$implies (a^2+b^2+c^2+d^2)^2 > ((a+b+c+d)^4)/16$$

Now I have proved that both LHS and RHS are greater than $$((a+b+c+d)^4)/16$$

still unable to prove LHS > RHS. Please help with this.

inequality

share|cite|improve this question

edited Apr 2 at 17:38

Madavan Viswanathan

asked Apr 2 at 17:30

Madavan ViswanathanMadavan Viswanathan

447

$endgroup$

add a comment | 

1

$begingroup$

I am trying to prove the inequality:

$$(a+b+c+d)(a^3+b^3+c^3+d^3) > (a^2+b^2+c^2+d^2)^2$$ given $a,b,c,d$ are positive and unequal.

starting from LHS

since AM of mth power > mth power of AM

$$(a^3+b^3+c^3+d^3)/4 > ((a+b+c+d)/4)^3$$

multiplying both sides by $(a+b+c+d)$

$$(a+b+c+d)(a^3+b^3+c^3+d^3)/4 > (a+b+c+d)^4/(4^3)$$

$implies$
$$(a+b+c+d)(a^3+b^3+c^3+d^3) > (a+b+c+d)^4/16$$

now taking expression on the RHS and using AM of mth power > mth power of AM

$$(a^2+b^2+c^2+d^2)/4 > ((a+b+c+d)/4)^2$$

squaring both sides

$$(a^2+b^2+c^2+d^2)^2/16 > (a+b+c+d)^4/(4^4)$$

$$implies (a^2+b^2+c^2+d^2)^2 > ((a+b+c+d)^4)/16$$

Now I have proved that both LHS and RHS are greater than $$((a+b+c+d)^4)/16$$

still unable to prove LHS > RHS. Please help with this.

inequality

share|cite|improve this question

edited Apr 2 at 17:38

Madavan Viswanathan

asked Apr 2 at 17:30

Madavan ViswanathanMadavan Viswanathan

447

$endgroup$

add a comment | 

$begingroup$

I am trying to prove the inequality:

$$(a+b+c+d)(a^3+b^3+c^3+d^3) > (a^2+b^2+c^2+d^2)^2$$ given $a,b,c,d$ are positive and unequal.

starting from LHS

since AM of mth power > mth power of AM

$$(a^3+b^3+c^3+d^3)/4 > ((a+b+c+d)/4)^3$$

multiplying both sides by $(a+b+c+d)$

$$(a+b+c+d)(a^3+b^3+c^3+d^3)/4 > (a+b+c+d)^4/(4^3)$$

$implies$
$$(a+b+c+d)(a^3+b^3+c^3+d^3) > (a+b+c+d)^4/16$$

now taking expression on the RHS and using AM of mth power > mth power of AM

$$(a^2+b^2+c^2+d^2)/4 > ((a+b+c+d)/4)^2$$

squaring both sides

$$(a^2+b^2+c^2+d^2)^2/16 > (a+b+c+d)^4/(4^4)$$

$$implies (a^2+b^2+c^2+d^2)^2 > ((a+b+c+d)^4)/16$$

Now I have proved that both LHS and RHS are greater than $$((a+b+c+d)^4)/16$$

still unable to prove LHS > RHS. Please help with this.

inequality

share|cite|improve this question

edited Apr 2 at 17:38

Madavan Viswanathan

asked Apr 2 at 17:30

Madavan ViswanathanMadavan Viswanathan

447

$endgroup$

share|cite|improve this question

edited Apr 2 at 17:38

Madavan Viswanathan

asked Apr 2 at 17:30

Madavan ViswanathanMadavan Viswanathan

447

share|cite|improve this question

edited Apr 2 at 17:38

Madavan Viswanathan

asked Apr 2 at 17:30

Madavan ViswanathanMadavan Viswanathan

447

asked Apr 2 at 17:30

Madavan ViswanathanMadavan Viswanathan

447

asked Apr 2 at 17:30

Madavan ViswanathanMadavan Viswanathan

447

2 Answers
2

active

oldest

votes

3

$begingroup$

Hint: Use Cauchy inequality: $$(x_1+x_2+...+x_n)(y_1+y_2+...+y_n)geq (sqrtx_1y_1+...+sqrtx_ny_n)^2$$

share|cite|improve this answer

answered Apr 2 at 17:42

Maria MazurMaria Mazur

50.5k1361126

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks very much , that nailed the problem is a single statement , amazing.
  $endgroup$
  – Madavan Viswanathan
  Apr 2 at 17:48
  

  
  
  
  

add a comment | 

1

$begingroup$

Hint: You will get the left-hand side minus the right-hand side:
$$a^3b+a^3c+a^3d-2,a^2b^2-2,a^2c^2-2,a^
2d^2+ab^3+ac^3+ad^3+b^3c+b^3d-2,b^2c^
2-2,b^2d^2+bc^3+bd^3+c^3d-2,c^2d^2+cd
^3
$$ Now you need to combine like terms:
$$a^3b-2a^2b^2+ab^2=ab(a^2-2ab+b^2)$$ and so on.

share|cite|improve this answer

answered Apr 2 at 17:40

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79.6k42867

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks very much , got it.
  $endgroup$
  – Madavan Viswanathan
  Apr 2 at 17:47
  

  
  
  
  

add a comment | 

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3

$begingroup$

Hint: Use Cauchy inequality: $$(x_1+x_2+...+x_n)(y_1+y_2+...+y_n)geq (sqrtx_1y_1+...+sqrtx_ny_n)^2$$

share|cite|improve this answer

answered Apr 2 at 17:42

Maria MazurMaria Mazur

50.5k1361126

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks very much , that nailed the problem is a single statement , amazing.
  $endgroup$
  – Madavan Viswanathan
  Apr 2 at 17:48
  

  
  
  
  

add a comment | 

3

$begingroup$

Hint: Use Cauchy inequality: $$(x_1+x_2+...+x_n)(y_1+y_2+...+y_n)geq (sqrtx_1y_1+...+sqrtx_ny_n)^2$$

share|cite|improve this answer

answered Apr 2 at 17:42

Maria MazurMaria Mazur

50.5k1361126

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks very much , that nailed the problem is a single statement , amazing.
  $endgroup$
  – Madavan Viswanathan
  Apr 2 at 17:48
  

  
  
  
  

add a comment | 

$begingroup$

Hint: Use Cauchy inequality: $$(x_1+x_2+...+x_n)(y_1+y_2+...+y_n)geq (sqrtx_1y_1+...+sqrtx_ny_n)^2$$

share|cite|improve this answer

answered Apr 2 at 17:42

Maria MazurMaria Mazur

50.5k1361126

$endgroup$

share|cite|improve this answer

answered Apr 2 at 17:42

Maria MazurMaria Mazur

50.5k1361126

answered Apr 2 at 17:42

Maria MazurMaria Mazur

50.5k1361126

answered Apr 2 at 17:42

Maria MazurMaria Mazur

50.5k1361126

1

$begingroup$

Hint: You will get the left-hand side minus the right-hand side:
$$a^3b+a^3c+a^3d-2,a^2b^2-2,a^2c^2-2,a^
2d^2+ab^3+ac^3+ad^3+b^3c+b^3d-2,b^2c^
2-2,b^2d^2+bc^3+bd^3+c^3d-2,c^2d^2+cd
^3
$$ Now you need to combine like terms:
$$a^3b-2a^2b^2+ab^2=ab(a^2-2ab+b^2)$$ and so on.

share|cite|improve this answer

answered Apr 2 at 17:40

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79.6k42867

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks very much , got it.
  $endgroup$
  – Madavan Viswanathan
  Apr 2 at 17:47
  

  
  
  
  

add a comment | 

1

$begingroup$

Hint: You will get the left-hand side minus the right-hand side:
$$a^3b+a^3c+a^3d-2,a^2b^2-2,a^2c^2-2,a^
2d^2+ab^3+ac^3+ad^3+b^3c+b^3d-2,b^2c^
2-2,b^2d^2+bc^3+bd^3+c^3d-2,c^2d^2+cd
^3
$$ Now you need to combine like terms:
$$a^3b-2a^2b^2+ab^2=ab(a^2-2ab+b^2)$$ and so on.

share|cite|improve this answer

answered Apr 2 at 17:40

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79.6k42867

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks very much , got it.
  $endgroup$
  – Madavan Viswanathan
  Apr 2 at 17:47
  

  
  
  
  

add a comment | 

$begingroup$

Hint: You will get the left-hand side minus the right-hand side:
$$a^3b+a^3c+a^3d-2,a^2b^2-2,a^2c^2-2,a^
2d^2+ab^3+ac^3+ad^3+b^3c+b^3d-2,b^2c^
2-2,b^2d^2+bc^3+bd^3+c^3d-2,c^2d^2+cd
^3
$$ Now you need to combine like terms:
$$a^3b-2a^2b^2+ab^2=ab(a^2-2ab+b^2)$$ and so on.

share|cite|improve this answer

answered Apr 2 at 17:40

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79.6k42867

$endgroup$

share|cite|improve this answer

answered Apr 2 at 17:40

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79.6k42867

answered Apr 2 at 17:40

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79.6k42867

answered Apr 2 at 17:40

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

79.6k42867