MHB How can positive numbers be used to prove an inequality challenge?

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The inequality challenge asserts that for positive numbers a, b, and c, the expression 8(a^3 + b^3 + c^3) is greater than or equal to the sum of the cubes of their pairwise sums, specifically (a+b)^3 + (a+c)^3 + (b+c)^3. Several inequalities are referenced, including a^3 + b^3 ≥ a^2b + b^2a, b^3 + c^3 ≥ b^2c + c^2b, and a^3 + c^3 ≥ a^2c + c^2a, which collectively support the main inequality. The discussion highlights contributions from participants who successfully engaged with the problem. Overall, the thread emphasizes the application of algebraic identities and inequalities to prove the stated inequality.
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If $a,\,b,\,c$ are positive numbers, show that $8(a^3+b^3+c^3)\ge (a+b)^3+(a+c)^3+(b+c)^3$.
 
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anemone said:
If $a,\,b,\,c$ are positive numbers, show that $8(a^3+b^3+c^3)\ge (a+b)^3+(a+c)^3+(b+c)^3----(1)$.
if (1) is true then expanding and simplifying :
$8(a^3+b^3+c^3)\ge 2a^3+2b^3+2c^3+3a^2b+3ab^2+3b^2c+3bc^2+3a^2c+3ac^2$
$2(a^3+b^3+c^3)\ge a^2b+ab^2+b^2c+bc^2+a^2c+ac^2
=ab(a+b)+bc(b+c)+ca(a+c)$
$=a^2(b+c)+b^2(c+a)+c^2(a+b)---(2)$
now we only have to prove (2) ,I will have a rest ,hope someone can finish it
 
we have $a^3 + b^3 – a^2 b – b^2 a$
= $a^3 - a^2 b – b^2 a + b^3$
= $a^2(a-b)- b^2(a-b) = (a^2-b^2) (a-b) = (a+b)(a-b)^2> = 0$

Hence
$a^3 + b^3 > = a^2 b + b^2 a$

Multiply by 3 and add $a^3 + b^3$ on both sides

$4(a^3 + b^3) >= a^3 + b^3 + 3(a^2 b + b^2 a) > = (a+b)^3$
$4(a^3 + b^3) >= (a+b)^3$ .. (1)

Similarly

$4(b^3 + c^3) >= (b+c)^3$ ... (2)
$4(c^3 + a^3) >= (c+a)^3$ ...(3)

Adding (1), (2), (3) we get the result
 
Last edited:
kaliprasad said:
we have $a^3 + b^3 – a^2 b – b^2 a$
= $a^3 - a^2 b – b^2 a + b^3$
= $a^2(a-b)- b^2(a-b) = (a^2-b^2) (a-b) = (a+b)(a-b)^2> = 0$

Hence
$a^3 + b^3 > = a^2 b + b^2 a$

Multiply by 3 and add $a^3 + b^3$ on both sides

$4(a^3 + b^3) >= a^3 + b^3 + 3(a^2 b + b^2 a) > = (a+b)^3$
$4(a^3 + b^3) >= (a+b)^3$ .. (1)

Similarly

$4(b^3 + c^3) >= (b+c)^3$ ... (2)
$4(c^3 + a^3) >= (c+a)^3$ ...(3)

Adding (1), (2), (3) we get the result
$a^3 + b^3 \geq a^2 b + b^2a$---(i)
$b^3 + c^3 \geq b^2 c + c^2 b$---(ii)
$a^3 + c^3 \geq a^2 c + c^2 a$---(iii)
(i)+(ii)+(iii) is just the result of (2) of my previous post
 
Thanks to Albert and kaliprasad for participating and well done for cracking the problem in such a nice way!

Solution suggested by other:

We begin by computing

$\begin{align*} 4(a^3+b^3)-(a+b)^3&=(a+b)(4(a^2-ab+b^2)-(a+b)^2)\\&=(a+b)(3a^2-6ab+3b^2)\\&=3(a+b)(a-b)^2\\&\ge0 \end{align*}$

where the inequality holds since $a$ and $b$ are assumed to be positive, so $a+b>0$ and of course $(a-b)^2 \ge 0$.

Thus, $4a^3+4b^3>(a+b)^3$ and similarly, $4a^3+4c^3>(a+c)^3$ and $4b^3+4c^3>(b+c)^3$.

Adding these three inequalities we obtain

$8a^3+8b^3+8c^3>(a+b)^3+(a+c)^3+(b+c)^3$,

which is precisely what we wanted.
 
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