MHB Proof of $a^{2a}\times b^{2b}\times c^{2c}>a^{b+c}\times b^{c+a}\times c^{a+b}$

  • Thread starter Thread starter Albert1
  • Start date Start date
  • Tags Tags
    Inequality Proof
Click For Summary
The discussion centers on proving the inequality \( a^{2a} \times b^{2b} \times c^{2c} > a^{b+c} \times b^{c+a} \times c^{a+b} \) under the condition that \( a > b > c > 0 \). Participants explore various mathematical approaches and techniques to establish this inequality, emphasizing the significance of the relationship between the variables. Several hints and strategies are proposed, including the application of logarithmic properties and inequalities. The conversation highlights the complexity of the proof and the need for a rigorous mathematical foundation. Ultimately, the goal is to validate the inequality through logical reasoning and mathematical principles.
Albert1
Messages
1,221
Reaction score
0
given :
$a>b>c>0$
prove :
$a^{2a}\times b^{2b}\times c^{2c}>a^{b+c}\times b^{c+a}\times c^{a+b}$
 
Mathematics news on Phys.org
Albert said:
given :
$a>b>c>0$
prove :
$a^{2a}\times b^{2b}\times c^{2c}>a^{b+c}\times b^{c+a}\times c^{a+b}$
hint:
let:
$A=a^{2a}\times b^{2b}\times c^{2c}$
$B=a^{b+c}\times b^{c+a}\times c^{a+b}$
prove:$\dfrac {A}{B}>1$
 
Albert said:
hint:
let:
$A=a^{2a}\times b^{2b}\times c^{2c}$
$B=a^{b+c}\times b^{c+a}\times c^{a+b}$
prove:$\dfrac {A}{B}>1$
for $a>b>c>0$ we have :
$\dfrac {A}{B}=a^{(a-b)}a^{(a-c)}b^{(b-a)}b^{(b-c)}c^{(c-a)}c^{(c-b)}$
$=(\dfrac{a}{b})^{a-b}(\dfrac{b}{c})^{b-c}(\dfrac{a}{c})^{a-c}>1^0\times1^0\times 1^0=1$
 
Last edited:

Similar threads

MHB Min Value of $\dfrac{a+3c}{a+2b+c}$+$\dfrac{4b}{a+b+2c}$+$\dfrac{8c}{a+b+3c}$
  • · Replies 1 ·
Replies
1
Views
2K
B Complete the square, yes, but what does it mean?
  • · Replies 19 ·
Replies
19
Views
3K
MHB Proving Inequality: $a^{2a}b^{2b}c^{2c} > a^{b+c}b^{c+a}c^{a+b}$
  • · Replies 2 ·
Replies
2
Views
1K
MHB Inequality Challenge: Prove $\ge 0$ for All $a,b,c$
  • · Replies 2 ·
Replies
2
Views
2K
MHB Positive Integers: Evaluate $a+b+c$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Inequality involving a, b, c and d
  • · Replies 1 ·
Replies
1
Views
866
I Prove that a triangle with lattice points cannot be equilateral
  • · Replies 1 ·
Replies
1
Views
2K
MHB How can we prove the inequality challenge for positive real numbers?
  • · Replies 3 ·
Replies
3
Views
2K
B When are two vectors collinear and coplanar?
  • · Replies 10 ·
Replies
10
Views
2K
MHB Prove Algebra Challenge: $(x,y,z,a,b,c)$ Equation
  • · Replies 2 ·
Replies
2
Views
1K