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
AI Thread 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:

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Similar threads

MHB Min Value of $\dfrac{a+3c}{a+2b+c}$+$\dfrac{4b}{a+b+2c}$+$\dfrac{8c}{a+b+3c}$
Replies
1
Views
2K
B Complete the square, yes, but what does it mean?
Replies
19
Views
3K
MHB Proving Inequality: $a^{2a}b^{2b}c^{2c} > a^{b+c}b^{c+a}c^{a+b}$
Replies
2
Views
1K
MHB Inequality Challenge: Prove $\ge 0$ for All $a,b,c$
Replies
2
Views
1K
MHB Positive Integers: Evaluate $a+b+c$
Replies
1
Views
1K
MHB Inequality involving a, b, c and d
Replies
1
Views
845
I Prove that a triangle with lattice points cannot be equilateral
Replies
1
Views
2K
MHB How can we prove the inequality challenge for positive real numbers?
Replies
3
Views
2K
B When are two vectors collinear and coplanar?
Replies
10
Views
2K
MHB Prove Algebra Challenge: $(x,y,z,a,b,c)$ Equation
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top