MHB Proving Inequality: $a^{2a}b^{2b}c^{2c} > a^{b+c}b^{c+a}c^{a+b}$

Click For Summary
The inequality \( a^{2a}b^{2b}c^{2c} > a^{b+c}b^{c+a}c^{a+b} \) is to be proven under the condition that \( a > b > c > 0 \). The discussion confirms that the inequality holds true given the specified conditions. Participants express satisfaction with the proof provided. The focus remains on the mathematical validation of the inequality rather than alternative perspectives. The conclusion emphasizes the correctness of the inequality under the stated assumptions.
Albert1
Messages
1,221
Reaction score
0
given:$a>b>c>0$

prove:$a^{2a}b^{2b}c^{2c}>a^{b+c}b^{c+a}c^{a+b}$
 
Mathematics news on Phys.org
Albert said:
given:$a>b>c>0$

prove:$a^{2a}b^{2b}c^{2c}>a^{b+c}b^{c+a}c^{a+b}$

Dividing both sides of the inequality by the expression on the right gives an equivalent inequality

$\displaystyle \left(\frac{a}{b}\right)^{a-b} \left(\frac{b}{c}\right)^{b-c} \left(\frac{a}{c}\right)^{a-c} > 1$,

which holds since the bases $\frac{a}{b}, \frac{b}{c}, \frac{a}{c}$ are all greater than 1 and the exponents $a-b, b-c, a-c$ are all positive.
 
Euge said:
Dividing both sides of the inequality by the expression on the right gives an equivalent inequality

$\displaystyle \left(\frac{a}{b}\right)^{a-b} \left(\frac{b}{c}\right)^{b-c} \left(\frac{a}{c}\right)^{a-c} > 1$,

which holds since the bases $\frac{a}{b}, \frac{b}{c}, \frac{a}{c}$ are all greater than 1 and the exponents $a-b, b-c, a-c$ are all positive.
very good ,you got it !
 

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 ·
Replies
1
Views
2K
MHB Proof of $a^{2a}\times b^{2b}\times c^{2c}>a^{b+c}\times b^{c+a}\times c^{a+b}$
  • · 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
High School Complete the square, yes, but what does it mean?
  • · Replies 19 ·
Replies
19
Views
3K
MHB Inequality involving a, b, c and d
  • · Replies 1 ·
Replies
1
Views
886
MHB How can we prove the inequality challenge for positive real numbers?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Prove the inequality (a^3-c^3)/3≥abc((a-b)/c+(b-c)/a)
  • · Replies 3 ·
Replies
3
Views
2K
MHB Prove Algebra Challenge: $(x,y,z,a,b,c)$ Equation
  • · Replies 2 ·
Replies
2
Views
1K
High School When are two vectors collinear and coplanar?
  • · Replies 10 ·
Replies
10
Views
2K