Is this inequality true? Prove or Disprove it!

  • Thread starter Thread starter twoflower
  • Start date Start date
  • Tags Tags
    Inequality
AI Thread Summary
The inequality (A+B)^{p} ≤ p(A^{p}+B^{p}) is under scrutiny for its validity, particularly for irrational values of p. Initial attempts to prove it using the binomial theorem were unsuccessful, suggesting the need for additional conditions. Testing specific values, such as A=0, indicates that p must be greater than or equal to 1 for the inequality to hold. Further examples demonstrate that without restrictions on A, B, and p, the inequality does not consistently apply. Clarifying these parameters is essential for a proper proof or disproof of the inequality.
twoflower
Messages
363
Reaction score
0
I've encountered this nice-looking inequality:

<br /> \left(A+B\right)^{p} \le p\left(A^{p}+B^{p}\right)<br />

(p can irrational as well)

but I can't find a way to prove or disprove its correctness. I've tried using the binomial theorem, but it didn't seem it would lead me to the finish.

Could someone please tell me how to prove that?

Thank you very much!
 
Physics news on Phys.org
Yeah, looks nice. But there must be other premises. Taking A=0 shows you need p>=1. e.g.
 
What other info do you have?
For example, (2+3)^3 <= 3(2^3 + 3^3)
Nope, doesn't work.

(0 + 1)^(power) isn't going to be less than that power*(0^power + 1^power)

So, there must be some restriction on A, B, and p that you haven't stated.
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
View full post »

Thread 'Making the denominator of a certain fraction real'

Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
View full post »

Similar threads

Mathematical Induction with binomial sum
Replies
9
Views
2K
Solving an inequality for a change of variables
Replies
15
Views
1K
Manipulating an inequality in the bisection method
Replies
2
Views
2K
Proving Inequalities: Tips and Strategies for Success
Replies
10
Views
2K
Prove/Disprove: p(a∩b) ≤ q^2 with a,b Independant
Replies
20
Views
3K
Prove or disprove that there is a rational number x and an i
Replies
5
Views
3K
How Do You Prove Trigonometric Identities with Minimal Equations?
Replies
34
Views
4K
Proving Inequality $$4x^4 + 4y^3 + 5x^2 + y + 1 \ge 12xy$$
Replies
16
Views
2K
Chernoff Bounds using importance sampling identity
Replies
0
Views
2K
I Zero raised to a positive real number
Replies
12
Views
2K

Hot Threads

Recent Insights

Back
Top