Prove Inequality: 1/p + 1/q=1, u,v >= 0

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In summary, the given problem can be solved by using the weighted AM-GM inequality and rearranging equations to show that p+q=pq. Then, the second inequality can be simplified by removing the fractions.
  • #1
ehrenfest
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Homework Statement


Let p and q be positive real numbers such that

1/p + 1/q=1

Prove that if [itex]u\geq 0[/itex] and [itex]v \geq 0[/itex], then

[tex]uv \leq \frac{u^p}{p}+\frac{v^q}{q}[/tex]


Homework Equations





The Attempt at a Solution


I am really stumped. Is there like a famous inequality that I need to use here that I am forgetting?
This vaguely reminds me of the AM-GM inequality...
 
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  • #2
It certainly works when u=v=1...
 
  • #3
I think I got it, but it was tricky.

First, try to prove that [tex]\frac{1}{p}+\frac{1}{q}=1[/tex] can be rearranged to show that [tex]p+q=pq[/tex]

Then start to work on rearranging the second inequality, in order to remove the fractions.
 
  • #4
CrazyIvan said:
I think I got it, but it was tricky.

First, try to prove that [tex]\frac{1}{p}+\frac{1}{q}=1[/tex] can be rearranged to show that [tex]p+q=pq[/tex]

Well, that comes from just multiplying the first equation by pq.



CrazyIvan said:
Then start to work on rearranging the second inequality, in order to remove the fractions.

So, I have been trying this and not getting anywhere. I tried plugging in p q = p+q. Can you be more specific?
 
  • #5

What is the meaning of "Prove Inequality: 1/p + 1/q=1, u,v >= 0"?

The inequality 1/p + 1/q = 1, where u and v are greater than or equal to 0, is a mathematical expression that shows the relationship between two numbers, p and q. It states that the sum of the reciprocals of p and q is equal to 1.

How can I prove this inequality?

There are several ways to prove this inequality, depending on the level of mathematical understanding. One method is to use algebraic manipulation to transform the equation into a form that is easier to prove. Another method is to use graphical representations or numerical examples to demonstrate the inequality.

What are the implications of this inequality?

This inequality has various implications in mathematics and other fields such as economics and physics. It is commonly used in optimization problems, probability theory, and the study of power laws. In economics, it is known as the Pareto principle, which states that a small number of individuals hold a large proportion of wealth or resources.

Can this inequality be extended to more than two variables?

Yes, this inequality can be extended to multiple variables, such as 1/x1 + 1/x2 + 1/x3 + ... + 1/xn = 1, where xi is greater than or equal to 0. This is known as the generalized form of the inequality and is often used in mathematical proofs and applications.

Are there any real-life applications of this inequality?

Yes, this inequality has various real-life applications, such as in economics, where it is used to study income distribution and wealth inequality. It is also used in physics to describe the relationship between pressure, volume, and temperature in gases, known as the ideal gas law. Additionally, it has applications in probability theory, where it is used to calculate the probability of events.

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