AM-GM Inequality problem.

1. Nov 1, 2009

hsong9

1. The problem statement, all variables and given/known data
Let X1,X2,...,Xn be positive real numbers. Show that

((x15+...+xn5 )/ n)1/5 >= ((x14+...+xn4 )/ n)1/4

2. Relevant equations

3. The attempt at a solution
I have tried by taking logarithms. Is it right approach?
Or.. It can be applied to AM-GM-HM inequality?
how?

Last edited: Nov 1, 2009
2. Nov 1, 2009

jbunniii

I think you can prove it using Jensen's inequality, if you know it.

3. Nov 1, 2009

hsong9

I know Jensen's inequality, but I am not sure how it will work for my question.
Can you give me some hints?
Thanks

4. Nov 1, 2009

jbunniii

OK, one form of Jensen's inequality is

$$\phi\left(\frac{\sum y_i}{n}\right) \leq \frac{\sum \phi(y_i)}{n}$$

which is true provided that $\phi$ is a convex function.

Hint: let

$$y_i = x_i^4$$.

Then what convex function $\phi$ would work? Hint: You need something that will change powers of 4 into powers of 5.

Last edited: Nov 1, 2009
5. Nov 1, 2009

jambaugh

You could raise both sides to the 20th power and expand!?! Your first step however would be to factor out the n's. They should disappear.

Try maybe some variable substitutions? $$x_k = y_k^p$$ for some strategically chose value of $$p$$

(I'm guessing here so my suggestions may not be helpful. But I don't think the logarithms will be helpful due to the sums inside.)

6. Nov 1, 2009

hsong9

Thanks for your hints, but I am not sure yet.
If I have x^4 = y and x^5 = y^(5/4), then
I get the inequality which is just the same with the problem.
how do I get the inequality of Jensen? (without logarithms?)

Thanks

7. Nov 1, 2009

jbunniii

I'm not sure that I understand your question. Let

$$y_i = x_i^4$$

Now what $\phi$ did you choose? If you choose the right one, then after a little algebraic manipulation you can prove that your inequality is true. I just worked it out here at my desk; it took 4 lines.

If you are asking how to prove Jensen's inequality, the finite case is easy (yet very clever) and can be found under the heading "Proof 1 (finite form)" here:

http://en.wikipedia.org/wiki/Jensen's_inequality

8. Nov 1, 2009

hsong9

You mean I have to make some convex function
and then apply into the inequality.
Finally, I will get the answer. Right?

hmm, If so, I will think about the function.

9. Nov 1, 2009

hsong9

I think f(x) = x^(5/4) is fine as your hint.
And also f(x) is convex as condition of x_{i} --> positive real numbers.
Right?

10. Nov 1, 2009

jbunniii

Yes, that's correct. (In fact, x^(5/4) isn't even defined for negative x, unless you allow it to take on imaginary values.)

11. Nov 1, 2009

jambaugh

I was thinking more along the lines x = y^(5/4) or x = y^(4/5). Substitute then apply an appropriate power to both sides of the equation and you should get the Jensen form.

12. Nov 1, 2009