AM-GM Inequality problem.

[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?

[jbunniii]

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?

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

[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

[jbunniii]

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

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.

[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.)

[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

[jbunniii]

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

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%27s_inequality

[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.

[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?

[jbunniii]

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?

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.)

[jambaugh]

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

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.

[hsong9]

I got the answer.
It's only 4-5lines.

Thanks!