Prove inequality for fifth and fourth power means

[hsong9]

Homework Statement

Let X1,X2,...,Xn be positive real numbers. Show that

((x₁⁵+...+x_n⁵ )/ n)^1/5 >= ((x₁⁴+...+x_n⁴ )/ n)^1/4

Homework Equations

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]

Homework Statement

Let X1,X2,...,Xn be positive real numbers. Show that

((x₁⁵+...+x_n⁵ )/ n)^1/5 >= ((x₁⁴+...+x_n⁴ )/ n)^1/4

Homework Equations

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's_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!