Comparing Polynomials: Determining the Larger Sum of Exponentiated Coefficients

[phonic]

Dear all,

Does anyone know how to determine which of the following sum is larger? Thanks a lot!
$$\sum_{i=1}^n a_i^{1/3}\sum_{i=1}^n a_i$$

$$n\sum_{i=1}^n a_i^{4/3}$$

$$a_i>0 \forall i$$

This is two polynomial of the same order. It is not clear to determine which one is larger, if I take the derivatives.

 

[JasonRox]

Dear all,

Does anyone know how to determine which of the following sum is larger? Thanks a lot!
$$\sum_{i=1}^n a_i^{1/3}\sum_{i=1}^n a_i$$

$$n\sum_{i=1}^n a_i^{4/3}$$

$$a_i>0 \forall i$$

This is two polynomial of the same order. It is not clear to determine which one is larger, if I take the derivatives.

Try proving by induction.

 

[mathman]

There is no general answer. It is easy to compare the two sums if all a's are the same. If =1, the sums are the same. If a>1, the first sum is larger. If a<1, the second sum is larger.

 

[JasonRox]

There is no general answer. It is easy to compare the two sums if all a's are the same. If =1, the sums are the same. If a>1, the first sum is larger. If a<1, the second sum is larger.

That's exactly what I was thinking.

Basically it depends on the sequence of $a_i$. Like whether or not it has certain boundaries like you mentionned.