Inequality of exponent

  • Thread starter phonic
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  • #1
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Dear all,

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

[tex]
n\sum_{i=1}^n a_i^{4/3}
[/tex]

[tex]
a_i>0 \forall i
[/tex]

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

Answers and Replies

  • #2
JasonRox
Homework Helper
Gold Member
2,314
3
phonic said:
Dear all,

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

[tex]
n\sum_{i=1}^n a_i^{4/3}
[/tex]

[tex]
a_i>0 \forall i
[/tex]

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.
 
  • #3
mathman
Science Advisor
7,942
496
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.
 
  • #4
JasonRox
Homework Helper
Gold Member
2,314
3
mathman said:
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 [itex]a_i[/itex]. Like whether or not it has certain boundaries like you mentionned.
 

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