1

Another probably stupid question of mine: For real $a_i$'s and $b_i$'s: If $\sum a_i+\sum b_i>\sum a_i$, does it follow that $\sum a_i^k+\sum b_i^k\geq \sum a_i^k$, $k\geq 1$ and if so how can it be shown? It should be noted, that all the $a_i$'s are positive and that $\sum b_i$ is positive as well, although the $b_i$'s are not necessarily positive.

Any help appreciated. Thank you very much in advance.

john stepanov
  • 93

1 Answers1

2

What you are really asking here is this:

If $\sum b_i > 0$, does it follow that $\sum b_i^k\geq 0$? (Can you see this is equivalent to your question?)

The answer is a resounding no. For example, take $b_1=b_2=b_3=\frac12$ and $b_4=-1$. Then $\sum b_i = \frac12>0$, but $\sum b_i^3 = -\frac58$.

5xum
  • 123,496
  • 6
  • 128
  • 204
  • Thank you, not seeing the equivalent statement was really stupid. However, I can see now, that it does work for k being an even integer. – john stepanov Feb 18 '14 at 12:00
  • Well, for even values of $k$, you always have $\sum b_i^k \geq 0$, regardless of $\sum b_i$. – 5xum Feb 18 '14 at 12:04