Does an inequality of infinite sums imply another?
Suppose $A, B$ are (infinite) sets of Euclidean vectors with $\sum_{\substack{y \in A}}||y||^{-2} \leq \sum_{\substack{z \in B}}||z||^{-2}$ Can we conclude $\sum_{\substack{y \in A}}e^{-\tau||y||^2} \leq \sum_{\substack{z \in B}}e^{-\tau||z||^2}$, for all positive real numbers $\tau$?
Intuitively this seems true, but there may be something I'm not taking into account. I can raise the bounty if the question proves difficult.
Fredrick
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Martin
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The answer is accepted.
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