difference between expected values of distributions let us assume two distributions $p$ and $p'$ over the set of naturals $N$. 
Is it true the following property?
$\sum_{n \in N} p(n) \cdot n \le \sum_{n \in N} p'(n) \cdot n$
IFF 
for all $0 \le u \le 1$
$\sum_{n \in N} p(n) \cdot u^{n} \ge \sum_{n \in N} p'(n) \cdot u^n$
Thanks for your help!

Call $X$ a random variable with distribution $p$ and $Y$ a random variable with distribution $p'$, then one considers the assertions:


*

*$E[X]\leqslant E[Y]$ 

*$E[u^X]\geqslant E[u^Y]$ for every $u$ in $[0,1]$


Assertion 2. implies assertion 1. because $X=\lim\limits_{u\to1}\frac1{1-u}\cdot(1-u^X)$ and the limit is monotonous.
Assertion 1. does not imply assertion 2., as witnessed by the case  $E[u^X]=\frac23+\frac13u$ and $E[u^Y]=\frac34+\frac14u^2$.
