Limit of $\frac{(\sum_{1} ^ n a_j)^p}{n^{p - 1} \sum_{j = 1} ^ n a_j ^ p}$ where $\frac{\sum a_j}{n} \to \infty$ Let $a_1, a_2, ...$ be a sequence of positive numbers such that $\frac{\sum_{j = 1} ^ n a_j}{n} \to \infty$ as $n \to \infty$. What can we say about the behavior of $$\frac{(\sum_{j = 1} ^ n a_j)^p}{n^{p - 1} \sum_{j = 1} ^ n a_j ^ p}$$ as $n \to \infty$ for $p > 1$? If I have my way, it should go to $0$ under these conditions, but I haven't been able to show this, I suspect because I'm missing some inequality. I can use Jensen's inequality to show that the expression above is in $[0, 1]$, but I guess I need something sharper to get it to go to $0$; maybe some inequality that leaves a factor of $\sum a_j / n$ in the denominator to exploit.
 A: Not neccesarily zero: 
Let $a_{j}=j$. Then $$\sum_{j\leq n}a_{j}\sim\frac{n^{2}}{2}\ \text{and}\ \sum_{j\leq n}a_{j}^{p}\sim\frac{1}{p+1}n^{p+1}$$so that $$\frac{\left(\sum_{j\leq n}a_{j}\right)^{p}}{n^{p-1}\sum_{j\leq n}a_{j}^{p}}\sim\frac{p+1}{2^{p}}.$$
Hölder Mean: We can write the terms in a slightly nicer way.  Notice $$\frac{\left(\sum_{j\leq n}a_{j}\right)^{p}}{n^{p-1}\sum_{j\leq n}a_{j}^{p}}=\frac{\left(\frac{\sum_{j\leq n}a_{j}}{n}\right)^{p}}{\left(\frac{\sum_{j\leq n}a_{j}^{p}}{n}\right)}=\left(\frac{M_{1}(a_{1},\dots,a_{n})}{M_{p}(a_{1},\dots,a_{n})}\right)^{p}$$ where $M_{r}(\boldsymbol{x})$ is the generalized Hölder mean. For $p>1$ we know that $M_{1}(\boldsymbol{x})\leq M_{p}(\boldsymbol{x})$ so that the above quantity is always in $[0,1]$.
Sequence going to zero: We cannot say too much more, such as a lower bound, as it is possible to construct a sequence which goes to zero.
Let $a_{j}=e^{j}$. Then $$\sum_{j\leq n}a_{j}\sim e^{n}\ \text{and}\ \sum_{k\leq n}a_{j}^{p}\sim\frac{1}{p}e^{np}.$$ Consequently $$\frac{\left(\sum_{j\leq n}a_{j}\right)^{p}}{n^{p-1}\sum_{j\leq n}a_{j}^{p}}\sim\frac{\left(e^{n}\right)^{p}}{n^{p-1}\left(\frac{1}{p}e^{np}\right)}\sim\frac{p}{n^{p-1}}\rightarrow0.$$
Hope that helps,
