What is the upper bound for the given function f(t,p)?

[phonic]

Dear members,

I try to find the upper bound of the following function. Can anybody gives a hint? Thanks!

<br /> f(t,p)=\sum_p \frac{p(1-p)}{t^5}[p^4(9t^4-81t^3+225t^2-274t+120)+p^3(-12t^4+129t^3-400t^2+524t-240)+<br />
<br /> \mbox{\hspace{2cm}}p^2(4t^4-59t^3+ 216t^2-311t+150)+p(7t^3-36t^2+59t-30)+(t-1)^2]<br />
where
<br /> t=1,2,3,..<br />
<br /> \sum_p p = 1<br />

The problem is to find the function g(t) that
<br /> f(t,p) \leq g(t)<br />
It seems that
<br /> g(t)\sim 1/t<br />
Is it possible to find a better bound?

 

[fresh_42]

No. Not as long as no additional conditions on $p$ in dependency of $t$ are given. You can assume the worst case of $\sum_p=p=1$ and get $f(t,1)=O(t^{-1})$ so all you can do is finding a better constant.