1
vote
1answer
32 views

Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\displaystyle \int_0^{s}f(t)\,\mathrm dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all ...
11
votes
6answers
399 views

Asymptotic behaviour of a multiple integral on the unit hypercube

A few days ago I found an interesting limit on the "problems blackboard" of my University: $$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu = 1.$$ The correct claim, ...
2
votes
1answer
83 views

How to asymptotically estimate a lower bound of this function?

The function is given as $$f(x)\geq \sum_{i=1}^{[x/2]}f(i)+1$$ The boundary condition is $f(0)=0$. What I can get is this function grows faster than any polynomial function, and grows slower than ...