3
votes
2answers
59 views

Prove that $\lim\limits_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0$.

I conjecture that for any $\epsilon>0$, we have $$ \lim_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0 $$ where $\Gamma(x,a) = \int_a^\infty t^{x-1}e^{-t} \mathrm{d}t$ denotes the ...
5
votes
2answers
284 views

Proving Inequality with the Greatest Integer Function

Show that $$[(m+n)x]+[(m+n)y] \ge [mx+(n-1)y]+[my+(n-1)x]$$ where $m,~n \in \Bbb{N}$ and $0\le x,~y < 1$. I've tried everything for about half a day and still couldn't figure it out. ...
0
votes
0answers
73 views

Representation of an equality

I know that I keep asking the similar problems with a little modification but it is really important to me to make sure that I am at the right track. This is my previous question link. Since we can ...
2
votes
0answers
160 views

Help with an integral inequality involving an incomplete beta function

I would like to determine if the following inequality is true: ...