<s>[INST] Upper bound on $ \binom{a}{m+1}\sum ^m_{j=0} \binom{a-m-1}{j}/\binom{b}{j+m+1}$ Given $a,b,m$ such that $0<2m<a<b$.
I would like to find out upper bound of $$S = \binom{a}{m+1}\sum ^m_{j=0} \frac{\binom{a-m-1}{j}}{\binom{b}{j+m+1}}$$
Anyone can help me please?
Thank you so much.
 [/INST] I begin by using this identity: $\int_0^1 t^{\alpha} (1-t)^{\beta}=\frac{!}{\alpha+\beta+1}\frac{1}{\binom{\alpha+\beta}{\alpha}}$, for some positive $\alpha$ and $\beta$. This can be alternatively expressed as follows: for some positive $n$ and $k$, we have $\frac{1}{\binom{n}{k}}= (n+1) \int_0^1 t^{k}(1-t)^{n-k} dt$. Applying this to the right hand side, we have: 
$$S=(b+1) \binom{a}{m+1} \sum_{j=0}^m \binom{a-m-1}{j} \int_0^1 t^{j+m+1}(1-t)^{b-j-m-1} dt\\
=  (b+1) \binom{a}{m+1} \int_0^1 (1-t)^{b-m-1} t^{m+1} \left[\sum_{j=0}^m \binom{a-m-1}{j} \left( \frac{t}{1-t}\right)^j \right] dt \\
\leq (b+1) \binom{a}{m+1} \int_0^1 (1-t)^{b-m-1} t^{m+1} \left[\sum_{j=0}^{a-m-1} \binom{a-m-1}{j} \left( \frac{t}{1-t}\right)^j \right] dt
\\
= (b+1) \binom{a}{m+1} \int_0^1 (1-t)^{b-m-1} t^{m+1}  \left[ \left(1+ \frac{t}{1-t}\right)^{a-m-1} \right] dt
\\
= (b+1) \binom{a}{m+1} \int_0^1  t^{m+1} (1-t)^{b-a} \\ 
=  \frac{b+1}{b-a+m+2}    \frac{ \binom{a}{m+1}}{\binom{b-a+m+1}{m+1}}
\\
=(b+1)\frac{a!(b-a)!}{(b-a+m+2)!(a-m-1)!}
$$ 
In case I have made arithmetic mistakes, I am sure you can pick it up and mend it, since I hope the main logistics are clear
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