Upper bound for Hypergeometric Function We would like to find an upper bound on the following function:
$\left(\frac{\omega_1}{\omega_2}\right)^{(\alpha_1-1)} \frac{\Gamma(\alpha_1+\alpha_2-1)}{\Gamma(\alpha_1)\Gamma(\alpha_2)} {}_2F_1\left(\alpha_1-1,\alpha_1+\alpha_2-1,\alpha_1,-\frac{\omega_1}{\omega_2}\right)$
for all $\omega_1,\omega_2 > 0$ and $\alpha_1,\alpha_2 >1$.
From the way we derived this function, we know that it is smaller than 1, but we would like to find an upper bound (expressed in the parameters $\omega_1,\omega_2,\alpha_1,\alpha_2$) that is as tight as possible.
Thank you in advance for all possible suggestions and insights!
Sebas   
 A: This is the regularized incomplete beta function $I_x(a,b)=B(x;a,b)/B(a,b)$ for 
$$
x=\omega_1/(\omega_1+\omega_2),\qquad a=\alpha_1-1,\qquad b=\alpha_2. 
$$
Recall that $B(a,b)=B(1;a,b)$ with
$$
B(x;a,b) = \int_0^x t^{a-1}\,(1-t)^{b-1}\,\mathrm{d}t.
$$
Note that $0\le I_x(a,b)\le 1$ or every $0\le x\le 1$ and that the function $x\mapsto I_x(a,b)$ increases from $I_0(a,b)=0$ to $I_1(a,b)=1$. Furthermore, when $a$ et $b$ are positive integers, this is the cumulative distribution function of a random variable with binomial distribution, an interpretation which may be used to estimate $I_x(a,b)$. More precisely, $I_x(a,b)=P(X\le b-1)$ where the distribution of $X$ is binomial $(a+b-1,1-x)$. 
See here. (Not knowing what the OP does and does not know, it is difficult to determine what to add to the present answer.)
For example, if $\alpha_1\to+\infty$ and $\alpha_2\to+\infty$ with $\alpha_1/\alpha_2\to\varrho$, and if $\omega_1$ and $\omega_2$ are fixed, then by the law of large numbers the limit of the quantity written by the OP is $1$ if $\omega_1>\varrho\omega_2$ and $0$ if $\omega_1<\varrho\omega_2$.
