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