inequalities for Gaussian hypergeometric function I am looking for inequalities for a special case of the Gaussian hypergeometric function $$
_2F_1\left(1, a, b+1, \tfrac{1}{c}\right),
$$
where $a \geq b > 0$ and $c > 1$ are positive real numbers.
Could someone share me some related references?
 A: By the second transformation in http://dlmf.nist.gov/15.8.E1, we find
\begin{align*}
F\!\left( {1,a;b + 1;\frac{1}{c}} \right) &= \left( {\frac{c}{{c - 1}}} \right)^a F\!\left( {b,a;b + 1;\frac{1}{{1 - c}}} \right) \\ & = \left( {\frac{c}{{c - 1}}} \right)^a F\!\left( {a,b;b + 1;\frac{1}{{1 - c}}} \right) \\ &= \left( {\frac{c}{{c - 1}}} \right)^a b\int_0^1 {\frac{{t^{b - 1} }}{{\left( {1 + \frac{t}{{c - 1}}} \right)^a }}dt} 
\end{align*}
where I used http://dlmf.nist.gov/15.6.E1 in the last step. Now
$$
\int_0^1 {\frac{{t^{b - 1} }}{{\left( {1 + \frac{t}{{c - 1}}} \right)^a }}dt}  \le \int_0^1 {t^{b - 1} dt}  = \frac{1}{b}.
$$
And so
$$
F\!\left( {1,a;b + 1;\frac{1}{c}} \right) \le \left( {\frac{c}{{c - 1}}} \right)^a 
$$
Note that I took into account the slight difference between $F$ and $\mathbf F$ (see http://dlmf.nist.gov/15.2.i) in the DLMF. It is also seen from the integral formula that the upper bound is asymptotically sharp for large positive $c$.
A consequence of Theorem 2.1 in http://doi.org/10.1137/19M1262498 is that, under the conditions in the question,
\begin{multline*}
F\!\left( {1,a;b + 1;\frac{1}{c}} \right) \\ = \left( {\frac{c}{{c - 1}}} \right)^a \frac{b}{{\Gamma (a)}}\left( {\sum\limits_{n = 0}^N {\frac{{\Gamma (a + n)}}{{(b + n)n!}}\frac{1}{{(1 - c)^n }}}  + \theta _N (a,b,c)\frac{{\Gamma (a + N)}}{{(b + N)N!}}\frac{1}{{(1 - c)^N }}} \right)
\end{multline*}
where $0<\theta _N (a,b,c)<1$ and $N$ is any non-negative integer. Since $c>1$, this will give you successive upper and lower bounds as you increase $N$.
