Consider the classical hypergeometric function $F(5/4,3/4; 2, z)$ for $z\in (0,1]$. Is this bounded by some real number (independent of $z$)?
I'm aware of Euler's formula:
$F(5/4,3/4; 2, z) = \frac{1}{\Gamma(5/4)\Gamma(3/4)}\int_0^1 t^{-1/4} (1-t)^1/4 (1-tz)^{-5/4} dt.$
The best I can do using this formula is $F(5/4, 3/4; 2, z) \leq \frac{4}{ 3\Gamma(5/4) \Gamma(3/4)}\frac{1}{1-z}. $ This is not good though, because it blows up as $z$ tends to $1$.
Any suggestions?
Maybe it is easy to show that $F$ is strictly increasing on $(0,1]$ and continuous on $\mathbf{R}$. Then, we simply need to estimate $F(5/4,3/4;2,1)$.