Conditions for univalency of $_2F_1(a,b;c,z)$ on $|z|<1$ have been established in Univalence of Gaussian and confluent hypergeometric functions, see theorem 4.

Conditions for univalency of $_2F_1(a,b;c,z)$ on $|z|<1$ have been established in Univalence of Gaussian and confluent hypergeometric functions, see theorem 4. A sufficient condition is $-2\leq a<0$, $-1\leq b$, $b\neq 0$ and $$c>\max(2+|a+b|,1-ab).$$ Further conditions are in Univalence and Convexity Properties for Gaussian Hypergeometric Functions.