On quotients and differences of hypergeometric functions

For Gaussian hypergeometric functions $F(x)=F(a,b;c;x),$ $a,b,c>0,$ we consider the quotient $Q_F(x,y)=(F(x)+F(y))\mathrm{/}F(z)$ and the difference $D_F(x,y)=F(x)+F(y)-F(z)$ for $0<x,y<1$ with $z=x+y-xy\mathrm{.}$ We give best possible bounds for both expressions under various hypotheses about the parameter triple $(a,b;c)\mathrm{.}$