  R. K\"ustner proved in his 2002 paper that the function $w_{a,b,c}(z)=$
$F(a+1,b;c;z)/F(a,b;c;z)$ maps the unit disk $|z|<1$ onto a domain convex in
the direction of the imaginary axis under some condition on the real parameters
$a,b,c.$ Here $F(a,b;c;z)$ stands for the Gaussian hypergeometric function. In
this paper, we study the order of convexity of $w_{a,b,c}.$ In particular, we
partially solve the problem raised by the afore-mentioned paper by K\"ustner.
