The first few terms of the Maclaurin series can be obtained explicitly:

$$1-x-{\frac { \left( d-1 \right)  \left( K-1 \right) }{2\,Kd+2}}{x}^{2
}-{\frac { \left( d-1 \right)  \left( d-2 \right)  \left( K-1 \right) 
 \left( K-2 \right) }{ \left( 6\,Kd+6 \right)  \left( Kd+2 \right) }}{
x}^{3}+O \left( {x}^{4} \right) 
$$

The coefficient of $x^2$ is not especially small, but of course on a small
interval around $0$ the $1-x$ dominates.  I guess the question is really to 
estimate $h''(x)$ on the interval $[0,.1]$.