My previous answer, in much simplified form: take $g(z):=\frac{1-\sqrt{1-z}}{z}$ and $$h(z):=f(z)g(z)=\frac{1-\sqrt{1-z}}{z}\,\sqrt{\frac{1+z}{2}}
=\frac{\sqrt{1+z}-\sqrt{1-z^2}}{z\sqrt2}.$$ 
Then $g$ and $h$ are pgf's.