= Solution
Differentiate once less than might seem necessary: writing $v=h'$ turns the equation into
$$
\frac{v'(z)}{v(z)}=-\frac4\kappa\left(\frac1z+\frac1{z-1}\right).
$$
Uniqueness for this first-order linear equation also covers the solution $v\equiv0$. Thus every solution has $h'(z)=C[z(z-1)]^{-4/\kappa}$. Since $4/\kappa<1$, the derivative is integrable at $1$. The specified boundary value fixes the additive constant, giving
$$
\boxed{h(z)=C\,s_\kappa(z),\qquad
s_\kappa(z)=\int_1^z[v(v-1)]^{-4/\kappa}\,dv,\qquad C>0.}
$$
The strict positivity forces $C>0$; $C=0$ gives the zero function. These are all the positive solutions. The function $s_\kappa$ is the increasing <scale function of a one-dimensional diffusion> for the ratio process.
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