 Solution

Let $H=z^{1-\alpha -{\alpha }^{\prime }}(1-z{)}^{1-\gamma -{\gamma }^{\prime }}W$. The exponent-sum relation gives $2-\alpha -{\alpha }^{\prime }-\gamma -{\gamma }^{\prime }=1+\beta +{\beta }^{\prime }$. Therefore the assumed boundedness of $z^{1+\beta +{\beta }^{\prime }}W$ at infinity makes $H$ bounded there, up to a fixed nonzero branch factor. By part (ii), $H$ is entire and hence bounded also on compact sets. Liouville theorem makes it a constant $A$, givingIndependence forces $A\ne 0$: if the Wronskian vanished, uniqueness for the second-order linear equation at an ordinary point would make the two solutions dependent. In fact the differential identity in part (ii) already proves constancy; the boundedness hypothesis supplies the requested Liouville route as well.