= Solution
Let $H=z^{1-\alpha-\alpha'}(1-z)^{1-\gamma-\gamma'}W$. The exponent-sum relation gives $2-\alpha-\alpha'-\gamma-\gamma'=1+\beta+\beta'$. Therefore the assumed boundedness of $z^{1+\beta+\beta'}W$ 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 $A$, giving
$$
\boxed{W(z)=Az^{\alpha+\alpha'-1}(1-z)^{\gamma+\gamma'-1}.}
$$
Independence forces $A\ne0$: 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.
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