The mapping properties of solutions of the differential equation \(zF_\lambda(z)+\lambda \psi F_\lambda (z)=\lambda \phi(z)\), \(F_\lambda(0)=0\), \(\lambda >0\), where \(\psi(z)=1+cz+\cdots\) is a function with positive real part in \(D=\{z:| z| 0\). When \(\phi(z)=z/(1+\gamma z)\), \(\psi(z)=1/(1+\gamma z)\) (\(| \gamma| \leq 1\)), the author has shown that \( F_\lambda \) is convex for \(\lambda >0\) and \( F_{\lambda_1} \prec F_{\lambda_2} \prec I\) for \(0<\lambda_1< \lambda_2\) where \(I(z)=z\). The importance of convexity is shown by an example where \(F_\lambda\) is a starlike solution of the above differential equation while \( F_{\lambda_1}\not\prec F_{\lambda_2} \) whenever \(0<\lambda_1< \lambda_2\).