Summary: Let \(\mathcal{A}\) be the class of all analytic functions \(f\) in the open unit disk \(\mathbb{U}\) of the form \(f(z)=z+\sum^\infty_{k=2}a_kz^k\). For \(\lambda>0\), \(\operatorname{Re}c>0\) and \(\alpha<1\), two subclasses \(\mathcal{P}(\lambda)\) and \(\mathcal{S}^*_\alpha\) of \(\mathcal{A}\) are introduced. In this paper, we find suitable conditions on \(\lambda\), \(c\) and \(\alpha\) such that for each \(f\in\mathcal{P}(\lambda)\) satisfying \((z/f(z))*F(1,c;c+1;z)\neq 0\) for all \(z\in\mathbb{U}\) for all \(z\in\mathbb{U}\), the function \[ G(z)=\frac{z}{(z/f(z))*F(1,c;c+1;z)}\quad (z\in\mathbb{U}) \] belongs to \(\mathcal{P}(\lambda')\), \(\mathcal{S}^*_\alpha\) or \(\mathcal{S}^*(\alpha)\). Here \(S^*(\alpha)\) denotes the usual class of starlike of order \(\alpha\) (\(0\leq\alpha<1\)) in \(\mathbb{U}\). We also determine necessary conditions so that \(f\in\mathcal{P}(\lambda)\) implies that \[ \left| \frac{zG'(z)}{G(z)}-\frac{1}{2\beta} \right| < \frac{1}{2\beta},\quad |z|<r, \] where \(r=r(\lambda,c;\beta)\) will be specified.