On certain analytic functions defined by differential inequality

Abstract: For the family of analytic functions $f(z)$ in the open unit disk $\mathbb{D}$ with $f(0)=f'(0)-1=0$, satisfying the differential equation \begin{equation*} zf'(z) - f(z) = \dfrac{1}{2} z^2 \phi(z), \quad |\phi(z)| \leq 1, \end{equation*} we obtain radii of convexity, starlikeness, and close-to-convexity of partial sums of $f(z)$. We also study the generalization of this family having the form \begin{equation*} zf'(z)-f(z) = \lambda z^2 \phi(z), \quad |\phi(z)| \leq 1, \end{equation*} where $\lambda > 0,$ and obtain some useful properties of these functions.