Analytic and geometric properties of open door functions

In this paper, we study analytic and geometric properties of the solution $q(z)$ to the differential equation $q(z)+z{q}^{\mathrm{\prime }}(z)\mathrm{/}q(z)=h(z)$ with the initial condition $q(0)=1$ for a given analytic function $h(z)$ on the unit disk $\mathrm{\mid }z\mathrm{\mid }<1$ in the complex plane with $h(0)=1.$ In particular, we investigate the possible largest constant $c>0$ such that the condition $\mathrm{\mid }\mathrm{\Im }[zf\mathrm{"}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)]\mathrm{\mid }<c$ on $\mathrm{\mid }z\mathrm{\mid }<1$ implies starlikeness of an analytic function $f(z)$ on $\mathrm{\mid }z\mathrm{\mid }<1$ with $f(0)=f^{\mathrm{\prime }}(0)-1=0.$