Region of variability for spiral-like functions with respect to a boundary point

Abstract

For $\mu\in\mathbb C$ such that ${\rm Re\,}\mu>0$ let ${\mathcal F}_{\mu} $ denote the class of all non-vanishing analytic functions $f$ in the unit disk $\mathbb{D}$ with $f(0)=1$ and $$ {\rm Re} \bigg(\frac{2\pi}{\mu}\, \frac{zf'(z)}{f(z)}+ \frac{1+z}