Region of variability of subclasses of univalent functions (Q326906)

Summary: Let \(\mathcal{G}(\alpha)\) and \(\mathcal{F}(\alpha)\) denote the subclasses of locally univalent, normalized analytic functions \(f\) in the unit disk \(|z|<1\) satisfying the condition \[ \operatorname{Re}\bigg(1 + \frac{zf''(z)}{f'(z)}\bigg) <1 + \frac{\alpha}{2} \] and \[ \operatorname{Re}\bigg(1 + \frac{zf''(z)}{f'(z)}\bigg) > \frac{\alpha}{2} - 1 \] for some \(0<1\alpha\leq 1\) respectively. For any fixed \(z_{0}\) in the unit disk and \(\lambda \in [0,1)\), we determine the region of variability for \(\log f'(z_{0})\) when \(f\) ranges over the class \(\{f\in\mathcal{G}(\alpha):f''(0) = -\alpha\lambda\}\) and \(\{f\in\mathcal{F}(\alpha):f''(0)=(4-\alpha)\lambda\}\).