Geometric properties of a class of analytic functions defined by a differential inequality (Q275630)

Summary: Let \(\mathcal{A}\) be the class of analytic functions \(f\) defined in the open unit disk \(E\) and normalized by \(f(0) = f '(0) - 1 = 0\). For \(f(z) / z \neq 0\) in \(E\), let \(\mathcal{M} \left(\alpha, \lambda\right) : = \{f \in \mathcal{A} : | - \alpha z^2(z / f(z))'' + f'(z)(z / f(z))^2 - 1 | \leq \lambda,\;z \in E \}\), where \(\lambda > 0\) and \(\alpha \in \mathbb{R} \backslash [- 1 / 2,0]\). In the present paper, we find conditions under which functions in the class \(\mathcal{M}(\alpha, \lambda)\) are starlike of order \(\gamma\), \(0 \leq \gamma < 1\).