Starlikeness Criteria for a Generalized Class of Univalent Functions Defined by Second-Order Differential Inequalities

Let \mathcal{A} denote the class of functions analytic in the unit disk \mathbb{D}, normalized by f(۰)=۰ and f'(۰)=۱. We study the starlikeness of a generalized class of univalent functions defined by a second-order differential inequality involving the operator (z/f(z))^{\mu}, where ۰<\mu<۱. This class extends the well-known class \mathcal{M}(\lambda) introduced by Obradovi\'c and Ponnusamy and contains several previously studied subclasses as special cases. By applying methods of differential subordination and refined analytic estimates, we obtain explicit sufficient conditions on the parameter \lambda ensuring that functions in this class are starlike in \mathbb{D}. In particular, sharp bounds for \lambda are derived to guarantee inclusion in the classes \mathcal{S}^* and \mathcal{S}^*(\beta) and corresponding starlikeness results are established for an associated integral operator. These findings generalize and refine earlier starlikeness criteria for univalent functions defined by second-order differential inequalities.

کلیدواژه ها:

Univalent functions ، Starlike functions ، Differential inequalities ، Differential subordination ، Geometric function theory