Geometric radii for Salagean-Komatu type functions

Abstract

In this paper, we investigate the geometric properties of the subclass \(TS_{k}^{a,\delta}(\hbar,l)\) of analytic functions with negative coefficients, defined via a composite differential operator combining the Salagean and Komatu operators. Using the coefficient characterization of Murthy and Reddy together with Silverman’s exact coefficient criteria for negative coefficients, we establish sharp radii of starlikeness and convexity of order \(\alpha\) \((0 \le \alpha < 1)\) for functions belonging to this class. The classical radii are obtained as immediate corollaries. We also record a sharp radius of close-to-convexity of order \(\alpha\) as an application and show that, whenever the corresponding radius is strictly less than \(1\), the extremal problem is attained by a one-term function. The methodology presented here yields a rigorous radius theory for this recently introduced operator-defined class.