A New Multivalent Analytic Functions With Negative Coefficients in Distortion Bounds and Its Extremal Properties

This work investigates a new subclass SMⁿb, β, τ, p(μ) of multivalent analytic functions with negative coefficients, characterized by a generalized multiplier transformation operator Mβ, τ, pⁿ. Various properties of this subclass are investigated, including coefficient bounds, extreme points of extremal functions, distortion bounds, radii of convexity and starlikeness, and sharp lower bounds for partial sums. Additionally, graphical representations of extremal functions are provided to visually illustrate the theoretical results. These findings contribute to the ongoing development of geometric function theory.