Extremal problems on a unified family of k-uniformly starlike andconvex functions with complex order

Abstract

This paper introduces and studies a new subclass of analytic functions, denoted $\mathcal{TU}_g(\kappa,b,\delta,\lambda)$, defined via the Hadamard product with a fixed function $g$. Governed by parameters $\delta \in [0, 1]$, $\kappa \geq 0$, $b \in \mathbb{C} \setminus {0}$, and $\lambda \in [0, 1)$, this class generalizes the concepts of $k$-uniformly starlike and convex functions of complex order. We establish several foundational properties for this class, including sharp coefficient bounds, radii of starlikeness and convexity, and extreme points. Furthermore, we prove subordination results, resolve Silverman's conjecture on integral means inequalities, and investigate the invariance under integral operators. Our results unify and extend numerous established theorems in geometric function theory.