Exploring q-Fibonacci Numbers in Geometric Function Theory: Univalence and Shell-like Starlike Curves

Emphasising their connection with shell-like star-like curves, this work investigates a new subclass of star-like functions defined by q-Fibonacci numbers and q-polynomials. We study the geometric and analytic properties of this subclass, including the computation of intervals of univalence and nonunivalence for some functions. Moreover, we define a sufficient condition for functions in this subclass to satisfy the criteria of the famous class of analytic functions with positive real components. This work improves our understanding of the link between Fibonacci-type sequences and the geometric properties of analytic functions by using subordination ideas and the features of q-Fibonacci sequences. Emphasising the possibility for diverse research in combinatorial and analytical mathematics, the results offer fresh insights and support further study on the applications of calculus in geometric function theory.