In this paper, we introduce and investigate a novel subfamily of Bazilevič functions, defined using the $m$-leaf polynomial and the principle of subordination. For this subfamily, we establish several properties, including subordination results, inclusion relationships, and sharp coefficient bounds. Furthermore, we derive a general Fekete–Szegö inequality, providing a solution to this classical problem for functions within the new subfamily.

This article introduces an efficient spectral collocation framework for numerically solving the time-fractional Huxley equation. New basis functions of shifted Dickson polynomials of the first kind are introduced and employed. To achieve this, new formulas for the shifted polynomials are derived, including a series representation, an inverse formula, and expressions for both integer and fractional derivatives, which together with the collocation method serve as the foundation of the proposed numerical algorithm for converting the equation with its governing conditions into a non-linear algebraic system. A convergence and error analysis of the proposed method is also provided. We present numerical results and compare them with existing methods to illustrate the high accuracy of the proposed algorithms and their applicability.