In this paper, we introduce new Ma-Minda classes \(\mathcal {S}_H^*\) and \(\mathcal {C}_H\) associated with a new Carathéodory function \(\begin{aligned} \varphi _H(z) = 1 + z + \frac{1}{3} z^2 - \frac{1}{9} z^3. \end{aligned}\) For these classes, several characteristic properties are established, including structural formulas, growth and distortion results, radius estimates, inclusion results, and initial Taylor-Maclaurin coefficients. Additionally, we define subclasses of bi-univalent functions related to \(\varphi _H\) and involving fractional derivatives, for which we provide coefficient bounds, investigate the Fekete-Szegö functional, and calculate the upper bound of the second Hankel determinant. Furthermore, graphical illustrations are included to support the results discussed throughout the paper.
