In this paper, we establish the initial Taylor-Maclaurin coefficients for normalized analytic functions \(f\,\) in the open unit disk. We also assume that \(f\,\) and its inverse \(g=f\,^{-1}\) satisfy the following conditions \(\begin{aligned} \rm{e}^{\rm{i}\phi }\left[ f\,^{\prime }\,(z)\right] ^{\tau }\left[ \frac{z}{f\,(z)}\right] ^{\nu }&\prec \psi (z)\cos \phi +\rm{i}\sin \phi \quad \text {and} \quad \rm{e}^{\rm{i}\phi }\left[ g'(z)\right] ^{\tau }\left[ \frac{z}{g(z)}\right] ^{\nu }&\prec \psi (z)\cos \phi +\rm{i}\sin \phi , \end{aligned}\) for \(-\pi /2<\phi <\pi /2\) , where \(\psi \) is a univalent function whose range is symmetric with respect to the real axis, and \(\tau \) and \(\nu \) are non-zero real numbers. We also examine other classes of related functions and establish connections with previously known results.
