On Variability Regions and Coefficient Inequalities for Exponentially Convex Janowski Functions

The current study concerns explicitly obtaining regions of variability described as \(\mathcal{R}_{\rho}[\alpha,A,B;z_0]:=\{\log f'(z_0)+\alpha f(z_0):f\in\mathcal{E}_{\rho}[\alpha,A,B]\}\) for the subclass \(\mathcal{E}[\alpha,A,B]\) of \(\alpha\)-exponentially convex Janowski analytic functions. Also, sharp initial coefficient bounds are obtained, and the corresponding Fekete-Szegö functional is solved to answer a recently posed open problem partially. The work presented herein generalizes and unifies various known results reported earlier. Geometric illustrations for various involved parameters support the current findings.