Generalized Analytic Function Classes Using an Extended Two-Factor q-Differential Operator: Fekete–Szegő Problem

Abstract

In this paper, we introduce an extended two-factor q-differential multiplier operator denoted by ℳ_q,λ,β^n,mf(z). Using this operator, we define the associated subclasses X_q,λ,β^n,m(φ) and Y_q,λ,β^n,m(φ,δ) of analytic functions. We obtain upper bounds for the initial coefficients a₂, a₃ and Fekete–Szegő inequalities for the functional |a₃ − μa₂²| for functions belonging to these subclasses. The bounds are derived by reducing the coefficient problem to standard Carathéodory estimates. The estimate for X is sharp, whereas for Y sharpness is established at the Carathéodory level and actual extremality is conditional on solvability of the defining differential equation. Additional consequences include third coefficient estimates, a logarithmic coefficient inequality, an upper bound for the second Hankel determinant, and a classical limiting theorem.