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

In this paper, we introduce an extended two-factor q-differential multiplier operator denoted by ℳq,,n,mf(z). Using this operator, we define the associated subclasses Xq,,n,m() and Yq,,n,m(,) of analytic functions. We obtain upper bounds for the initial coefficients a2, a3 and FeketeSzeg inequalities for the functional |a3 a22| for functions belonging to these subclasses. The bounds are derived by reducing the coefficient problem to standard Carathodory estimates. The estimate for X is sharp, whereas for Y sharpness is established at the Carathodory 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.