A Unified Approach to the Generalized Fekete–Szegö Functional in Complex Banach Spaces

The generalized Fekete–Szegö functional for a nonempty class F⊆ S F ⊆ S is given by f | a₃(f)-λ [ a₂(f)] ²| -μ | a₂(f)| , f ⟼ a 3 ( f ) - λ a 2 ( f ) 2 - μ a 2 ( f ) , where f∈ F f ∈ F , λ ∈ C λ ∈ C , μ ≥ 0 μ ≥ 0 , and a₂(f)=f''(0)/2,\ a₃(f)=f'''(0)/6 a 2 ( f ) = f ′ ′ ( 0 ) / 2 , a 3 ( f ) = f ′ ′ ′ ( 0 ) / 6 are, respectively, the second and third coefficients in the power expansion of f . This functional was studied on the classes S and K in [10, 35]. A sharp estimate for the generalized Fekete–Szegö functional on S^ S ⋆ is missing from the known literature. In this paper, we present a unified approach to the generalized Fekete–Szegö functional on subclasses of starlike, convex, and α α -convex functions on the unit disc U . Also, we propose an extension of the generalized Fekete–Szegö functional for nonempty classes of normalized locally biholomorphic mappings on the unit ball B B