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

Abstract The generalized Fekete–Szegö functional for a nonempty class $$\\mathcal {F}\\subseteq S$$ F ⊆ S is given by $$ f\\longmapsto \\left| a_3(f)-\\lambda \\left[ a_2(f)\\right] ^2\\right| -\\mu \\left| a_2(f)\\right| , $$ f ⟼ a 3 ( f ) - λ a 2 ( f ) 2 - μ a 2 ( f ) , where $$f\\in \\mathcal {F}$$ f ∈ F , $$\\lambda \\in \\mathbb {C}$$ λ ∈ C , $$\\mu \\ge 0$$ μ ≥ 0 , and $$a_2(f)=f''(0)/2,\\ a_3(f)=f'''(0)/6$$ 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 $$S^\\star $$ 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 $$\\alpha $$ 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 $$\\mathbb {B}$$ B