A note on the Fekete-Szego¨ Conjecture for Usual Subclasses of Univalent Functions

One of the most fundamental problems in the theory of univalent functions is the Bieberbach problem, also known as the Bieberbach conjecture, which claims that the modulus of the nth Taylor coefficient of a function f in class S is bounded by n (that is, f ∈ S, n ≥ 2 (n ∈ N = {1, 2, 3, . . . }), |cn| ≤ n). In general, studies on the Bieberbach conjecture, the proof of which occupied mathematicians for a long time, have evolved—as a result of the intellectual efforts in the field—from certain bounds on the modulus of individual Taylor coefficients to the investigation of bounds on the modulus of functionals formed by combinations of these coefficients. In the present article, the Fekete–Szego conjecture, ¨ which involves one such important function, is considered, and certain bounds are determined for a parameter η with 0 ≤ η ≤ 1.