A general approach to the Fekete-Szegö problem. (English) Zbl 1132.30007

Choi, Jae Ho; Kim, Yong Chan; Sugawa, Toshiyuki

J. Math. Soc. Japan 59, No. 3, 707-727 (2007).

The study offers a new method to solve the Fekete-Szegő problem for classes of close-to-convex functions defined in terms of subordination. The essence of this problem is concerned with inequalities for the coefficient functional
\[ \Lambda_\mu(f)= a_3-\mu a^2_2={1\over 6}\biggl(f'''(0)- {3\mu\over 2} [f''(0)]^2\biggr) \] which corresponds to subclasses of normalized univalent functions in the unit disk, that is \(f(z)= z+ a_2z^2+ a_3z^3+\cdots+ \mu\in[0, 1]\). The obtained results are applied to the class of strongly close-to-convex functions.

Reviewer: Witold Pedrycz (Edmonton)

MSC:

30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50	Coefficient problems for univalent and multivalent functions of one complex variable

Keywords:

univalent function; subordination; coefficient bound