Estimates for a class of analytic and univalent functions connected with quasi-subordination

Abstract

Geometric Function Theory, an active field of studied that has its roots in complex analysis, has gained an impressive attention from many researchers. This occurs largely because it deals with the study of geometric properties of analytic (and univalent) functions where many of its applications spread across many fields of mathematics, mathematical physics and engineering. Notable areas of application include conformal mappings, special functions, orthogonal polynomials, fluid flows in physics and engineering designs. The investigations in this paper are on a subclass of analytic and univalent functions defined in the unit disk Ω and denoted by Q^a_q(m). The definition of the new class encompasses some well-known subclasses of analytic and univalent functions such as the classes of starlike functions, Yamaguchi functions, and Ma-Minda functions. Two key mathematical principles involved in the definition of the class are the principles of Taylor’s series and quasi-subordination. Some of the investigations carried out on functions f ∈ Q^a_q(m) are however, the upper estimates for some initial bounds, the solution to the well-known Fekete-Szego problem and the upper estimate for a Hankel determinant.