Properties of a generalized class of analytic functions with coefficient inequality

Abstract

Let (βn) n≥2 be a sequence of nonnegative real numbers and δ be a positive real number. We introduce the subclass A(βn, δ) of analytic functions, with the property that the Taylor coefficients of the function f satisfies ∑ ∞ n≥2 βn|an| ≤ δ , where f (z) = z + ∑ ∞ n=2 anz n. The class A(βn, δ) contains nonunivalent functions for some choices of (βn) n≥2. In this paper, we provide some general properties of functions belonging to the class A(βn, δ) , such as the radii of univalence, distortion theorem, and invariant property. Furthermore, we derive the best approximation of an analytic function in such class by using the semiinfinite quadratic programming. Applying our results, we recover some known results on subclasses related to coefficient inequality. Some applications to starlike and convex functions of order α are also mentioned.