Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains

Abstract

Let $A$ be the class of functions $f(z) = z + ∑_{k = 2}^{ ∞} a_k z^k$ analytic in an open unit disc $∆$. We use a generalized linear operator closely related to the multiplier transformation to study certain subclasses of $A$ mapping $∆$ onto conic domains. Using the principle of the differential subordination and the techniques of convolution, we investigate several properties of these classes, including some inclusion relations and convolution and coefficient bounds. In particular, we get many known and new results as special cases.