A single-valued function f(z) is said to be univalent in a domain if it never takes on the same value twice, that is, if f(z₁) = f(z₂) for implies that z₁ = z₂. A set is said to be starlike with respect to the line segment joining w₀ to every other point lies entirely in . If a function f(z) maps onto a domain that is starlike with respect to w₀, then f(z) is said to be starlike with respect to w₀. In particular, if w₀ is the origin, then we say that f(z) is a starlike function. Further, a set is said to be convex if the line segment joining any two points of lies entirely in . If a function f(z) maps onto a convex domain, then we say that f(z) is a convex function in .

By using a certain linear operator defined by a Hadamard product or convolution, several interesting subclasses of analytic functions in the unit disk are introduced and studied systematically. The various results presented here include, for example, a number of coefficient estimates and distortion theorems for functions belonging to these subclasses, some interesting relationships between these subclasses, and a wide variety of characterization theorems involving a certain functional, some general functions of hypergeometric type, and operators of fractional calculus. Some of the coefficient estimates obtained here are fruitfully applied in the investigation of certain subclasses of analytic functions with fixed finitely many coefficients.

The object of the present paper is to derive some properties of neighborhoods of analytic functions with negative coefficients in the open unit disk.