NEIGHBORHOODS OF CERTAIN CLASSES OF ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS

By making use of the familiar concept of neighborhoods of analytic functions, the author proves several inclusion relations associated with the (n,δ)-neighborhoods of various subclasses defined by Salagean operator. Special cases of some of these inclusion relations are shown to yield known results.


Introduction
Let T( j) denote the class of functions of the form a k z k a k ≥ 0; j ∈ N = {1, 2,...} (1.1) which are analytic in the open unit disc U = {z : |z| < 1}.Following [5,8], we define the ( j,δ)-neighborhood of a function f (z) ∈ A( j) by In particular, for the identity function e(z) = z, we immediately have The main object of this paper is to investigate the (j,δ)-neighborhoods of the following subclasses of the class T( j) of normalized analytic functions in U with negative coefficients.
For a function f (z) ∈ T( j), we define (1.4) The differential operator D n was introduced by Sȃlȃgean [9].With the help of the differential operator D n , we say that a function f (z) ∈ T( j) is in the class T j (n,m,α) if and only if Re for some α (0 ≤ α < 1), and for all z ∈ U.

Neighborhood for the class T j (n,m,α)
For the class T j (n,m,α), we need the following lemma given by Sekine [11].
Applying the above lemma, we prove the following.
It is easy to see the following.
The result is sharp.
The result is sharp.
From the above lemmas, we see that R j (n,α) ⊂ P j (n,α).
Putting j = 1 in Theorem 2.2, we have the following.
Putting j = 1 in Theorem 3.5, we have the following.
Putting j = 1 in Theorem 4.1, we have the following.