International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 528570, 10 pages

doi:10.1155/2012/528570

Research Article

Neighborhoods of Certain Multivalently Analytic Functions

Serap Bulut

Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 İzmit-Kocaeli, Turkey

Received 28 March 2012; Accepted 16 May 2012

Academic Editor: Hernando Quevedo

Copyright © 2012 Serap Bulut. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions. Among the various results obtained here for each of these function classes, we derive the coefficient bounds, distortion inequalities, and other interesting properties and characteristics for functions belonging to the classes introduced here.

1. Introduction and Definitions

Let be the set of real numbers, and let be the set of complex numbers,

Let denote the class of functions of the form

which are analytic and -valent in the open unit disk

Denote by the Hadamard product (or convolution) of the functions and , that is, if is given by (1.2) and is given by

then

In [1], the author defined the following general class.

Definition 1.1. Let the function . Then we say that is in the class if it satisfies the condition

where

is given by (1.4), and denotes the falling factorial defined as follows:

Various special cases of the class were considered by many earlier researchers on this topic of geometric function theory. For example, reduces to the function class

(i) for , , and , studied by Mostafa and Aouf [2],(ii) for , and , studied by Srivastava et al. [3],(iii) for , , and , studied by Prajapat et al. [4],(iv) for , , , and , studied by Srivastava and Bulut [5],(v) for , , , , and , studied by Ali et al. [6].

Definition 1.2. Let the function . Then we say that is in the class if it satisfies the following nonhomogenous Cauchy-Euler differential equation (see, e.g., [7, page 1360, Equation (9)] and [5, page 6512, Equation (1.9)]):

where

Setting , , , and in Definition 1.2, we have the special class introduced by Mostafa and Aouf [2].

Following the works of Goodman [8] and Ruscheweyh [9] (see also [10, 11]), Altıntaş [12] defined the -neighborhood of a function by

It follows from the definition (1.11) that if

then

The main object of this paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes

Apart from deriving coefficient bounds and distortion inequalities for each of these classes, we establish several inclusion relationships involving the -neighborhoods of functions belonging to the general classes which are introduced above.

2. Coefficient Bounds and Distortion Theorems

Lemma 2.1 (see [1]). Let the function be given by (1.2). Then is in the class if and only if

where

Remark 2.2. If we set , , and in Lemma 2.1, then we have [2, Lemma 1].

Lemma 2.3 (See[1]). Let the function given by (1.2) be in the class . Then, for , one has

where is defined by (2.2).

Remark 2.4. If we set , , and in Lemma 2.3, then we have [2, Lemma  2].

The distortion inequalities for functions in the class are given by the following Theorem 2.5.

Theorem 2.5. Let a function be in the class . Then

and in general

where is defined by (2.2).

Proof. Suppose that . We find from the inequality (2.3) that

which is equivalent to (2.5) and

which is precisely the assertion (2.6).

If we set , , and in Theorem 2.5, then we get the following.

Corollary 2.6. Let a function be in the class . Then

and in general

where is defined by (2.2).

The distortion inequalities for functions in the class are given by Theorem 2.7 below.

Theorem 2.7. Let a function be in the class . Then

and in general

where is defined by (2.2).

Proof. Suppose that a function is given by (1.2), and also let the function be occurring in the nonhomogenous Cauchy-Euler differential equation (1.9) with of course

Then we readily see from (1.9) that

so that

Moreover, since , the first assertion (2.3) of Lemma 2.3 yields the following inequality:

and together with (2.19) and (2.18) it yields that

Finally, in view of the following sum:

the assertion (2.12) of Theorem 2.7 follows at once from (2.20) together with (2.21). The assertion (2.13) can be proven by similarly applying (2.17), and (2.19)–(2.21).

Remark 2.8. If we set , , , and in Theorem 2.7, then we have [2, Theorem 1].

3. Neighborhoods for the Classes and

In this section, we determine inclusion relations for the classes

involving -neighborhoods defined by (1.11) and (1.13).

Theorem 3.1 (see [1]). If and

then

where and are given by (1.12) and (2.2), respectively.

Remark 3.2. If we set , , and in Theorem 3.1, then we have [2, Theorem 2].

Theorem 3.3. If and

then

where and are given by (1.11) and (2.2), respectively.

Proof. Suppose that . Then, upon substituting from (2.16) into the following coefficient inequality:

we obtain

Since , the assertion (2.4) of Lemma 2.3 yields

Finally, by making use of (2.4) as well as (3.8) on the right-hand side of (3.7), we find that

which, by virtue of the sum in (2.21), immediately yields

Thus, by applying the definition (1.11), we complete the proof of Theorem 3.3.

Remark 3.4. If we set , , , and in Theorem 3.3, then we have [2, Theorem  3].

Acknowledgment

The present paper was supported by the Kocaeli University under Grant no. HD 2011/22.

References

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