Abstract

We use the familiar convolution structure of analytic functions to introduce two new subclasses of multivalently analytic functions of complex order, and prove several inclusion relationships associated with the -neighborhoods for these subclasses. Some interesting consequences of these results are also pointed out.

1. Introduction and Preliminaries

Let denote the class of functions of the form which are analytic and -valent in the open unit disk If is given by (1.1) and is given by then the Hadamard product (or convolution) of and is defined (as usual) by

We denote by the subclass of consisting of functions of the form which are -valent in

For a fixed function defined by we introduce a new class of functions belonging to the subclass of which consists of functions of the form (1.5), satisfying the following inequality:

We note that there exist several interesting new (or known) subclasses of our function class . For example, if in (1.7), we obtain the class studied very recently by Prajapat et al. [1]. On the other hand, if the coefficients in (1.6) are chosen as follows: and is replaced by in (1.4) and (1.5), then we obtain the class of -valently analytic functions (involving the multiplier transformation operator defined in [2]) which was studied recently by Srivastava et al. [3]. Also, if we set in (1.7) and if the arbitrary sequence in (1.6) is selected as follows: also if is replaced by in (1.4) and (1.5), then we obtain the class of -valently analytic functions (involving the familiar Ruscheweyh derivative operator) investigated by Raina and Srivastava [4]. Further, when in (1.7), then reduces to the class studied recently by Ali et al. [5]. Moreover, when with the parameters being so chosen that the coefficients in (1.6) satisfy the following condition: then the class transforms into a (presumably) new class defined by

The operator involved in (1.14), is the Dziok-Srivastava linear operator (see for details [6]; see also [7, 8]) which contains such well-known operators as the Hohlov linear operator, Saitoh generalized linear operator, Carlson-Shaffer linear operator, Ruscheweyh derivative operator as well as its generalized version, the Bernardi-Libera-Livingston operator, and the Srivastava-Owa fractional derivative operator. One may refer to [7] or [6] for further details and references for these operators. The Dziok-Srivastava linear operator defined in [6] has further been generalized by Dziok and Raina [7] (see also [8, 9]).

Following a recent investigation by Frasin and Darus [10], let then a -neighborhood of the function is defined by It follows from the definition (1.16) that if then We observe that where and denote, respectively, the -neighborhoods of the function defined by Ruscheweyh [11] and Silverman [12].

Finally, for a fixed function let denote the subclass of consisting of functions of the form (1.5) which satisfy the following inequality:

The object of the present paper is to investigate the various properties and characteristics of functions belonging to the above-defined subclasses of -valently analytic functions in . Apart from deriving coefficient inequalities for each of these function classes, we establish several inclusion relationships involving the -neighborhoods of functions belonging to these subclasses.

2. Coefficient Bound Inequalities

We begin by proving a necessary and sufficient condition for the function to be in each of the classes

Theorem 2.1. Let be given by (1.5). Then is in the class if and only if

Proof. Assume that Then, in view of (1.5)–(1.7), we get which yields Putting in (2.4), the denominator expression on the left-hand side of (2.4) remains positive for and also for all . Hence, by letting , through real values, inequality (2.4) leads to the desired assertion (2.2) of Theorem 2.1.
Conversely, by applying the hypothesis (2.2) of Theorem 2.1, and letting we find that Hence, by the maximum modulus principle, we infer that which completes the proof of Theorem 2.1.

Remark 2.2. In the special case when Theorem 2.1 corresponds to a result given recently by Srivastava et al. [3, Theorem 1, page 3]; Theorem 2.1 yields the result given recently by Raina and Srivastava [4, Theorem 1, page 3]; Theorem 2.1 reduces to the result of Orhan and Kamali [13, Lemma 1, page 57]; Theorem 2.1 gives a recently established result due to Ali et al. [5, Theorem 1, page 181].

The following results concerning the class of functions can be proved on similar lines as given above for Theorem 2.1.

Theorem 2.3. Let be given by (1.5). Then is in the class if and only if

Remark 2.4. Making use of the same substitutions as mentioned above in (2.6), Theorem 2.3 yields another known result due to Srivastava et al. [3, Theorem 2, page 4]. Also, using the same substitutions as mentioned above in (2.8), we get the result of Orhan and Kamali [13, Lemma 2, page 58].

3. Inclusion Properties

We now obtain some inclusion relationships for the function classes involving the -neighborhood defined by (1.18).

Theorem 3.1. If and then

Proof. Let . Then, in view of assertion (2.2) of Theorem 2.1, and the given condition we get which implies that Applying the assertion (2.2) of Theorem 2.1 again (in conjunction with (3.5)), we obtain Hence, which by virtue of (1.18) establishes the inclusion relation (3.3) of Theorem 3.1.

In the analogous manner, by applying the assertion (2.10) of Theorem 2.3 instead of the assertion (2.2) of Theorem 2.1 to the functions in the class , we can prove the following inclusion relationship.

Theorem 3.2. If and then

Remark 3.3. Applying the parametric substitutions listed in (2.6), Theorems 3.1 and 3.2 would yield the known results due to Srivastava et al. [3, Theorem 3, page 4; Theorem 4, page 5]. Also, using substitutions (as mentioned above in (2.8)) in Theorems 3.1 and 3.2, we get the results due to Orhan and Kamali [13, Theorem 1, page 58; Theorem 2, page 59].

4. Neighborhood Properties

This concluding section determines the neighborhood properties for each of the classes which are defined as follows.

A function is said to be in the class if there exists a function such that Analogously, a function is said to be in the class if there exists a function such that inequality (4.2) holds true.

Theorem 4.1. If and then

Proof. Suppose that We then find from (1.16) that which readily implies that
Next, since , we have in view of (3.5) that so that provided that is given by (4.3). Thus, by the above definition, where is given by (4.3), which proves Theorem 4.1.