On Certain Subclass of Meromorphic Spirallike Functions Involving the Hypergeometric Function

We introduce and investigate a new subclass M m 1(θ, λ, η) of meromorphic spirallike functions. Such results as integral representations, convolution properties, and coefficient estimates are proved. The results presented here would provide extensions of those given in earlier works. Several other results are also obtained.


Introduction
Let M denote the class of functions of the form Let , ∈ M, where is given by (1) and is defined by Then the Hadamard product (or convolution) * of and is defined by Let P denote the class of functions given by which are analytic in U and satisfy the condition For which is real with | | < /2, 0 ≤ < 1, we denote by MS * ( , ) and MK * ( , ) the subclasses of ∈ M which are defined, respectively, by R ( ( ) ( ) ) < − cos ( ∈ U * ) , ( ) ) < − cos ( ∈ U * ) .
By setting = 0 in (7), we get the well-known subclasses of ∈ M consisting of meromorphic functions which are starlike and convex of order (0 ≤ < 1), respectively. For some recent investigations on meromorphic spirallike functions and related topics, see, for example, the earlier works [1][2][3][4] and the references cited therein.
For > 1, Wang et al. [5] and Nehari and Netanyahu [6] introduced and studied the subclass M( ) of M consisting of functions satisfying Let A be the class of functions of the form 2 The Scientific World Journal which are analytic in U. A function ∈ A is said to be in the class S * ( , , ) if it satisfies the condition The function class S * ( , , ) is introduced and studied recently by Orhan et al. [7]. An analogous of the class S * ( , , ) has been studied by Murugusundaramoorthy [8].
From the definition of the operator H [ , ] , it is easy to observe that where is a positive number for all ∈ N. Recently, Aouf [11], Liu and Srivastava [12], and Raina and Srivastava [13] obtained many interesting results involving the linear operator H [ , ]. In particular, for we obtain the following linear operator: which was introduced and investigated earlier by Liu and Srivastava [14] and was further studied in a subsequent investigation by Srivastava et al. [15]. It should also be remarked that the linear operator H [ , ] is a generalization of other linear operators considered in many earlier investigations (see, e.g., [16][17][18]).
The following lemma will be required in our investigation.
Proof. From (19), we have Combining (21), we find that Thus, for ≥ 2, we deduce from (22) that This completes the proof of Lemma 2.

Main Results
We begin by proving the following integral representation for the class M ( , , ).
Next, we derive a convolution property for the class M ( , , ).
Combining (1) and (37), we have Evaluating the coefficient of in both sides of (38) yields By observing the fact that | | ≤ 2 for ∈ N, we find from (39) that In order to prove that we use the principle of mathematical induction. Note that Therefore, assume that Combining (41) Hence, by the principle of mathematical induction, we have as desired. By means of Lemma 2 and (42), we know that (20) holds. Combining (47) and (20), we readily get the coefficient estimates asserted by Theorem 5.
In what follows, we present some sufficient conditions for functions belonging to the class M ( , , ).
If we take = 1 − ( − 1) cos in Theorem 7, we obtain the following result.
then it belongs to the class M ( , , ).
Proof. In virtue of Corollary 8, it suffices to show that condition (52) holds. We observe that which is equivalent to This completes the proof of Theorem 9.