1. Introduction and Definitions
Let denote the class of functions of the form
which are analytic in the open unit disk . Also let f and g be analytic in with . Then we say that f is subordinate to g in , written or , if there exists the Schwarz function w, analytic in such that , and . We also observe that
if and only if
whenever g is univalent in .
For functions , given by
we define the Hadamard product (or convolution) of and by
Making use of the principle of subordination between analytic functions, Bhoosnurmath and Devadas  considered the subclasses and
of the class for and as following (see
also  and  ):
We note that
where the classes and are introduced and studied by many authors (see   and  ). Furthermore, denote the α-spirallike functions studied by Spacek  , which are univalent in .
With a view to define the Srivastava-Attiya transform, we recall here a general Hurwitz-Lerch zeta function, which is defined in  by the following series:
For further interesting properties and characteristics of the Hurwitz-Lerch Zeta and other related functions see   and  .
Recently, Srivastava and Attiya  have introduced the linear operator , defined in terms of the Hadamard product by
The operator is now popularly known in the literature as the Srivastava-Attiya operator. Various class-mapping properties of the operator (and its variants) are discussed in the recent works of Srivastava and Attiya  , Liu  , Murugusundaramoorthy  , Yuan and Liu  and others.
It is easy to observe from (1.1) and (1.4) that
We note that:
2) (see Alexander  );
3) (see Flett  );
4) (see Jung et al.  );
5) (see Sǎlǎgean  ).
It is easily verified from (1.6) that
Next, by using the linear operator , we introduce the following new
classes of analytic functions for , , and
It follows from the definitions (1.8) and (1.9) that
In this article, we investigate some convolution properties and coefficient estimates for the classes and . Furthermore, several inclusion properties and relevant connections of the results presented here with those obtained in earlier works are also discussed.
2. Convolution Properties and Coefficient Estimates
Unless otherwise mentioned, we will assume in the reminder of this paper that
, and . In order to establish our convolution
properties, we shall need the following lemmas due to Bhoosnurnath and Devadas   .
Lemma 2.1 (  ). The function defined by (1.1) is in the class if and only if
Lemma 2.2 (  Lemma 3 with n = 1). The function defined by (1.1) is in the class if and only if
We begin by proving the following theorem.
Theorem 2.3 The function defined by (1.1) is in the class if and only if
Proof. From Lemma 2.1, we find that if and only if
where M is given by (2.2). Then, by applying (1.6), the left hand side of (2.5) becomes
which completes the proof of Theorem 2.3.
Theorem 2.4 The function defined by (1.1) is in the class if and only if
Proof. From Lemma 2.2, we observe that if and only if
where N is given by (2.4). Then, by using (1.6), the left hand side of (2.6) may be written as
which evidently proves Theorem 2.4.
Next, we determine coefficients estimates for a function of the form (1.1) to be in the classes and .
Theorem 2.5 Let and . The function defined by (1.1) is in the class if its coefficients satisfy the condition
by virtue of Theorem 2.3, we conclude that . Thus, the proof of Theorem 2.5 is completed.
By using arguments similar to those above with Theorem 2.4, we can prove the following theorem.
Theorem 2.6 Let and . The function defined by (1.1) is in the class if its coefficients satisfy the condition
3. Inclusion Properties and Applications
To prove the inclusion properties for the classes and , we shall require the following lemma due to Eenigenburg et al.  .
Lemma 3.1 (  ). Let be convex univalent in with for all . If is analytic in with , then
implies that .
By applying Lemma 3.1, we prove
Theorem 3.2 Let and . If
Proof. Let for and , and set
where is analytic in with . By applying the identity (1.7), we obtain
Making use of the logarithmic differentiation on both side in (3.3), we have
Since the function is convex univalent in with , from (3.1) we see that
Thus, by using Lemma 3.1 and (3.4), we observe that in , so that . This completes the proof of theorem 3.2.
Theorem 3.3 Let and . Suppose that (3.1) holds for all . Then
Proof. Applying (1.10) and Theorem 3.2, we observe that
which evidently proves Theorem 3.3.
Putting and in Theorem 3.2 and 3.3, we have the following corollary.
Corollary 3.4 Suppose that and
Finally, we consider the generalized Bernardi-Libera-Livingston integral operator defined by (cf.   and  )
Theorem 3.5 Let , and . Suppose that
If , then .
Proof. If we set
where is analytic in with . By virtue of (3.5), we observe that
In view of (3.7) and (3.8), we have
By using same argument as in the proof of Theorem 3.2 with (3.6), we conclude that . This evidently completes the proof of Theorem 3.5.
Theorem 3.6 Let , and . Suppose that (3.6) holds for all . If , then .
Proof. By using Theorem 3.4, it follows that
which completes the proof of Theorem 3.6.
This work was supported by Daegu National University of Education Research grant in 2017.
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