#### Abstract

By making use of the differential subordination analytic functions, we investigate inclusion relationships among certain classes of analytic and -valent functions defined by generalized linear operator.

#### 1. Introduction

Let denote the class of functions of the form which are analytic and valent in the open unit disc

For functions , given by (1) and given by the Hadamard product (or convolution) of and is defined by

For , we say that the function is subordinate to denoted by , if there exists a Schwarz function , that is, with and , such that for all . It is well known that, if the function is univalent in , then is equivalent to and (see [1, 2]).

Now, we introduce the linear multiplier operator by It is easily verified from (4) that By specializing the parameters , , , , and , we obtain the following operators:(i) (see );(ii) (see [4, 5]); (iii) (see [6, 7]); (iv) (see ); (v) (see [9, 10]);(vi) (see ); (vii) (see ); (viii) (see ); (ix) (see ).

By using the multiplier operator , we define the following classes of functions.

Definition 1. For fixed parameters and , with and , we say that the function is in the class if it satisfies the following subordination condition: We note that(i) was studied by Cho et al. ; (ii) was studied by Aouf et al. ; (iii) was studied by Patel .
For complex numbers , , and , the Gaussian hypergeometric function is defined by where and . The series (8) converges absolutely for , hence it represents an analytic function in (see [18, Chapter 14]).
If , from the fact that , we deduce that the image is symmetric with respect to the real axis, and that maps the unit disc onto the disc . If , the function maps the unit disc onto the half plan , hence we obtain the following.

Remark 2. The function is in the class if and only if

Definition 3. The function is in the class if it satisfies the following the inequality: where ; from (9) and (10) it follows, respectively, that We note that when and , the class was studied by Aouf et al. .
Let us consider the first-order differential subordination
A univalent function is called its dominant, if for all analytic functions that satisfy this differential subordination. A dominant is called the best dominant, if for all dominants . For the general theory of the first-order differential subordination and its applications, we refer the reader to [1, 2].
The object of the present paper is to obtain several inclusion relationships and other interesting properties of functions belonging to the subclasses and by using the theory of differential subordination.
To establish our main results, we will require the following lemmas.

Lemma 4 (see ). Let , and let be a convex function with

If is analytic in , with , then

Lemma 5 (see ). Let , and consider the integral operator defined by where the powers are the principal ones.
If then the order of starlikeness of the class , that is, the largest number such that , is given by the number , where
Moreover, if , where and with , then where

Lemma 6 (see ). Let   be analytic in with and for , and let with , .

(i) Let and satisfy either or . If satisfies then and this is the best dominant.

(ii) Let be such that , and if satisfies then and this is the best dominant.

#### 2. Inclusion Relationships

Unless otherwise mentioned, we assume throughout this paper that and the power is the principal one.

Theorem 7. Let (1)Supposing that for all , then .(2)Moreover, if we suppose in addition that then where the bound is the best possible.

Proof. Let , and put the function is analytic in , with and . Differentiating (29) logarithmically with respect to , we have then, using (5) in (30), we obtain By differentiating both sides of (31) logarithmically with respect to and multiplying by , we have Combining (32) together with , we obtain that the function satisfies the Briot-Bouquet differential subordination as follows:
Now we will use Lemma 4 for the special case and . Since is a convex function in , a simple computation shows that whenever (25) holds, we have ; that is, . If in addition, we suppose that the inequality (26) holds, then all the assumptions of Lemma 5 are verified for the above values of , , and . Then it follows the inclusion , where the bound given by (28) is the best possible.
From Theorem 7, according to the definitions (7) and (11), we deduce the next inclusions.

Corollary 8. Let , such that (25) holds.(1)Supposing that for all , then (2)If we suppose in addition that (26) holds, then where is given by (28). As a consequence of the last inclusion, one has .
For the special case , Theorem 7 reduces to the following.

Corollary 9. Let .(1)Supposing that for all , then(2)If we suppose in addition that then where the bound is the best possible.

Theorem 10. Let , where and , then for , where

Proof. Since , the function given by is analytic in with and . Using (5) in (42) and taking the logarithmic differentiation in the resulting equation, we obtain If we denote , then and and substituting in (43) we obtain hence By using the well-known results  together with the inequality (45), we get Since the right hand side term of the inequality (47) is nonnegative whenever is given by (41), using the fact that the real part of an analytic function is harmonic, we deduce that for .
For a function , let the integral operator defined by Saitoh  and Saitoh et al.  From (4) and (48), we have

We now prove the next result.

Theorem 11. Let and (i)Supposing that for all , then (ii)Moreover, if we suppose in addition that then where the bound is the best possible.

Proof. Let , and suppose that for all . Let then is analytic in , with and . Taking the logarithmic differentiation in (55), we have Now, by using (49) in (56), we obtain By differentiating in both sides of (57) logarithmical with respect to and multiplying by , we have Since , from (58), we obtain that the function satisfies the Briot-Bouquet differential subordination
Now we will use Lemma 4 for the special case and ; we have , that is, . If we suppose in addition that the inequality (52) holds, then all the assumptions of the Lemma 5 are satisfied for ,  , and , hence it follows the inclusion , and the bound given by (54) is the best possible.

Taking in Theorem 11, we obtain the next corollary.

Corollary 12. Let and .(1)Supposing that for all , then(2)If we suppose in addition that then where the bound is the best possible.

Theorem 13. Let , and let with and . Suppose that If with for all , then implies where is the best dominant.

Proof. Let us put then is analytic in , with and for all . By differentiating both sides of (68) logarithmical with respect to and using (5), we have Now the assertions of Theorem 13 follows by using Lemma 6 for the special case .

Putting and , in Theorem 13, we obtain the following corollary.

Corollary 14. Assume that satisfies either If with for all , then implies and is the best dominant.

#### 3. Properties Involving the Multiplier Operator

Theorem 15. If , then for all , with ,   and , the next subordination holds

Proof. If , from (7) it follows that Moreover, the function defined by (74) and the function given by are convex in . By combining a general subordination theorem [25, Theorem 4] with (74), we get For every analytic function in with , we have and thus, from (76) and (77), we deduce This last subordination implies and by simplification, we get the assertion of Theorem 15.

Corollary 16. If , then for the next inequalities hold All of the estimates asserted here are sharp.

Proof. Taking and in (73) and using the definition of subordination, we obtain where is analytic function in with and for . According to the well-known Schwarz's theorem, we have for all .(i)If , then we find from (83) that (ii)If , we can easily obtain This proves the inequality (80) for . Similarly, we can prove the other inequalities in (80) and (81). Now, for and , we observe from (83) that and for , (82) is a direct consequence of (83).
It is easy to see that all of the estimates in Corollary 16 are sharp, being attained by the function defined by

Remark 17. (i) Putting and in our results, we obtain the results obtained by Aouf et al. .
(ii) By specializing the parameters , and , we obtain various results for different operators defined in Section 1.

#### Acknowledgment

The authors thank the referees for their valuable suggestions which led to improvement of this paper.