#### Abstract

We introduce a new subclass of analytic functions in the unit disk , defined by using the generalized hypergeometric functions, which extends some previous well-known classes defined by different authors. Inclusion results, radius problems, and some connections with the Bernardi-Libera-Livingston integral operator are discussed.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the unit disk .

The* convolution* (or the* Hadamard product*) of two functions and , where is given by (1) and , , is defined as

For the complex parameters with , where , , and , we define the* generalized hypergeometric function * as follows (see [1, 2]): where (or the* shifted factorial*) is defined by

Using the function defined byLiu and Srivastava [3] introduced and studied the properties of the linear operator , defined by the Hadamard (or convolution) productwhere the function is analytic and -valent in the punctured unit disk and has the form . Note that linear operator was motivated essentially by the work of Dziok and Srivastava [4].

Corresponding to the function defined bywe introduce a function given bywhere .

Analogous to defined by (6), we now define the linear operator on as follows:For convenience, we write

*Remark 1. *The linear operator includes various other linear operators which were considered in some earlier works:(i)In particular, for , , and , we obtain the linear operator which was defined by Hohlov [5].(ii)Moreover, putting , , , and , we obtain the well-known Carlson-Shaffer operator (see [6, 7]).

From definitions (8) and (9), using notation (10), it is easy to prove the differentiation formula

We note that the operator is closely related to the Choi-Saigo [8] operator for analytic functions, which includes the integral studied by Liu [9] and K. I. Noor and M. A. Noor [10].

Let be the class of functions , analytic in the unit disk , satisfying the condition andwhere , , and . This class was introduced in [11], and as a special case we note that the class was defined by Pinchuk in [12]. Moreover, is the class of analytic functions with the real part greater than .

*Remark 2. *(i) Like in [13, 14], it can easily be seen that the function , analytic in , with , belongs to if and only if there exists the functions such that(ii) It is known from [15] that the class is a convex set.

We will assume throughout our discussions, unless otherwise stated, that , , , with , , and all the powers represent* the principal branches*; that is, .

Using the linear operator , we will define the following classes of analytic functions.

*Definition 3. *Let , , and let be a complex number such that . A function is said to be in the class if and only ifwhere and satisfies the condition

*Remark 4. *From the above definition, the following subclasses of emerge as special cases:(i)For, , , , , and , we haveand this class was studied by Chen [16].(ii)When, , and , then reduces to the class studied by Noor [13].(iii)For, , , , and , we obtain the classwhich was studied by Ponnusamy and Karunakaran [17].

We will use the following lemmas to prove our main results.

Lemma 5 (see [18]). *If is an analytic function in , with , and if is a complex number satisfying , , thenimplieswhere is given byand is an increasing function of , and . The estimate is sharp in the sense that the bound cannot be improved.*

Lemma 6 (see [19]). *Let , , and let be a complex-valued function satisfying the following conditions: *(i)* is continuous in a domain .*(ii)* and .*(iii)*, whenever and .**If is an analytic function in , such that for all , and for all , then for all .*

In this paper, we investigate several properties of the class , like inclusion results and radius problems; moreover, a connection with the Bernardi-Libera-Livingston integral operator is also discussed.

#### 2. Main Results

Theorem 7. *If , then .*

*Proof. *Let an arbitrary function , and denotewhere is analytic in , with , and satisfies condition (15). From part (i) of Remark 2, we have that , if and only ifwhere .

Using the differentiation formula (11) together with (15), after an elementary computation, we obtainwhere is given by (15).

Now, using the representation formula (22), we haveSince , from relations (23) and (24), it follows thatand using the substitution, , the above relation becomesTo prove our result, we need to show that (26) implies , . We will define the functional by taking , and , and thus we haveIt is easy to see that the first two conditions of Lemma 6 are satisfied; hence, we proceed to verify condition (iii). Since , that is, for all , , it follows thatwhenever . Using Lemma 6, we conclude that , for , which completes our proof.

Theorem 8. *If , then .*

*Proof. *If we consider an arbitrary function , then , whereAccording to Theorem 7, we haveand a simple computation shows thatSince the class is a convex set (see (ii) from Remark 2), it follows that the right-hand side of (31) belongs to for , which implies that .

Now, let us define the operator byFor , the operator was introduced by Bernardi [20], while the special case was previously studied by Libera [21] and Livingston [22].

Theorem 9. *If , is given by (32), and with , thenimplies thatwhere is given by (20), with .*

*Proof. *Differentiating relation (32), we haveand using definition (9), this impliesIf we letaccording to part (i) of Remark 2, we need to prove that is of the formwhere .

Using (36), from the above relation, we have Thus, from part (i) of Remark 2, it follows thatand from Lemma 5, we conclude that , , with given by (20) and .

The next result deals with the converse of Theorem 7.

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

*Proof. *For arbitrary , let us define the function as in (43). Thus, it follows thatand satisfies the conditionFrom part (i) of Remark 2, we have that (42) holds if and only ifwhere .

Using the above representation formula, similar to the proof of Theorem 7, we deduce that and substituting , , we finally obtain where .

To prove our result, we need to determine the value of , such thatwhenever.

Using the well-known estimates for the class [23], that is, and according to (43), we obtainfor all and .

A simple computation shows that () if and only ifAssuming that (50) holds, from (49), we deduce thatfor . It is easy to see that the right-hand side of the above inequality is greater than or equal to zero if and only ifand combining this with (50), we obtain our result.

*Remark 11. *We note the following special case obtained from the above result: for , formula (41) reduces to

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work of the third author was entirely supported by the grant given by* Babeş-Bolyai* University, dedicated for* Supporting the Excellence Research 2015*. The first author, Badr S. Alkahtani, is grateful to King Saud University, Deanship of Scientific Research, College of Science Research Center, for supporting this project.