Subclasses of Analytic Functions Defined by Generalized Hypergeometric Functions

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.