Connections between Certain Subclasses of Analytic Univalent Functions Based on Operators

Abstract

In this paper, by applying the Hohlov linear operator, connections between the class ,  , and two subclasses of the class of normalized analytic functions are established. Also an integral operator related to hypergeometric function is considered.

1. Introduction

Let denote the family of functions that are analytic in the open unit disk with the normalization Let denote the subclass of functions in which are also univalent in . A well-known subclass of is the class ST (see, e.g., [1]) of starlike functions of the form (1), satisfying Re. Another class , introduced in [2], consists of functions satisfying Re Various properties of this class have been obtained in [2–4]. A related class has been recently considered in [5], initially introduced in [6]. A function of the form (1) is said to be in the class if

Theorem 1 (see [5]). A function of the form (1) is in the class if

Ponnusamy and Ronning [7] introduced and studied the class ,   of functions for which there exists a number such that . If the function of the form (1) belongs to the class , then For complex numbers ,  , and , the Gaussian hypergeometric function is defined by where is the Pochhammer symbol given by It is known that The Hadamard product (or convolution) of two functions defined by (1) and given by is defined by Hohlov [8] introduced a linear operator , corresponding to the Gaussian hypergeometric function which is defined by the convolution For a function of the form (1), we have The operator is a natural extension of several operators such as Alexander, Libera, Bernardi, and Carlson-Shaffer operators denoted, respectively, by ,  ,  , and .

Motivated by the work of Thulasiram et al. [9], in this paper, by applying the linear operator , we establish some interesting connections between the class and the classes ST and consisting of functions given by (1). Also we consider an integral operator related to the hypergeometric functions.

2. Main Results

In the sequel the function is given by (1).

Theorem 2. Let . Also let be a real number such that . If and if the inequality is satisfied, then

Proof. Let . Applying the well-known estimate due to Nevanlinna [10] for the coefficients of the functions , in view of Theorem 1, we need to prove that By virtue of the relation , and on writing and and using the fact that , we havewhich is satisfied by the hypothesis.

On setting , an improvement of the assertion of Theorem 2 is obtained as given in Theorem 3.

Theorem 3. Let . Also let be a real number such that . If , and if the inequality is satisfied, then

Theorem 4. Let . Also, let be a real number such that ,,. If and the inequality is satisfied, then .

Proof. Let be of the form (1) and let . By virtue of Theorem 1 and in view of (10), it remains to show that Using the inequality (4) and the relations and , we obtain that

On taking , an improvement of the assertion of Theorem 4 is obtained as given in Theorem 5.

Theorem 5. Let . Further let be a real number such that ,  . If and the inequality is satisfied, then

3. An Integral Operator

We now obtain results in connection with a particular integral operator [11] defined by where is given by (5).

Theorem 6. Let . Also let c be a real number such that . Let be given by (19). If the hypergeometric inequality is satisfied, then .

Proof. The function has the series representation given by In view of Theorem 1, it is enough to prove that Nowby hypothesis.

A result analogous to Theorem 6 can be stated for the class in Theorem 7.

Theorem 7. Let . Also let c be a real number such that . Let and be of the form (1). If the hypergeometric inequality is satisfied, then .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author Maisarah Haji Mohd is supported by USM Short Term Grant 304/PMATHS/6313192.