Coefficient Estimate of Biunivalent Functions of Complex Order Associated with the Hohlov Operator

Abstract

We introduce and investigate a new subclass of the function class of biunivalent functions of complex order defined in the open unit disk, which are associated with the Hohlov operator, satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients and for functions in this new subclass. Several, known or new, consequences of the results are also pointed out.

1. Introduction, Definitions, and Preliminaries

Let denote the class of functions of the following form: which are analytic in the open unit disk By we denote the class of all functions in which are univalent in . Some of the important and well-investigated subclasses of the class include, for example, the class of starlike functions of order in and the class of convex functions of order in . It is well known that every function has an inverse , defined by where

A function is said to be biunivalent in , if and are univalent in . Let denote the class of biunivalent functions in given by (1).

An analytic function is subordinate to an analytic function , written , provided that there is an analytic function defined on with and satisfying . Ma and Minda [1] unified various subclasses of starlike and convex functions for which either of the quantity or is subordinate to a more general superordinate function. For this purpose, they considered an analytic function with positive real part in the unit disk , , , and maps onto a region starlike with respect to 1 and symmetric with respect to the real axis. The class of Ma-Minda starlike functions consists of functions satisfying the subordination . Similarly, the class of Ma-Minda convex functions consists of functions satisfying the subordination .

A function is bi-starlike of Ma-Minda type or biconvex of Ma-Minda type, if both and are, respectively, Ma-Minda starlike or convex. These classes are denoted, respectively, by and . In the sequel, it is assumed that is an analytic function with positive real part in the unit disk , satisfying and , and is symmetric with respect to the real axis. Such a function has a series expansion of the form

The convolution or Hadamard product of two functions and is denoted by and is defined as where is given by (1) and . Here, in our present investigation, we recall a convolution operator due to Hohlov [2, 3], which indeed is a special case of the Dziok-Srivastava operator [4, 5].

For the complex parameters , , and , the Gaussian hypergeometric function is defined as where is the Pochhammer symbol (or the shifted factorial) defined as follows: For the positive real values , , and , by using the Gaussian hypergeometric function given by (7), Hohlov [2, 3] introduced the familiar convolution operator as follows: where Hohlov [2, 3] discussed some interesting geometrical properties exhibited by the operator . The three-parameter family of operators contains, as its special cases, most of the known linear integral or differential operators. In particular, if in (9), then reduces to the Carlson-Shaffer operator. Similarly, it is easily seen that the Hohlov operator is also a generalization of the Ruscheweyh derivative operator as well as the Bernardi-Libera-Livingston operator.

Recently, there has been triggering interest to study biunivalent function class and obtained nonsharp coefficient estimates on the first two coefficients and of (1). But the coefficient problem for each of the Taylor-Maclaurin coefficients, is still an open problem (see [6–11]). Many researchers (see [12–17]) have recently introduced and investigated several interesting subclasses of the biunivalent function class and they have found nonsharp estimates on the first two Taylor-Maclaurin coefficients and .

Motivated by the earlier work of Deniz [18] (see [19–21]) and Peng and Han [22], in the present paper, we introduce new subclasses of the function class of complex order , involving Hohlov operator , and find estimates on the coefficients and for functions in the new subclasses of function class . Several related classes are also considered, and connection to earlier known results are made.

Definition 1. A function given by (1) is said to be in the class , if the following conditions are satisfied: where the function is given by (4).
On specializing the parameters and , , and , one can state the various new subclasses of as illustrated in the following examples.

Example 2. For and , a function , given by (1), is said to be in the class , if the following conditions are satisfied: where , and the function is given by (4).

Example 3. For and , a function , given by (1), is said to be in the class , if the following conditions are satisfied: where , and the function is given by (4).
It is of interest to note that, for and , the class reduces to the following new subclasses.

Example 4. For and , a function , given by (1), is said to be in the class , if the following conditions are satisfied: where , and the function is given by (4).

Example 5. For and , a function , given by (1), is said to be in the class , if the following conditions are satisfied: where and the function is given by (4).
In the following section, we find estimates on the coefficients and for functions in the above-defined subclasses of the function class by employing the technique which is different from that used by earlier authors. Earlier authors investigated the coefficients of biunivalent functions mainly by using the following lemma.

Lemma 6 (see [23]). If , then for each , where is the family of all functions , analytic in , for which where

2. Coefficient Bounds for the Function Class

We begin by finding the estimates on the coefficients and for functions in the class .

Suppose that and are analytic in with , , and and suppose that It is well known that Thus, from (5), it follows that

Theorem 7. Let a function , given by (1), be in the class . Then where and are given by (10).

Proof. It follows from (12) that where and are given by (21) and (22), respectively.
Now, by equating the coefficients in (24), we get From (25) and (27), we find that which implies By adding (26) and (28) and by using (29) and (30), we obtain Now, by using (20) and (31), we get Hence, This gives the bound on as asserted in (23).
Next, in order to find the bound on , by subtracting (28) from (26), we get It follows from (20), (30), and (35) that By using (34), we obtain This completes the proof of Theorem 7.

By putting in Theorem 7, we have the following corollary.

Corollary 8. Let the function given by (1) be in the class . Then

By taking and , in Corollary 8, we get the following corollary.

Corollary 9. Let the function given by (1) be in the class . Then

By putting in Theorem 7, we have the following corollary.

Corollary 10. Let the function given by (1) be in the class . Then

By taking and , in Corollary 10, we get the following corollary.

Corollary 11. Let the function given by (1) be in the class . Then

3. Concluding Remarks

For the class of strongly starlike functions, the function is given by which gives and .

Remark 12. From Theorem 7, when and for the class [8], we get
On the other hand, if we take then .

Remark 13. From Theorem 7, when for the class , we get

Remark 14. By putting in Corollary 11 we obtain more accurate results corresponding to the results obtained in [19]. Further, by taking and is given by (43) (or by (45), the results obtained in Theorem 7 and Corollary 11 yield more accurate results than the results obtained in [15, 21].

Remark 15. If , , and with , then the operator turns into well-known Bernardi operator: and are the well-known Alexander and Libera operators, respectively. Further, if in (9), then immediately yields the Carlson-Shaffer operator . So, various other interesting corollaries and consequences of our main results (which are asserted by Theorem 7 above) can be derived similarly. The details involved may be left as an exercise for the interested reader.

Conflict of Interests

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