Application of Quasi-Subordination for Generalized Sakaguchi Type Functions

Abstract

The author constructs a new class of univalent functions applying the Ruscheweyh derivative. Moreover, the coefficient estimates including a Fekete-Szegö inequality of this class were determined.

1. Introduction and Definitions

Let be the unit disk and let be the class of functions analytic in , satisfying the conditionsThen each has the Taylor expansionMoreover, by we shall represent the class of all functions in which are univalent in Let be an analytic function in and , such thatwhere all coefficients are real. Also, let be an analytic and univalent function with positive real part in with , , and maps the unit disc onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor’s series expansion of such function is of the formwhere all coefficients are real and . Let be the class of functions consisting of form (5).

If the functions and are analytic in , then is said to be subordinate to , written asif there exists a Schwarz function , analytic in , withsuch that

In the year 1970, Robertson [1] introduced the concept of quasi-subordination. For two analytic functions and , the function is said to be quasi-subordinate to in and written asif there exists an analytic function such that analytic in andthat is, there exists a Schwarz function such that . Observe that if , then so that in . Also notice that if , then and it is said that it is majorized by and written by in . Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization (see, e.g., [1–3] for works related to quasi-subordination).

In [4], Sakaguchi introduced the class of starlike functions with respect to symmetric points in , consisting of functions that satisfy the condition , . Similarly, in [5], Wang et al. introduced the class of convex functions with respect to symmetric points in , consisting of functions that satisfy the condition For different parametric values, we get the classes studied in the literature by Frasin [6], Goyal et al. [7], and Owa et al. [8].

The Fekete-Szegö functional for normalized univalent functions of the form given by (3) is well known for its rich history in Geometric Function Theory. Its origin was in the disproof by Fekete and Szegö [9] of the 1933 conjecture of Littlewood and Paley that the coefficients of odd univalent functions are bounded by unity (see, for details, [9]). The Fekete-Szegö functional has since it received great attention, particularly in connection with many subclasses of the class of normalized analytic and univalent functions (see, e.g., [10–15]).

The object of the present paper is to introduce a new class of univalent functions applying the Ruscheweyh derivative, where Ruscheweyh [16] observed thatfor , where This symbol , , is called by Al-Amiri [17] the th order Ruscheweyh derivative of . We note that , , and

2. Preliminary Results

The study of special functions plays an important role in Geometric Function Theory in Complex Analysis and its related fields. Special functions can be categorized into three, namely, Ramp function, threshold function, and sigmoid function. The popular type among all is the sigmoid function because of its gradient descendent learning algorithm. It can be evaluated in different ways, most especially by truncated series expansion. The sigmoid function of the formis useful because it is differentiable. The Sigmoid function has very important properties, including the following (see [18]):(i)It outputs real numbers between and .(ii)It maps a very large input domain to a small range of outputs.(iii)It never loses information because it is a one-to-one function.(iv)It increases monotonically.

Lemma 1 (see [19]). Let the Schwarz function be given by then where

Lemma 2 (see [18]). Let be a sigmoid function and and then , , where is a modified sigmoid function.

Lemma 3 (see [18]). Let and then

3. Main Result and Its Consequences

Definition 4. A function is said to be in the class , if the following quasi-subordination holds: where with and

From the definition, it follows that if and only if there exists an analytic function with , such that If, in the subordination condition (19), , then the class is denoted by and the functions therein satisfy the condition that

Theorem 5. Let of the form (3) be in the class Then and for some and the result is sharp.

Proof. Let In view of Definition 4, we can writewhere the function is a modified sigmoid function given byCombining (4), (14), and (24), we obtainIn the light of (23) and (25), we getNow, (26) gives From (27), it follows thatFor some , we obtain from (28) and (29)Since given by (4) is analytic and bounded in , therefore, on using [20] (p 172), we have for some () On putting the value of from (31) into (30), we getIf in (32), we obtainIf in (32), letwhich is a polynomial in and hence analytic in , and maximum is attained at ,  (). We find thatwhich on using Lemma 1 shows that

For the case when , one has the following.

Corollary 6. Let of form (3) be in the class Thenand for some and the result is sharp.

Putting in Corollary 6, we obtain the following corollary.

Corollary 7. Let of form (3) be in the class Then and for some and the result is sharp.

Setting in Corollary 7, we have the following.

Corollary 8. Let of form (3) be in the class Then and for some and the result is sharp.

Setting in Corollary 7, we have the following.

Corollary 9. Let of form (3) be in the class Then and for some and the result is sharp.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This research is supported by the Scientific and Technological Research Council of Turkey (TUBITAK 2214A).