Abstract

The purpose of the current paper is to investigate some geometric properties of the class , called strongly Ozaki close-to-convex functions, such as strongly starlikeness and close-to-convexity. Further, we find sharp bounds on Fekete-Szegö functionals and logarithmic coefficients for functions belonging to the class , which incorporates some known outcomes as the specific cases.

1. Introduction and Preliminaries

Let denote the open unit dick in the complex plane . Let be the class of functions of the following normalized form: which are analytic in and represent by the class of all functions of , which are univalent in . Let denote the set of all analytic functions in that are satisfying the conditions of and for , i.e., , is considered as the family of Schwarz functions.

For two analytic functions and in the open unit dick , it is said that the function is subordinate to the function in , written , if there exists a Schwarz function such that for all . In particular, if the function is univalent in , the following equivalence holds:

We denote by the subclass of consisting of all for which is a starlike of order , with and denote by the subclass of consisting of all for which is a convex of order , with

Also, the subclass of a strongly starlike function of order in is defined as

Note that and are the class of starlike functions in and the class of convex functions in , respectively.

Furthermore, is denoted as the subclass of including functions such as close-to-convex of order if there is a function so that and we denote by the subclass of consisting of all of for which

Individually, is the class of close-to-convex functions in and is the subclass of close-to-convex functions in (see [1]). Here, we understand that is a number in

Recently, many authors have studied the families of analytic functions of the class and also investigated bound estimation problems, geometric property issues, and related topics for functions belonging to these families in [2–10] as well as in the references cited therein.

For example, Cho et al. [4] studied the majorization issue for a general well-known category of starlike functions, which was defined by Ma and Minda [11]. Also, they investigated the majorization issue for the various subclasses for different special functions . Moreover, estimates for the coefficients of majorized functions regarding the class were given. Further, Alimohammadi et al. [2] introduced a subclass of and extended the class , defined by Nunokawa et al. in [12], consisting of all satisfying and studied some geometric properties like close-to-convexity and strongly starlikeness. They determined sharp bounds of Fekete-Szegö functionals and logarithmic coefficients for this class. Kargar and Ebadian [9] considered the subclass of locally univalent functions in satisfying the inequality: for some .

Recently, Allu et al. [3], motivated essentially by the subclass , introduced the class and obtained sharp bounds for three first coefficients and the corresponding inverse coefficients for the functions of this class.

Definition 1 [3]. Let and . Then, is called strongly Ozaki close-to-convex if and only if

Note that the class was introduced in [9] and members of this class were called Ozaki close-to-convex functions. Also, was studied by Ponnusamy et al. [13]. Furthermore, .

It is remarkable that by means of the principle of subordination between analytic functions, the definition of the class can be rewritten as follows:

The present paper was undertaken to investigate some geometric features of the class such as close-to-convexity and strongly starlikeness. In addition, we found estimates for the coefficients and give sharp bounds on Fekete-Szegö functionals and logarithmic coefficients for functions belonging to the class , which incorporates some known outcomes as the specific cases.

2. Some Geometric Properties of the Class

In this section, we investigate some geometric properties like strongly starlikeness and close-to-convexity for the class to present the relation of this class with the well-known families of univalent functions. The key in proving is Nunokawa’s lemma [14] (see also [15]), and so in order to prove our result, we require the following lemmas.

We denote by the class of all complex-valued functions for which is univalent at each and for all where

Lemma 2 ([16], Lemma 2.2d (i)). Let with and let be analytic in with and . If is not subordinate to in , then there exist and such that :

Lemma 3 (see [14, 15]). Let the function given by be analytic in with and for all If there exists a point with for some then where where

Theorem 4. Let and where is given by . If satisfies the following condition: then

Proof. The result is proven by contradiction. Let and define the function by Then, it is concluded that is analytic in , , and for all . Indeed, if has a zero of order , then we have where is analytic in with Then, Hence, with , in the right hand of the above equality, the argument can properly take any value between and , which contradicts to (20).
Define the function by Then, , , and . Clearly, if and only if on . Suppose for some . Then, is not subordinate to . By applying Lemma 2, there exist and so that and . Thus, with and Therefore, Lemma 3 results in where and is stated by (17) or (18).
First, let . Then, we write , and so for , we have We define the function by Then, is a differentiable function on , and Now, we define as Then, , for , and So, the function has a negative value on and a positive value on .
According to the assumption and from (32), it follows that for all for . Also, this shows that the function defined by is nondecreasing on . Hence, Therefore, we get Now applying (30) and (37), we obtain which contradicts to (20).
Next, let . Then, we write . Thus, for and utilizing (37), we get which contradicts to (20).
From the above contradictions, it results in Hence, the proof is completed.

Corollary 5. Let where is given by and If , then .

Remark 6. According to the assumptions of Corollary 5, if , then implies and so it is well known that is univalent.