Research Article | Open Access
Zahra Maleki, Saeid Shams, Ali Ebadian, Ebrahim Analouei Adegani, "On Some Problems of Strongly Ozaki Close-to-Convex Functions", Journal of Function Spaces, vol. 2020, Article ID 6095398, 9 pages, 2020. https://doi.org/10.1155/2020/6095398
On Some Problems of Strongly Ozaki Close-to-Convex Functions
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 ). 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.  studied the majorization issue for a general well-known category of starlike functions, which was defined by Ma and Minda . 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.  introduced a subclass of and extended the class , defined by Nunokawa et al. in , 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  considered the subclass of locally univalent functions in satisfying the inequality: for some .
Recently, Allu et al. , 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 . Let and . Then, is called strongly Ozaki close-to-convex if and only if
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  (see also ), 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 (, 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 :
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.
Theorem 7. Let . If and then
Proof. The result is proven by contradiction. To prove our result, we set the function by
Then, is analytic in , , for all , and
If there is a point , then
with , and
Then, from Lemma 3, we have
where and is stated by (17) or (18).
For the case when and we have which contradicts to (41).
Next, for the case when and by applying the same method mentioned above, it can be concluded that which contradicts to (41).
As a result, from the above contradictions, we obtain and therefore, the proof is completed.
Corollary 8. Let , and . If , then .
Remark 9. >According to the assumptions of Corollary 8, if , then implies and so it is well known that is univalent.
3. Coefficient Bounds
In this section, we find sharp bounds on Fekete-Szegö functionals and logarithmic coefficients (see [17–22]) for functions belonging to the class . Also, we present a general problem of coefficients in this class. To prove our main results, some requirements are needed. We remark in passing that the logarithmic coefficients of are defined by the next form:
Ma and Minda  defined the class consisting of several well-known classes as follows: where in here, it is supposed that is a univalent function in with such that it has the following form:
Lemma 13 . Let be a convex function in with the form If function , then
Theorem 14. Let . Then, All the inequalities are sharp.
Proof. Let . From (11), it follows that
Now, it is enough to obtain the estimate of . By Lemma 10 with for , we obtain
where is given by Lemma 11,
First, if we consider
then it is clear that for and for . Therefore, we conclude for .
Also, regarding then it is clear that for . Thus, in the above relation, is equivalent to . Also, that is, holds for (see Figure 1). Therefore, considering the above results, we conclude for .
Next, considering it is clear that for , and according to the above computation, the inequality holds for . Therefore, for .
Finally, let us consider
It is clear that for . On the other hand, the inequality is equivalent to and this last inequality holds for . Therefore, for . Since , so for .
By applying the above four conclusions from (66), it follows that
Suppose that such that Then, the functions are of the following form: and the are extremal functions for diverse problems in the subclass . Hence, the extremal function for the absolute value of the coefficient is , and those for the absolute value of the coefficients and in the first relations are given as and , respectively, whereas in the second relations, it is given as .