Analytic Functions Related with Starlikeness
The aim of present investigation is to study a new class of analytic function related with the Sokol-Nunokawa class. We derived relationships of this class with strongly starlike functions and obtained many interesting results.
Let be the class of functions having the series representation,which are analytic in the open unit disk and normalized by and .
A function is in class of univalent function if and only if implies , for all , . An analytic function subordinate to an analytic function denoted as if and only if there exists an analytic function in with and , , such that , . In particular, if is univalent in , then if and only if and .
Let denote the class of analytic functions of the form such that in . A function is in class of normalize convex function if and only if . Also, a function is in class if and only if .
In 1991, Goodman  introduced the class of uniformly convex function . A function is in class if, for every circular arc with center in , the arc is convex. Also, a function is in class if, for every circular arc contained in with center in , the arc is starlike.
The classes and were studied by Ma and Minda [2, 3] and Ronning [4, 5]. They provided the analytic characterization of class and as follows:and
In 1999, Kanas et al.  introduced the class , , of -uniformly convex functions as follows:
Furthermore, the class of -uniformly starlike functions is defined as follows:
Many authors had investigated the interesting properties of the abovementioned classes. For details, see [6–13]. The class [14, 15] denotes the class of strongly starlike functions of order , which is defined as follows:
In 2015, Sokol et al.  introduced the class as follows:
For , , in a recent paper, Darus et al.  defined a new class as follows:
The class has some interesting relationship with other classes of univalent functions. Motivated from [17–21], we introduced a new class of analytic functions and found its relation with some other existing classes of analytic functions.
Definition 1. Let , . Then, is in class if and only ifFor special values of parameter , we obtained many known classes of analytic functions. For example, if we take , then , the class of convex functions.
2. Set of Lemmas
To prove our main results, we need the following lemmas.
Lemma 1. Let and and let be a complex-valued function satisfying the conditions: (i) is continuous in a domain .(ii) and .(iii) whenever and . If is a function that is analytic in such that and for , then . This result is due to Miller .
Lemma 2. Let and . Then, , where
Proof. Let . Then, . Since .
We formulate a functional by taking , , thus . Clearly, (i) is continuous in domain (ii)Consider . Then, , where we have used the fact that . This implies that , where .
Clearly, , if and only if and . From , .
From , we have . Now, using Lemma 1, we have the required result.
Lemma 3 (see ). If and the complex number satisfies , then the differential equation , has a univalent solution in given by
If is analytic in and satisfies , then , and is the best dominant.
Lemma 4 (see ). Let and be complex numbers. Then, for , .
Lemma 5. Let be analytic in of the form
, with in . If there exists a point , such that and for some , then we have , where , where . This result is the generalization of Nunokawa’s lemma [25, 26].
3. Main Results
In this section, we derived our main results.
Theorem 1. The function is in the class if and only if .
Proof. Let , then some simple calculation gives usBy using (9) and (12), we obtainUsing (12) and (13), it is clear that if and only ifIt is suffices to study , , and in (14).Then, equation (15) leads to a simple computation which shows that
orThe right-hand side of the above inequality (16), is seen to have minimum for and minimum value is . Hence, a necessary and sufficient conditions for (16) is or . This completes the proof.
Corollary 1. The function belongs to a class if and only if .
Remark 1. It is important to note that if , then .
Theorem 2. Let . Then, , where is the class of starlike functions of order , where is defined in (10) and and .
Proof. Let , where is analytic in unit disk and . Now, using (9), we obtain . This implies that , where . The above relation can be written in the following Briot Bouquet differential subordination . Now, using Lemma 3, we have this complete the proof.
Special case: for and , we have , that is, implies .
Theorem 3. Let of the form ; then, is strongly starlike of order , where .
Proof. Let then, is of the form . Now using the definition of class , we have .
If there exists a point , we have . Then, from Lemma 5, we have , where andFor the case , we obtainAlso, we have .
Then, from for , we haveTherefore, .
By using (18) and (19), we have , which is a contradiction, therefore, for . Similarly, we can prove the case , by using the same method as the above and will get a contradiction. This proves that is strongly starlike of order .
Theorem 4. If , then , where , with and .
Proof. Let , where is analytic in and . .
This implies that , where , . The above relation can be written in the following Briot–Bouquet differential subordination, . Using Lemma 4, for and , we have , which is required.
Remark 2. If , then and , where is defined in (10).
Theorem 5. Let and ; then, contains an open disk of radius .
Proof. Let be a complex number such that , for , then .
Since is univalent in so thatSince , thenUsing (21) in (20), , where is defined in (10).
Theorem 6. Let such that . Then,where is defined in (10).
Proof. As , and . Now, , and . Then, , this implies , where , and , in which is defined in (10).
In this article, we studied a new class of analytic functions. We derived certain necessary and sufficient condition, order of strongly starlike. We also pointed out some existing results in literature.
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
SGAS and SH came with the main thoughts and helped to draft the manuscript, AT wrote, reviewed, and edited the manuscript, AR proved the main theorems, and MD and SN are revised the paper. All authors read and approved the final manuscript.
The sixth author was supported by the Universiti Kebangsaan Malaysia grant GUP-2019-032.
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