Abstract
In this paper, we introduce a new subclass of analytic functions in open unit disc. We obtain coefficient estimates, extreme points, and distortion theorem. We also derived the radii of close-to-convexity and starlikeness for this class.
1. Introduction
Let denote the class of normalized analytic functions in open unit disc and having Taylor series of formSilverman [1] introduced and studied a subclass of consisting of functions of the form
A complex valued function is said to be univalent in if implies , for all , Let be the subclass of composed of univalent functions in . By , , and , we mean the well-known subclasses of that are starlike, convex, and close-to-convex functions, respectively; for detail see [2, 3].
In 1991, Goodman [2, 3] introduced classes and of uniformly convex and uniformly starlike functions, respectively. A function is uniformly convex if maps every circular arc contained in with center onto a convex arc. The function is uniformly starlike if maps every circular arc contained in with center onto a starlike arc with respect to . A more useful representation of and was given in [2, 3]; see also [4–7] as
and
In 2011, Noor et. al [8] introduced and studied a class , , as follows.
A function is said to be in the class if and only if
The abovementioned few classes were widely investigated by many authors in the last decades; see [4, 8–15] and the references cited therein. By taking motivation from cited work, we define a unified class of analytic functions as follows.
Definition 1. A function is said to be in the class if and only if
where , , ,
We further let .
Special Cases(i) [8].(ii) [8].(iii) [16].(iv)[16].(v) [14].
Throughout this paper , , , and , unless otherwise stated.
In the next section we shall present our major findings as theorems and corollaries.
2. Main Results
Theorem 2. Let be given by (1); then is in ifwhere
Proof. Assume that inequality (7) holds true. Then we have This expression is bounded above by if
When , we obtain the main result proved by Noor et at. [8] stated as follows.
Corollary 3. A function given by (1) is in the class , if it satisfies the following condition:
When , , then we have the following known result proved by Noor et at. [8] stated as follows.
Corollary 4. A function given by (1) is in the class , if it satisfies the following condition:
Theorem 5. A necessary and sufficient condition for any function given by (2) to be in the class is that
Proof. In view of Theorem 2, we need only to prove the necessity. If and is real, then, by relation (6), we have
Letting along the real axis, we obtain the required inequality (13).
Corollary 6. Let and be of the form (2). Then
Next, we give the growth and distortion theorem for the class
Theorem 7. Let the function given by (2) be in the class Then, for , we have
and
Proof. From Theorem 5, for , we have
For given by (2) and using the triangle inequality we have
and
Theorem 8. Let be of the form (1). Then, for , we have
Proof. By differentiating (2) and after some simplification we haveandAs so, from Theorem 5, or equivalentlyUsing (25) in (22) and (23) yields required inequality (23).
Theorem 9. Let , for , , , and If the functions are in the class then
is in class
Proof. The proof follows immediately by using Theorem 5.
Theorem 10. Let , , , and
Then is in the class if and only if it can be expressed in the form
Proof. Consider Then, by Theorem 5, we haveand hence
Conversely assume that given by (2) is in the class Then By settingand , we can see that can be expressed in the form (28), which completes the proof.
Theorem 11. Let the function of the form (2) be in the class Then is close to convex of order in ,
Proof. It is well-known that for if (see [12]), where is given by (33).
From (2), we have Clearly , ifAs , from Theorem 5, inequality (36) will be true if and this implies
Theorem 12. Let of form (2) be in the class Then is starlike of order in ,
Proof. To prove , for it is sufficient to show (see [12]). From (2) we have Thus , ifAs , (42) must hold if which gives us , where is given by (39).
Next we prove that the class is closed under generalized Bernardi-Livingston operator defined as(see [17]).
Theorem 13. If then
Proof. Considerusing (2) and, after integration, we obtainSince , so an easy calculation leads to and, therefore,
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors jointly work on results, and they read and approved the final manuscript.
Acknowledgments
The work here is supported by UKM grant: GUP-2017-064 and DPP-2015-FST.