Study of Generalized -Close-to-Convex Functions Related to Parabolic Domain

Abstract

The concepts of bounded boundary rotation and parabolic domain are used to define certain new classes of analytic functions such as , , , and . The difference of coefficients, necessary conditions, distortion bounds, radius problem, and several other interesting properties of these newly defined classes are also studied.

1. Introduction

Let denote the class of functions which are analytic in the open unit disc and are represented by the series expansion of the form

Two functions and , analytic in , we say is subordinate to , written or , if there exists a Schwartz function analytic in with , such that . If the function is univalent in , then we have the following equivalence:

For , be given by (1) and , the convolution (Hadamard product) of and is defined as

Kanas and Wisniowska [1, 2] defined the subclasses of related to the conic domains , where

The domain represents the right half plane, is a domain related to parabola, , represents hyperbola and an ellipse for

We shall deal with the domain and the function given below as plays the role of extremal function for .

Let denote the class of Caratheodory functions , analytic in with and satisfying the property .

Using the concept of -calculus, Ismail et al. [3] defined a subclass of consisting of functions called -starlike in and established that is the well-known class of starlike functions with respect to origin. In [4], a firm foothold of the usage of the -calculus in the context of geometric function theory was effectively established. For the study of various subclasses of , the quantum (or -) calculus has been used as an important tool (see [5–7]).

Jackson [8, 9] first defined, with some applications, the -derivative and -integral operators. For more and potentially useful survey-cum-expository review article, refer to [10] for interested researchers and scholars. For convenience, we provide here some basics and details of -calculus which we use in this study. Throughout this paper, we will assume that satisfies the condition . (i)Let and define the -number(ii)The -derivative (or -difference) of is defined in a given subset of byprovided that exists.

From (7), we note that for a differentiable function in a given subset of .

From (1) and (7), we observe that

By using a special case of general conic domain , concept of -calculus, and principle of subordination, we define as in [11] the class as follows.

Let be analytic in with . Then, if and ony if and is given by (5).

Geometrically, the function takes all values from the domain , which is defined as

It can easily be seen that , with , and implies that . Also, , . Also, and

We generalize the class as follows.

Definition 1. An analytic function is in the class if and only if there exist , such that

It is clear that .

Using the class , we now define the following classes of analytic function.

Definition 2. Let denote the class of all functions , with such that in and .
With , if and only if in . For , .
Special cases: (i), and is well-known class of functions of bounded boundary rotation (see [12])(ii)Letbe convex univalent in (see [13]).
We note that Thus, Definition 2 can be written as that is, in . (iii), where is the class of convex functions in . Similarly, is a subclass of the class of starlike functions in

Definition 3. Let with in . Then, , if there exists such that Special cases: (i)Choose . Then, . Then, reduces to a subclass of consisting of generalized close-to-convex functions. The class has been introduced and studied in [14], where is the class of close-to-convex functions (see [15])(ii)Let , where is given by (44). Then,In this case, we say that in .

Remark 4. (i)For , the class of-convex functions is defined asAs , and the class of -starlike functions, first introduced and studied in [3]. Also, if and only if , and it has been proved [3] that (ii)If , then is starlike of order for (see [13]).

Definition 5. Let with , for . Then, is said to belong to the class if and only if there exists a function such that, for , , ,

We note that and , where is the class of -close-to-convex functions.

2. Preliminaries

Lemma 6 (see [16]). Let be subordinate to . If is univalent in and is convex, then

Lemma 7 (see [17]). Suppose that , with , and , and such that , . If is analytic in with , then implies that .

Lemma 8 (see [18]). Let Then, for where is a constant.

Lemma 9 (see [19]). Let Then, for

Lemma 10 (see [20]). Let be analytic in , , , . If then

3. The Class

Using the well-known results, for example, see [21, 22], the following results can easily be proved.

Theorem 11. Let . Then, with , , and , .

When , , we obtain a known result. Also, for , (28) can be written as

That is,

Theorem 12. Let and let . Then, for , , , we have where .

We note that, with , is univalent for , . See [23].

Theorem 13. Let , , . Then, , with , contains the Schlicht disc :

Proof. Let with and .
For , , . Then, by Lemma 6, we have From Theorem 12, it follows that is univalent in .
Let be any complex number such that for Then, the function is analytic and univalent in .
Now, , and simple calculations yield that We use this bound for and the well-known sharp bound for the second coefficient of univalent functions and have this gives us The proof is complete.

For the permissible values of and , we obtain several interesting results as special cases of this result.

Theorem 14. Let . Then, where , , .

Proof. From Theorem 11, for , we have relation (30). Now, for , it follows that is starlike of order (see [13]). Also, it is known [24] that there exists such that Thus, we can write (30) as Now, using well-known distortion results for , we obtain (39) as required.

Theorem 15. Let = and , and let for , be defined by

Then, in (i)We note that (42) reduces to the generalized Bernardi operator for , when and (ii)By choosing , we prove this result for

Proof. From (42), it follows that that is, With simple computations, we obtain from (44) Now, using Lemma 7 along with (45), we have and consequently . This completes the proof.

Theorem 16. Let and with . Then, for , where is constant, and .

Proof. Since , it implies that , . We can write Then, First, we calculate, for , by using Theorem 11 in the following integral: , and .
Let . Then, by a result due to Golusin [25], there exists a with such that for , We use (50) and distortion result for and apply Holder’s inequality to have where .
Now, with subordination for , starlike function and Lemma 8 in (51), it follows that From (48), (49), and (52), it follows that Thus, choosing in (53), we have where and are constants.

As a special case, we note that, for , we have

When , , and in this case, if we choose , then

4. The Class

We discuss here some basic properties of .

Theorem 17. Let . Then, there exist , if and only if

Proof. From Definition 3, we can write The converse case follows easily, and the proof is complete.

Since , , we note that , and for the class , we refer to [22]. From Definition 3, we have

We now prove the following inclusion result.

Theorem 18. For , with ,

Proof. Let . Then, there exists such that Since , and , for , it follows that .

We now discuss a geometrical property for the class and investigate the behaviour of the inclination of the tangent at a point to the image of the circle , , and is any number of interval under the mapping of .

Let , Let and for , ,

Now, since we have

Therefore,

On the other hand,

So, the integral on the left hand side of (67) characterizes the increment of the angle of the tangent to the curve between the points and for .

We now prove the following.

Theorem 19. Let and . Then, for , , , we have where and .

Proof. From Definition 3, we can write where and .
Since, for , it is well known that Thus, the values of are contained in the circle whose diameter is the line segment from to . The circle is centered at the point and has the radius . So, attains its maximum at points where a ray from the origin is tangent to the circle, that is, when Thus, it follows that Now, from (69), (72), and Theorem 12, it follows that, for , ,

Remark 20. In [23], the class of functions if and only if, for , , ,

For is close-to-convex and hence univalent. For , need not to be finitely valent. We note that . It can easily be seen that consists of univalent functions for . Also, it can easily be seen that forms a subset of a linear-invariant family of order . For this, we refer to [26].

Theorem 21. Let and , and . Let be defined by

Then, in

Proof. As Therefore, This proves in .

Theorem 22. Let belong to the class . Then, for , , , where is a constant.

Proof. For , we can write It is known [27] that, for all , , we can write Thus, by using Cauchy theorem and using (79) and (80), we have where we have used Holder’s inequality and Lemma 9. Setting , in (81), we get

As a special case, when , , , , and in this case,

We now study a radius problem for .

Theorem 23. Let , and let . Then, for .

Proof. Let and . Then, by Definition 2, Using (12), it can be easily shown that for . Now, for , we can write and since in , the required result follows at once.

Special cases: (i)Let and . Then, , and in this case, is convex for (ii)Let , where is given by (13). Then,which implies is -convex in . Consequently, is -close-to-convex in .

5. The Class

Using Definition 5, we shall discuss the class in this section.

Theorem 24 (integral representation). A function if and only if there exists some function such that

Proof. From Definition 5, it follows that if and only if we can write Thus, and from this observation, the proof is immediate.

Theorem 25. Let . Then,

Proof. Let . Then, with , , we have We can write (91) as where .
We now show that .
Consider Let Then, This implies . Now, Since , in . Thus, from (92), it follows that .

Theorem 26. Let , , . Then, for , , we have where , .

Proof. Since , we can write That is Logarithmic differentiation of (99) and using Theorem 19, the required result immediately follows.

We note the following special cases. (i)Let . Then, for , , using Lemma 10, with and , we get(ii)If we choose in (i), then from (100) and a result due to Kaplan [15], it follows that is close-to-convex and hence univalent in

Theorem 27. Let . Then, is starlike of order , , for , where

Proof. By Definition 5, we have Since , , it is known [28] that there exists such that Using (103) in (102), with logarithmic differentiation and simple calculations, we obtain (see [14]) The right-hand side is positive for , where is given by (101).

If and , then is starlike of order in .

Data Availability

The data used to support the findings of this study are available from the corresponding authors upon request.

Conflicts of Interest

There is no conflict of interest.

Authors’ Contributions

The authors read and approved the final manuscript, and they contributed specially as the following; conceptualization was performed by K.I.N., A.M.S., and S.A.S.; methodology was performed by A.A.L. and S.A.K.; validation was performed by K.I.N., A.A.L., and A.M.S.; formal analysis was performed by S.A.S. and S.A.K.; investigation was performed by K.I.N. and A.M.S; writing (original draft preparation) was performed by S.A.S. and A.A.L.; writing (review and editing) was performed by A.A.L., K.I.N., and S.A.S.; and supervision was performed by K.I.N. and A.M.S. The authors agree with the contents of the manuscript.

Acknowledgments

The authors acknowledge the Taif University Researchers Supporting Project (number TURSP-2020/154), Taif University, Taif, Saudi Arabia.