Abstract
The aim of this particular article is at studying a holomorphic function defined on the open-unit disc for which the below subordination relation holds The class of such functions is denoted by The radius constants of such functions are estimated to conform to the classes of starlike and convex functions of order and Janowski starlike functions, as well as the classes of starlike functions associated with some familiar functions.
1. Introduction
To completely comprehend the mathematical concepts used throughout our key observations, some of the essential literature of the geometric function theory must be described and analyzed here. Let us begin with the symbol which describes the family of holomorphic (or analytic) functions in a subset of the complex plan having the following series expansion
Also, let the family of all univalent functions be denoted by and is a subset of the class Next, we define that the subordination between the function belongs to the class . Let Then, or the mathematical form of the subordination between and if a holomorphic function occurs in with the restriction and in such a way that hold. Further, if in then, the following relation holds:
Three significant subfamilies of , which are well studied and have nice geometric interpretations, are the families of starlike convex , and strongly starlike functions of order and , respectively. These families are defined as follows:
Particularly, the notations and represent familiar families of starlike and convex functions, respectively. These subfamilies of satisfy the following relationship
The reverse of the above relation hold only under certain restriction of the domain. That is; if in then, it was given in [1], Corollary, p. 98, that maps the disc onto a region which is star shaped about the origin for every . The constant is known as the radius of starlikness for the family . Also, given in [1], Corollary, p. 44, the radius of convexity for the families and is .
To make a radius statement for other things than starlikeness and convexity, we choose two subfamilies and of the set . The radius for the family , represented by , is the largest number such that for every and . Consequently, an alternative formulation of the radius of starlikeness for is that the radius for the family is
In 1992, Ma and Minda [2] considered the general form of the families as where is a holomorphic function with and has positive real part. Also, the function maps onto a star-shaped region with respect to and is symmetric about the real axis. They addressed some specific results such as distortion, growth, and covering theorems. In recent years, several subfamilies of the set were studied as a special case of the class . For example, (i)If we take with , then, we achieved the class which is described by the functions of the Janowski starlike class investigated in [3]. Furthermore, is the familiar starlike function family of order with (ii)The family with was developed in [4] by Sokol and Stankiewicz. The function maps the region onto the the image domain which is bounded by (iii)The class with was examined by Sharma and his coauthors [5] which consists of function in such a manner that is located in the region bounded by the cardioid given by (iv)The family with is studied in [6] while and were contributed by Raza and Bano [7] and Alotaibi et.al [8], respectively(v)By choosing we obtain the class which was established in [9]. The authors determined radius problems in this article for the defined class (vi)The class was explored recently in [10]. For such a class , the authors calculated Hankel determinant bounds of order three in [11]. Also, the class with was contributed by Mendiratta et al. [12] in which they investigated the radius problems(vii)The family with was introduced and studied by Raina and Sokół [13](viii)By considering the function we get the recently examined family introduced by Kumar and Arora [14]. They discussed relationships of this class with the already known classes. For more particular classes, see the articles [15–20]
In the present paper, we consider a trigonometric function with Also, one can easily obtain that By using this function, we define the below family of functions as
In other words, a function if and only if there exists a holomorphic function fulfilling such that
Now, we construct some examples of our newly described family . For this, consider the following functions
Since is univalent in , , and this implies that for each the relation holds. Thus, from (8), the functions corresponding to the functions , and respectively, belong to the family By taking in (8), we get the below function that plays a role of the extremal in many problems of the class
In this article, we work on determining the radius of starlikeness and convexity and radius for some subfamilies of starlike functions, mentioned above in which mostly have simple geometric interpretation. Besides these subfamilies, we also discuss the radius for some families of , whose functions have been expressed as a ratio between two functions.
2. Radii of Starlikeness and Convexity
In this portion, we examined the radius of starlikeness and convexity for the family
Theorem 1. The radius for the family is with
Proof. If then, by virtue of (7), a Schwarz function exists with such as Now, let with After easy computation, we get The equation has five roots in , namely, and . Since it is sufficient to show that . Furthermore, we can see that , , and Thus, we have whenever The radius of starlikeness of order , for the family , is the smallest positive root of
Taking in the above Theorem 1, we obtain the following corollary.
Corollary 1. The radius, for the family , is
Theorem 2. The radius , for the family is where is the smallest root of the equation and is such that
Proof. If , then, a holomorphic function occurs with and , such that
By simple computation, it gives
From (18), we get
Assume that with for calculating the minumum value of the right side of the last inequality. A simple calculation reveals that
where and It is easy to observe that
Consequently, we have
Now, consider that
The equation attained has five roots in , namely, and . Also, ; it is enough to consider only those roots which lie in . Furthermore, we seen that and; therefore
Hence,
Also,
Using the above facts along with the well-known inequality of Schwarz functions we have
Using (19), we obtain
The last inequality is true if with holds.
Hence, radius for the family is the minumum of and , where is the smallest positive root of the equation
and is such that
Corollary 2. The radius, for the family , is
Remark 1. The result in the last Theorem is not the best one. Considering the function described by (11) provides a sharp result. For the function , we have and
3. Radius Problems
To address our main results in this portion, first, we consider a few well-known families as follows.
Also, for
If we put , for then, the family is reduced to and Let the family contains the functions satisfying that for Furthermore, let
Ali et al. [21] recently studied the below families and calculated radii for certain families. Further, they achieved the conditions on and such that In this portion, radii for the family of Janowski starlike function and some other geometrically described families are explored. To get our results, we employ the following lemmas.
Lemma 1 [22]. If then, for
Lemma 2 [23]. If then, for
In particular, if then, for ,
The aim of the following lemma is at finding the largest and the smallest disks centered at and (1,0), respectively, such that the domain , where , is contained in the smallest disk and contains the largest disk.
Lemma 3. Let And Then, the following inclusions holds
Proof. Since with we have with First, we consider the square of the distance from to a point on the boundary of , which is given by To show that is the largest disk contained in it is sufficient to show that . But since therefore, we consider the range . Now, it can easily be obtained that has three roots , and The root is dependent on . The graph of shows that it is decreasing in and increasing in the interval . Hence, the minimum of is calculated on either or . A computation provides Thus, we get Therefore, we can write that or equivalently For the circle of the minimum radius centered at which contains we find the maximum distance from to a point on the boundary of and the square of this distance function is given by It is easy to verify that achieves its maximum value at which is Hence, the radius of the smallest disk which contains is ☐
In the following examples, we apply Lemma 3 to find the necessary and sufficient conditions for two specific functions that belong to the family
Example 1. (a)The function if and only if (b)The function if and only if
Proof. (a)We know that if and only if . Since we get , whenever The function maps onto the disk Since then, from Lemma 3 the above disk will be contained in if The above two inequalities give respectively. Thus, we have (b)Logarithmic differentiation of the functionyields that maps onto the disk since The disk above is contained in , in Lemma 3, whenever The above two inequalities give respectively. Thus, we have This completes the required proof.
Theorem 3. The sharp radius for the family is given by
Proof. Suppose that Consider the function described by Using logarithmic differentiation, we get Implementing Lemma 1 we have According to Lemma 3 if the following inequality holds, the image of under the function lies on disk : or equivalently Thus, radius of is the smallest positive root of in (0,1). Assume the function Then, it is clear to see that in the unit disk Hence, and Further, assures the sharpness of the results since at , we obtain This completes the proof.
Theorem 4. The sharp radius for the family is given by
Proof. Let and describe a function where Then, According to the definition of we get Utilizing Lemma 1 and Lemma 2, we conclude that Since it follows from Lemma 3 and (78) that the function if the following holds: or equivalently, the inequality holds. Thus, the radius for the class is the smallest positive root of Now, assume the functions described by Then, we get Furthermore, it is obvious that in the unit disk . Therefore, The function described in (83), at satisfies that Hence, the verified result is sharp.
Theorem 5. The radius for the family is given by where
Proof. Let Then, by Lemma 2 we get where center of the disk is Applying Lemma 3 it is easy to see that for and we achieved After some simple calculation, we have In addition, if for and from (89), we get Implementing Lemma 3 with leads to , if For we get Thus, from (89) and Lemma 3 we see that if the following holds: or equivalently, if This completes the proof.☐
Theorem 6. Let If either (a) and or(b) and holds, then,
Proof. Let Since using Lemma 2, we get
Therefore, either or
For , using Lemma 3 we see that if the following holds:
which, upon simplification, reduces to the condition stated in
For again, applying Lemma 3 we see that if the following holds:
which, upon simplification, reduces to the condition stated in (b).☐
Theorem 7. The sharp radii for the families , , , and are
Proof. (1)Let then,Thus, for we have For For checking the sharpness of the result, we assume the function described by Since it follows that and at and we see that and hence, the result is sharp (2)For function then,Thus, for we get For For checking the sharpness, assume the function described by where Since and from the definition of at we have and hence, the sharpness of the result is verified (3)Let , then,Therefore, for we get For For checking the sharpness, assume the function described by Since it follows that and at and we get Hence, the result is sharp (4)Let Then, Thus, for we get for For checking the sharpness of the result, consider the function defined by Since it follows that and at we have Hence, the result is sharp (5)For function we haveTherefore, for we have For we obtain For checking the sharpness of the result, we assume the function where Since it follows that and at and we have Hence, the verified result is sharp (6)Let Then, by Lemma 2 for we haveObviously, Hence, by Lemma 3 the above disk contains in so Simple calculation gives For checking the sharpness, assume the function defined as Since it follows that at we get Hence, this verified that the result is sharp. (7)Let ThenTherefore, for it gives For For checking the sharpness, assume the function defined as Since it follows that and at and we have This shows that the result is sharp (8)Supposing that we haveThus, for we get For For checking the sharpness, assume the function described as where Since it follows that and at and we have Hence, this showed that the result is sharp (9)Supposing that then,Thus, for we have For To show the sharpness of the result, we assume the function described by where Since it follows that and and we have and hence, the sharpness of the result is verified. (10)Let Then,Thus, for , we easily get For Now, we choose the following function to confirm its sharpness Since it follows that and at and we have This result is sharp.
4. Functions Defined in terms of the Ratio of Functions
Now, for the following families, we will talk about the radius problem. For brevity, we shall denote them by
Theorem 8. The sharp radii for function in the families , respectively, are
Proof. (1)Let and describe the function byThen, obviously Since it follows from Lemma 1 that for For checking the sharpness of the result, we assume the functions Thus, obviously, and hence, A computation shows that at Hence, the result is sharp (2)Let Describe the function byThen, and . Since it follows from Lemma 1 that For Thus, for For checking the sharpness of the result, assume the functions Then, obviously, and hence, The sharpness is obvious, since at we get (3)Let Describe the functions byThen, We know that if and only if and therefore, Using Lemma 1 we have Applying Lemma 3, we obtain For checking the sharpness, consider the functions From the definition of and we get Hence, Now, at we get This result is sharp.
Data Availability
No data are used.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors have equally contributed to complete this manuscript.
Acknowledgments
The research was supported by UKM grant: GUP-2019-032, Malaysia.