Abstract
Let be a nonvanishing analytic function in the open unit disc with . Consider the class consisting of normalized analytic functions whose ratios , , and are each subordinate to for some analytic functions and . The radius of starlikeness of order is obtained for this class when is chosen to be either or . Further, starlikeness radii are also obtained for each of these two classes, which include the radius of Janowski starlikeness, and the radius of parabolic starlikeness.
1. Two Subclasses of Normalized Analytic Functions
Let denote the class of normalized analytic functions in the unit disc . A prominent subclass of is the class consisting of functions such that is a starlike domain with respect to the origin. Geometrically, this means the linear segment joining the origin to every other point lies entirely in . Every starlike function in is necessarily univalent.
Since does not vanish, every function is locally univalent at . Further, each function mirrors the identity mapping near the origin and thus, in particular, maps small circles onto curves which bound starlike domains. If is also required to be univalent in , then it is known that maps the disc onto a domain starlike with respect to the origin for every (see [1], Corollary, p. 98). The constant cannot be improved. Denoting by the class of univalent functions , the number is commonly referred to as the radius of starlikeness for the class .
Another informative description of the class is its radius of convexity. Here, it is known that every maps the disc onto a convex domain for every ([1], Corollary, p. 44). Thus, the radius of convexity for is .
To formulate a radius description for other entities besides starlikeness and convexity, consider in general two families and of . The -radius for the class , denoted by , is the largest number such that for every and . Thus, for example, an equivalent description of the radius of starlikeness for is that the -radius for the class is .
In this paper, we seek to determine the radius of starlikeness and certain other -radius, for particular subclasses of . Several widely studied subclasses of have simple geometric descriptions; these functions are often expressed as a ratio between two functions. Among the very early studies in this direction is the class of close-to-convex functions introduced by Kaplan [2] and Reade’s class [3] of close-to-starlike functions. Close-to-convex functions are necessarily univalent, but not so for close-to-starlike functions.
In this paper, we examine two different subclasses of functions in satisfying a certain subordination of ratios. Interestingly, these classes contain nonunivalent functions. An analytic function is subordinate to an analytic function , written , iffor some analytic self-map in with . The function is often referred to as a Schwarz function.
Now, let be a nonvanishing analytic function in with . The classes treated in this paper consist of functions whose ratios , , and are each subordinate to for some analytic functions and :
When is the constant one function, then the class contains functions satisfying the subordination of ratios
When satisfies and , or their variants, these functions have earlier been studied, notably by MacGregor in [4–7] and Ratti in [8, 9]. For related investigations, see [10, 11] and several recent references therein. Under the present context, this amount to choosing or some other appropriate choices of .
In this paper, two specific choices of the function are made: and .
The class : this is the class given by
This class is nonempty: let be given by
Then, and , so that . The function will be shown to play the role of an extremal function for the class . Since vanishes at , the function is nonunivalent, and thus, the class contains nonunivalent functions. Incidentally, demonstrates the radius of univalence for is at most . In Theorem 1, the radius of starlikeness for is shown to be , whence has radius of univalence .
The following is a useful result in investigating the starlikeness of the class .
Lemma 1. Let . Then, satisfies the sharp inequalities
Proof. If , then for some Schwarz function . The well-known Schwarz lemma shows that andTherefore,for , that is, for . Similarly, for .
Since , the inequality (8) readily showsfor . This proves (7). The inequalities are sharp for the function defined by .
For , let and . Then, and
Since , we deduce from (7) and (11) thatfor each function . Sharp growth inequalities also follow from (6):for each . Crude distortion inequalities can readily be obtained from (12) and the growth inequality; however, finding sharp estimates remain an open problem.
The class : this class is defined by
Let be given by
Evidently, , , so that , and the class is nonempty. Similar to , the function plays the role of an extremal function for the class . The Taylor series expansion for is
Comparing the second coefficient, it is clear that is nonunivalent. Hence, the class contains nonunivalent functions. The derivative vanishes at , which shows the radius of univalence for is at most . From Theorem 1, the radius of starlikeness is shown to be , and so the radius of univalence for is .
Lemma 2. Every satisfies the sharp inequalities
Proof. Let . Since for some Schwarz self-map satisfying , it follows thatThe inequalities become equality for the function defined by respectively at and .
The function also satisfies the sharp inequality (see [1], Corollary, p. 199)From , we conclude thatThis inequality is sharp for the function defined by when . It is also sharp in the remaining interval for the function , where is the extremal function for which equality holds in (20).
For , let and . Then, and
Since , estimates (18) and (22) show thatfor each function . It also follows from (17) thatholds for each function and that these estimates are sharp.
In this paper, we shall adopt the commonly used notations for subclasses of . First, for , let denote the class of starlike functions of order consisting of functions satisfying the subordination
Thus,
The case corresponds to the classical functions whose image domains are starlike with respect to the origin. Various other starlike subclasses of occurring in the literature can be expressed in terms of the subordinationfor suitable choices of the superordinate function . When is chosen to be , , the subclass derived is denoted by . Functions are known as Janowski starlike functions. When , the subclass is denoted by , and its functions are called parabolic starlike functions.
In Section 2 of this paper, the radius of starlikeness of order , Janowski starlikeness, and parabolic starlikeness are found for the classes , with . Section 3 deals with the determination of the -radius for the class with , for certain other subclasses occurring in the literature. These classes are associated with particular choices of the superordinate function in (27). As mentioned earlier, the -radius for a given class , denoted by , is the largest number such that for every and . It will become apparent in the forthcoming proofs that there are common features in the methodology of finding the -radius for each of these subclasses.
2. Starlikeness of Order , Janowski Starlikeness, and Parabolic Starlikeness
The first result deals with the -radius (radius of starlikeness of order ) for the classes and . This radius is shown to equal the -radius, where is the subclass containing functions satisfying . The latter condition also implies that .
Theorem 1. Let . The radii of starlikeness of order for and are(i),(ii).
Proof. (i)The function is a decreasing function on . Further, the number is the root of the equation . For and , the inequality (12) readily yields At , the function given by yields Thus, This proves that the and radii for are the same number .(ii)Consider , . The number is clearly the root of the equation . Since is decreasing, then for . It follows from (23) that for ,Evaluating the function at yieldsHence,This proves that the and radii for the class are the same number .
Next, we find the -radius (Janowski starlikeness) for and . Recall that consists of analytic functions satisfying the subordination , .
Theorem 2. (i)Every is Janowski starlike in the disc for . If , then .(ii)The radius of Janowski starlikeness for is .
Proof. Since , the results in the case follow from Theorem 1. We now prove the results when .(i)Let and write . Then, (12) shows that for . For , then . For , we first show that the disc is contained in the images of the unit disc under the mapping . As , the image is the disc given by Silverman ([12], p. 50–51) has shown that the disc if and only if . With the choices , , , and , then . This proves that radius is at least . To prove sharpness, consider the function given by . Evidently, . For , evaluating at , then . This shows that proving sharpness in the case .(ii)Let and . It follows from (23) that for . For , we see that the disc is contained in the disc , as in the proof of (i). This proves that radius is at least . The result is sharp for the function given by the function .
The function given bymaps into the parabolic region
The class is the class of uniformly convex functions introduced by Goodman [13]. The corresponding class introduced by Rønning [14] is known as the class of parabolic starlike functions. The class consists of functions satisfying
Evidently, every parabolic starlike function is also starlike of order 1/2. The radius of parabolic starlikeness for the classes and is given in the next result.
Corollary 1. The radius of parabolic starlikeness for and is respectively equal to its radius of starlikeness of order 1/2. Thus,(i),(ii).
Proof. Shanmugam and Ravichandran ([15], p. 321) proved thatfor . Choosing , this implies that . Every parabolic starlike function is also starlike of order 1/2, whence the inclusion . Therefore, for any class , readily .
When , , Theorem 1 gives . This shows that . Since and from Theorem 1, it follows that and .
3. Further Radius of Starlikeness
In this section, we find the -radius for the class with , for certain other widely studied subclasses . These are associated with particular choices of the superordinate function in (27).
Denote by the class associated with in (27). This class was introduced by Mendiratta et al. [16], and it consists of functions satisfying the condition . The following result gives the radius of exponential starlikeness for the classes and .
Corollary 2. The -radius for the class iswhile that of is
Proof. Mendiratta et al. ([16], Lemma 2.2) proved thatfor , and this inclusion with gives . It was also shown in ([16], Theorem 2.1 (i)) that . Therefore, , which, as a consequence of Theorem 1, established the result.
Corollary 3 investigates the radius of cardioid starlikeness for each class and . The class , where in (27), was introduced and studied in [17]. Descriptively, provided lies in the region bounded by the cardioid .
Corollary 3. The following are the -radius for the classes and :(i),(ii).
Proof. Sharma et al. [17] proved that for , and this inclusion with gives . Thus, for . To complete the proof, we demonstrate for .(i)Evaluating the function at gives Thus, .(ii)Similarly, at , the function yieldsThis proves that .
In 2019, Cho et al. [18] studied the class consisting of functions satisfying the condition . We find the -radius for the classes and .
Corollary 4. The following are the -radius for each class and :(i),(ii).
Proof. It was proved in [18] that for , where . For , this implies that . Thus, for . The proof is completed by demonstrating for .(i)Evaluating the function at gives Thus, .(ii)Similarly, at , the function yieldsThis proves that .
Consider next the class introduced by Raina and Sokół in [19]. Functions provided lies in the region bounded by the lune . The result below gives the radius of lune starlikeness for each class and .
Corollary 5. The following are the -radius for each class and :(i),(ii).
Proof. It was shown by Gandhi and Ravichandran ([20], Lemma 2.1) that for . Choosing , the inclusion gives . Thus, for . We complete the proof by demonstrating for .(i)Evaluating the function at gives Thus, .(ii)Similarly, at , the function yieldsThis proves that .
As a further example, consider next the class , where , . This class associated with a rational function was introduced and studied by Kumar and Ravichandran in [21].
Corollary 6. The following are the -radius for the classes and :(i),(ii).
Proof. It was shown in [21] that for . This inclusion with gives . Thus, for . We next show that for .(i)At , the function yields Thus, .(ii)Evaluating at givesThus, .
The class , where , was introduced and studied by Wani and Swaminathan in [22]. Geometrically, provided lies in the region bounded by the nephroid: a 2-cusped kidney-shaped curve .
Corollary 7. The following are the -radius for the classes and :(i),(ii).
Proof. It was shown in [22] that for . This inclusion with gives . This shows that for . We next show that for .(i)Evaluating the function at results in Thus, .(ii)Similarly, evaluating at yieldsThis proves that .
Finally, we consider the class introduced by Goel and Kumar in [23]. Here, is the modified sigmoid function that maps onto the region . Thus, provided the function maps onto the region lying inside the domain .
Corollary 8. The -radius for the class iswhile that of is
Proof. The inclusion holds for (see [23]). At , the set inclusion shows that . It was also shown in [23] that for . The desired result is now an immediate consequence of Theorem 1.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first author gratefully acknowledges the support from USM research university grant 1001.PMATHS.8011101. A preprint of an earlier version is available at https://arxiv.org/abs/2101.01617.