Properties of Certain Classes of Holomorphic Functions Related to Strongly Janowski Type Function

Abstract

Most subclasses of univalent functions are characterized with functions that map open unit disc onto the right-half plane. This concept was later modified in the literature with those mappings that conformally map onto a circular domain. Many researchers were inspired with this modification, and as such, several articles were written in this direction. On this note, we further modify this idea by relating certain subclasses of univalent functions with those that map onto a sector in the circular domain. As a result, conditions for univalence, radius results, growth rate, and several inclusion relations are obtained for these novel classes. Overall, many consequences of findings show the validity of our investigation.

1. Introduction

Let be the class of normalized analytic functions in the open unit disc of the formwhere and .

Let and be two analytic functions in . Then, is said to be subordinate to , denoted by , if there exist a Schwarz function in , under the conditions and , such that .

Let denote the subclass of of univalent functions in and , and represent the usual subclasses of that are convex, star-like, and close to convex in , respectively. A number of classes related with strongly star-like and strongly convex functions have been studied; for details, see [1–6]. The Janwoski-type function has been defined and studied in [7]. Here, in this study, we will define strongly Janowski-type function and will discuss some novel classes in relation with this function.

The strongly Janowski-type function is defined aswhere and . It is easy to see that this function is univalent and convex in the unit disc .

Definition 1. Letbe analytic in such that . Then, if and only if

Definition 2. Let . Then, if and only ifand the function if and only if .
It is obvious that, for , we get the classes , and , introduced by Janowski in [7].

Definition 3. Let . Then, , , if and only ifwhere , and . Or equivalentlywhere and .
It is easy to note that is the well-known class defined in [8]. For , we have the class .

Definition 4. Let be an analytic function of form (1). Then, if and only ifwhere , and .

For , we get the class introduced and studied by Noor [8]. Moreover, for and , we will get the class defined by Padmanabhan and Parvatham in [9]. Also, if we take , and , then we get the class introduced by Paatero in [10].

Definition 5. Let . Then, if and only if there exists a function such thatorwhere , , , and .

For , we get the class studied by Noor [11]. For , we get the well-known class of close-to-convex functions.

Definition 6. Let be of form (1). Then, if and only if there exists a function such thatwhere , and .

Remark 1. (i)For and , we obtain which implies .(ii)As , we have the class , and implies where .(iii)It can easily be seen that can be represented as the following integral:

Some recent work on the classes related to our newly defined classes can be seen in [12, 13].

Throughout the present investigation, , , and , unless otherwise stated.

2. Preliminaries

Lemma 1. (see [14]). Let and be two analytic function in such that . If is a convex univalent function in , then .

Lemma 2. Let . Then, .

Proof. Let . Then, by definition, . Now, with simple steps, we can show that . Therefore, by Noshiro Warschawski theorem given in [15], is univalent in . Moreover,Let . Then, is decreasing on , and therefore, . Hence, is a convex function. Now, by using (15), we get the required result.

2.1. Special Cases

Some special cases of Lemma 2 are stated as follows:(i)For , we have and (ii)For , we get and (iii)For and , we have

Lemma 3. ; that is, if , then , where .

Proof. Let . Then, . This implies there exists a Schwarz function such that and :Now,Hence, we get the required result.

Lemma 4. (see [16]). Let be convex in with . Suppose also that is analytic in with , . If is analytic in with , then

Lemma 5. Let ; then,(i), where (ii)(iii)

Proof. The first result can be easily proved by using Lemma 3.
(ii) Let . Then, there exists two functions such thatNow, let and ; then,which can be written asComparing the nth terms of the equality and then by taking modulus on both sides, we obtainUsing Lemma 2, we will get the required result.
(iii) Using Parseval’s identity, we havewhich gives the required result.

Corollary 1. (i)For , we have (ii)For , and , we have (iii)For , and , we have , which is the result for the well-known class of Caratheodory functions

Lemma 6. (see [17]). Let be of form (1). Then, , where and if and only if(i)There exists some such that(ii)There exists two star-like functions and such that

Lemma 7. Let be of form (1). If , then , where .

Proof. Let . Then, there exist , such thatBy using Lemma 3, , where . Therefore, , for . Hence, .

3. Main Results

In the following theorems, we demonstrate and analyze some novel features of newly defined classes using strongly Janowski-type functions.

Theorem 1. Let be of form (1) in the class . Then, there exist two analytic functions , such that

Proof. Let ; then, there exist , such thatThen, by using the definition of , we can say that there exist two functions , such thatSubstituting these values in equation (28) and after some simple steps and an integration, we obtainSince , by the Alexander relation, there exist , such that and . By putting these values in equation (30), we get the required result.

Theorem 2. Let ; then, , for , where .

Proof. Let ; then, by Theorem 1, there exists two functions such thatLogarithmic differentiation of equation (31) yieldsSince , where , therefore, there exists such that and . We have the well-known distortion result, for class ,So, by means of equation (32), we obtainSince , it follows that for .

Theorem 3. Let . Then, , for , where .

Proof. Let ; then, there exists , such thatOn the contrary, we can say thatThen, logarithmic differentiation yieldsThen, employing equation (34) and the well-known distortion result,where , we obtainNow, letThen,Therefore, has a root in [0, 1]. Let be the root; then,Hence, , for .

Corollary 2. For , and , we get the radius of convexity for the class of close-to-convex functions.

Corollary 3. By substituting , we get ; also, we havewhich is the radius of convexity for the class .

Theorem 4. Let ; then, for ,where .

Proof. Let . Then, by using Lemma 7, we can say that , where . Therefore, by using Lemma 6, there exist some such thatBy using Theorem 2.2 of [17], we haveCombining equations (45) and (46), we get the required result.

Remark 2. Goodman [18] defined the class of close-to-convex functions of order as follows.
A normalized analytic function defined in (1) is said to be close to convex of order if, for , , , and ,Then, by the criteria defined by Kaplan in [19], we deduce that is univalent if .
Hence, it is easy to note that the function is close to convex of order . Moreover, the function is univalent for or .
Some noteworthy cases for the univalency of subclasses of are stated below.(i)For , we have . So, in this case, is univalent for .(ii)For , we have and . So, is univalent, for .(iii)For , is univalent for .(iv)For , we have and is univalent for .

Hence, for different values of parameters, we get different classes of analytic functions of bounded boundary rotations and modified limits for the univalency of functions. Since the class defined by Paatero in [10] is univalent for , here we have discussed some classes of bounded boundary rotations, for which this limit has been improved.

Theorem 5. Let ; then, , where O(1) is a constant. , , and .

The proof is straightforward by using Lemma 6 and Theorem 2.4 of [20].

Some noteworthy cases of Theorem 5 are stated as follows.

Remark 3. (i)For , we have . So, . For , we get a constant.(ii)For , we have and . Here, for , we get and . For , we get a constant. Similarly, if we take , we have and .(iii)For , we have and . For , we get a constant.(iv)For , we have , and we get a constant for .

Theorem 6. Let ; then, for and , we havewhere .

Proof. Let ; then, there exists such thatLogarithmic differentiation of (49) and some manipulations yieldFor , we haveNow, since . thus, there exists such thatwhere . Logarithmic differentiation of equation (52) and simple computation yieldSince , so by Theorem 4, we haveFor , we haveBy combining equations (51) to (55), we will get the required result.

From Theorem 6, we can conclude the following important results.

Remark 4. (i)The necessary condition for is that the outward drawn normal on the image domain turns back at most .(ii)The function is univalent if , that is, .(iii)The functions in the class are close to convex of order , that is, .(iv)For , we get ; therefore, implieswhere and .

Theorem 7. Let and ; then, we havewhere and .

Proof. Let ; then,where . So, there exists some such thatEquation (58) can be written asso we haveNow, let . Then, , so by using (59), we haveLet such thatSince , it follows that . So, by using the Cauchy theorem, we haveBy using Lemma 6, equation (64) can be written aswhere and . Using the distortion result for given in [15], we haveThen, by applying noted Hölder inequality, we haveSince , by using Lemma 5, we obtainwhere O(1) represents the constant term. By taking , we have the required result.

Clearly, we see the following.

Remark 5. (i)For , we get (ii)For , and

Theorem 8. Let . Then,

Proof. Let . Then, there exists such thatwhere , so . Using the fact that , this implies there exists such thatwhere . Now, consider . Then, by logarithmic differentiation and simple calculations, we obtainSince implies , it can easily be shown that is a convex set and ; therefore, . Thus,where . Since, therefore, . Hence, . Thus, which gives the required inclusion.

Theorem 9. .

Proof. Let ; then, by Definition 5,where ; hence, there exists such thatwhere . So, equation (74) can be written aswhere and . Since and , therefore, and . Now, we define such thatwhere ; then, defined by equation (78) is convex [21]. Now, letThen, by combining equations (76)–(79) and some simple computation, we obtainwhere and since . Now, we write , ; it is obvious that , so equation (80) can be written asTherefore, by using Lemma 4, it follows that , and hence, .

4. Conclusion

In this study, taking into account the subordination technique, we have introduced certain new classes of analytic functions associated with strong Janowski-type functions. Furthermore, we have obtained several results, and their well-known special cases are apprehended in form of Remarks 2–5. We hope that this new methodology will stimulate futuristic research in the fascinating field of geometric function theory and differential subordination.

Data Availability

No statstical data has been used in this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

The authors worked jointly and equally on this manuscript. All the authors approved the final version of the manuscript.

Acknowledgments

The authors are thankful to the Rector of Comsats University Islamabad for providing a research-oriented environment.