Starlikeness Associated with Tangent Hyperbolic Function

Abstract

The primary objective of this study is to establish a class of starlike functions (symmetric under rotation) that are related to a tangent hyperbolic. Integral preserving properties along with the sufficiency criteria involving coefficients are investigated for the same class. Differential subordinations problems, which are linked to Janowski and tangent hyperbolic functions, are also discussed. We also utilize these findings to find sufficient conditions for the starlike functions.

1. Introduction and Motivations

The purpose of this portion is to provide certain fundamental ideas in geometric function theory that will assist in the understanding of our results. In this respect, we start by defining the most fundamental class , which contains functions that are holomorphic or regular or analytic in the subset of the complex numbers , as follows

Also, the subset set comprises all normalised univalent functions in . For the given functions , we say that is subordinate to , written appropriately as , if there exist a Schwarz function , which is holomorphic in with so that

Second, if in is univalent, then, the following equivalence holds true:

In the analysis of holomorphic functions, image domains are extremely important. Based on the geometry of image domains, holomorphic functions are divided into various classes. Ma and Minda [1] proposed an analytic function in 1992, which is normalised by representations and . Also, the region is a star-shaped about and is symmetric on the real line axis. Particularly, (i)If we takethen, we achieve the class given by which is described as the functions of the Janowski starlike [2]. For more applications, see [3–5]. (ii)The class listed belowillustrates that the image domain is bounded by the Bernoullis lemniscate’s right-half plan, provided by , see [6–8]. (iii)The class was given bywas examined by [9]. It is made up of functions arranged in such a way that is located within the cardioid defined by (iv)By picking , the family yields to the class that was examined by Cho et al. [10]. But on the other hand, the class provided bywas studied in [11]. Also, by choosing a particular function instead of , various subcollections of starlike functions have lately been recognized. For more details, see [12–18].

Consider the holomorphic function in with the normalization . In [6], Ali and his coauthors have achieved adequate restrictions on alpha such that, for implies

Many scholars have studied similar types of consequences in recent articles, for examples, see the articles contributed by Omar and Halim [7], Kumar et al. [19, 20], Haq et al. [21], Paprocki and Sokół [22], Raza et al. [23], Sharma and Ravichandran [24], and Tuneski [25].

Give the idea of the class of starlike functions associated with tan hyperbolic function that are described [26, 27]:

To illustrate our basic findings, we must first acquire the following Lemma.

Lemma 1 (see [28]). Let be a holomorphic function in with the normalization . If then, a number exists in such a way that .

We consider the following limitations to reduce repetition.

2. Sufficiency Criterion

Theorem 1. Let a holomorphic function with the normalization obeying with the following restriction Then

Proof. Let us suppose that Then is holomorphic in with the normalization . Further, assume where we select the logarithmic and square root principal branches of the functions . Then, is clearly analytic in with . Also, since To prove this result, we need to show that in . Now making use of equation (21), we get Now Therefore If we choose , then, we have Also, by Lemma 1, a number exists with Furthermore, we also assume that for . Then, the following can be easily get If , , then, easy computation gives us the following where After some simple simplification, we can easily get that , , and are the roots of and in . Furthermore, as we can say now that , and so, we have Thus Therefore, using (37) and (38) in (27), we obtain Now let Then This confirms that the function is increasing, and hence, , so Using (18), we get the above condition is now clearly a contradiction with the hypothesis Hence, , and this completes the required proof.

By putting in (17), the following result can be gotten easily.

Corollary 2. Let , satisfying with Then, .

If we select , in (45), then, the below corollary is achieved.

Corollary 3. Let and satisfying with Then, .

Theorem 4. Let a holomorphic function with the normalization obeying with the following restriction Then

Proof. Let us assume that Then, the function is analytic in with Using (21), we have and so Also, by Lemma 1 along with (37) and (38), we have Now, let Then Using (50), we obtain The above condition is showing a clear contradiction to our hypothesis. Therefore, the proof of this result is completed.

By putting in (49), one can get the following result.

Corollary 5. Let and satisfying with Then, .

If we select in (59), then, the below corollary will be obtained.

Corollary 6. If and justifying with Then, .

Theorem 7. Suppose that If is an analytic function described in with and obeying then

Proof. Consider Then, the function is analytic in with Putting the value of in the last expression, we achieve and so By applying Lemma 1, Now, let Then Using (63), we obtain A contradiction to the hypothesis (73) occurs. Therefore, the proof of this result is completed.

By putting in (64), the below result is easily deduced.

Corollary 8. Let and satisfying with Then, .

Selecting in (73), then, the following result is attained.

Corollary 9. If and satisfying with Then, .

3. Bernardi Integral Operator and Its Relationship

In the study of function theory, the rules of operators are crucial to understanding the geometric properties of holomorphic functions. The convolution of various holomorphic functions is used to introduce many differential and integral operators. The geometric characteristics of holomorphic functions and univalent functions can be explained in this form, which makes more mathematical research easier. In 1916, Alexander was the first person to start studying operators. Many integral operators were later introduced by Bernardi [29] and Libera [30] to explore the families of starlike, convex, and close-to-convex functions. A number of researchers have recently shown considerable interest in investigating some properties of certain operators. For the study of different operators, see [31–36].

The Bernardi [29] integral operator is described by

The properties related to mapping for the given function in the class under the integral operator defined in (77) are examined in this section of the article.

Theorem 10. Suppose that If then where the operator is provided by (77).

Proof. Consider a function given by where we select the logarithmic and square root principal branches of the functions. As the function is defined by Therefore, is a holomorphic function in with . To prove the required result, it is sufficient to show that in . Also, from equation (81), we have Differentiating the above equation logarithmically, we have Utilizing (77) gives By taking again differentiation of the above equation logarithmically, we have Now, we describe a function Note that is analytic in with . Further Assume that there exists a point so that By applying Lemma 1, there exists a number such that . Suppose that . Then where Let . Then, some easy computation shows that An easy computation shows that the equation has five roots in , namely, , , . Since , it is sufficient to assume , and this implies that Also, suppose with In the same way as above, some simple calculation of give as that it has five roots in , namely , , . Since , so we amuse without restriction that and it shows that Thus, Now where Now, let Then This confirms that the function is increasing, and hence, , so Now by (78), we get A contradiction to the hypothesis Hence, this is the required proof.

Theorem 11. Assume that If then where is the Bernardi integral operator defined in (77).

Proof. Let a function be described by where we select the logarithmic and square root principal branches of the functions . Then, is analytic in with . We need only to show that in . From (112), Further, let where is holomorphic in with . Now by utilizing (77), (113), and (114), we obtain Assume that there exists a point such that By application of Lemma 1, there exists a number such that . Consider . Then, we have Now, assume Then This confirms that is an increasing function and it has its minimum value at , so Now by (109), A contradiction to the hypothesis This is our required proof.

Theorem 12. Suppose that If then where is the Bernardi integral operator defined in (77).

Proof. This proof can be easily completed by applying the same procedures used in the preceding theorems.

4. Convolution Conditions and Its Consequences

Convolution (or Hadamard product) is a crucial strategy for solving many function theory problems, and as a result, this concept has become a cornerstone of this subject. This section’s main goal is to investigate the characteristics of convolution and their consequences for the class of starlike functions involving tan hyperbolic function. For , the convolution, indicated by , is defined by

Further, the below facts will be true only if ;

The first result employing these concepts is given below.

Theorem 13. If , then if and only if for all and also for .

Proof. Given that is analytic in . Satisfying for all that is for which we have seen that it is an equivalent to (128) for . The proof for is now completed. Also, from (14), analytic function exists with the property that and so that and if we take , , then Using the relation (127), we have and then some easy computation will yield which is the needed relationship.
For the converse, let (128) hold for and so for all . It shows is analytic in with . Further, we take for . Clearly, the conditions in (128) and (130) are equivalent, as a result, the relationship (130) is formed, it is obvious that . As a result, a related component of contains the simply connected domain The univalence of the function , together with the fact , illustrates that , and it implies that .

Theorem 14. Let . A sufficient and necessary condition for is that

Proof. It is proved in the above theorem that if and only if relation (128) holds. We can rewrite (128) as Hence, this is our required proof.

Theorem 15. If the function and satisfying the following inequality then, .

Proof. To establish this result, we need to prove the relationship (133). For this, assume where we have used inequality (135). Thus, by virtue of Theorem 14, the proof is completed.

5. Partial Sum Problems

The partial sum problems of some holomorphic functions in the class are investigated in this portion. If a function has the series form given in equation (1), then, the partial sum of is defined by

In 1992, Szegö [37] proved an interesting result which states that if , then

Researchers were motivated by this result to investigate the problem of partial sum for the subclasses of holomorphic, multivalent, and univalent functions. In [38], Silverman determined sharp lower bounds on the real parts of the quotients between the normalised starlike or convex function and their consequences of partial sums. Further, Srivastava et al. [39], Shiel-Small [40], Ruschewyh [41], Robertson [42], Singh [43], Owa et al. [44], and Ponnusamy et al. [45] have derived certain attractive results involving the partial sum.

Theorem 16. Let be given by equation (1) and satisfying (135). Then where This result is sharp.

Proof. To prove relation (139), write where Now if and only if Finally, to show relation (146), it is sufficient to establish that the left side of relation (146) is bounded above by and is equal to The last inequality is true because of relation (141). To show that the inequality (139) is sharp, let us assume the function Then for we have To derive inequality (141), let us consider where Now if the following inequality holds Finally, to obtain the inequality (153), it is sufficient to show that the left side of inequality (153) is bounded by and is equivalent to which is true due to relation (141).

6. Concluding Remarks and Observations

Here in our present investigation, we have studied many useful properties of certain starlike functions involving the tan hyperbolic function, which is symmetrical around the real axis. These properties include problems of Bernardi integral preservation and coefficient sufficiency criterion. Furthermore, we computed several conditions on in order to ensure that if for each

These findings are also used to demonstrate that the function belongs to the newly established family Furthermore, this class can readily prove coefficient bounds, Hankel determinant, partial sums inequalities, and many other problems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors equally contributed to this manuscript and approved the final version.

Acknowledgments

The first-named author (Huo Tang) was partly supported by the Natural Science Foundation of the People’s Republic of China under Grant 11561001, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14, the Natural Science Foundation of Inner Mongolia of the People’s Republic of China under Grant 2018MS01026, the Higher School Foundation of Inner Mongolia of the People’s Republic of China under Grant NJZY20200, the Program for Key Laboratory Construction of Chifeng University (no. CFXYZD202004), the Research and Innovation Team of Complex Analysis and Nonlinear Dynamic Systems of Chifeng University (no. cfxykycxtd202005), and the Youth Science Foundation of Chifeng University (no. cfxyqn202133).