Coefficient Bounds of Kamali-Type Starlike Functions Related with a Limacon-Shaped Domain

Abstract

In this article, we familiarize a subclass of Kamali-type starlike functions connected with limacon domain of bean shape. We examine certain initial coefficient bounds and Fekete-Szegö inequalities for the functions in this class. Analogous results have been acquired for the functions and and also found the upper bound for the second Hankel determinant .

1. Introduction

Denote by the class of analytic functions in the open unit disk . The Hankel determinants of are denoted by where . Hankel determinants are beneficial, for example, in viewing that whether the certain coefficient functionals related to functions are bounded in or not and do they carry the sharp bounds, see [1]. The applications of Hankel inequalities in the study of meromorphic functions can be seen in [2, 3]. In 1966, Pommerenke [4] inspected of univalent functions and valent functions as well as starlike functions. In [5], it is evidenced that the Hankel determinants of univalent functions satisfy where and depends only on . Later, Hayman [6] demonstrated that , (; an absolute constant) for univalent functions. Further, the Hankel determinant bounds of univalent functions with a positive Hayman index were determined by Elhosh [7], of -valent functions were seen in [8–10], and of close-to-convex and -fold symmetric functions were discussed in [11]. Lately, several authors have explored the bounds , of functions belonging to various subclasses of univalent and multivalent functions (for details, see [6, 12–27]). The Hankel determinant is the renowned Fekete-Szegö Functional (see [28, 29]) and ; second, Hankel determinant is presumed by .

An analytic function is subordinate to an analytic function , written as , if there is an analytic function with , satisfying .

Let be the class of functions with positive real part consisting of all analytic functions satisfying and .

Ma and Minda [30] amalgamated various subclasses of starlike and convex functions which are subordinate to a function with maps onto a region starlike with respect to 1 and symmetric with respect to real axis and familiarized the classes as below:

By choosing satisfying Ma-Minda conditions and that maps on to some precise regions like parabolas, cardioid, lemniscate of Bernoulli, and booth lemniscate in the right-half of the complex plane, several fascinating subclasses of starlike and convex functions are familiarized and studied. Raina and Sokół [31] considered the class for and established some remarkable inequalities (also see [32] and references cited therein). Gandhi in [33] considered a class with , , a convex combination of two starlike functions. Further, coefficient inequalities of functions linked with petal type domains were widely discussed by Malik et al. ([34], see also references cited therein). The region bounded by the cardioid specified by the equation was studied in [35]. Lately, Masih and Kanas [36] introduced novel subclasses and of starlike and convex functions, respectively. Geometrically, they consist of functions such that and , respectively, are lying in the region bounded by the limacon

Lately, Yuzaimi et al. [37] defined a region bounded by the bean-shaped limacon region as below:

Suppose that is the function defined by is preferred so that the limacon is in the bean shape [37]. Motivated by this present work and other aforesaid articles, the goal in this paper is to examine some coefficient inequalities and bounds on Hankel determinants of the Kamali-type class of starlike functions satisfying the conditions as given in Definition 1.

Definition 1. Let be analytic and for , we let the class aswhere as in (9).

We include the following results which are needed for the proofs of our main results.

Lemma 2 see [38]. Suppose that , , then and the outcome is sharp for the functions formulated by

Lemma 3 see [30]. Suppose that , . Then, (i)For or , we haveEquality occurs when or one of its rotations. (ii)For , the equality exists when or one of its rotations(iii)For , the equality happens whenor one of its rotations.

Lemma 4 see [39]. If and is given by then for some with and .

Theorem 5. Let the function be given by (1) then

Proof. Since , there exists an analytic function with and in such that Define the function by or, equivalently then is analytic in with and has a positive real part in . By using (20) together with (9), it is evident that Since and equating coefficients of from (21) to (22), we get Now by applying Lemma 2, we get and also, where . Now by applying Lemma 2, we get To show these bounds are sharp, we define the function , with by Clearly, the function . This completes the proof.

Theorem 6. Let the function be given by (1) and for any then

Proof. Let the function be given by (1), as in Theorem 5, from (23) to (24), we have where . Now by Lemma 2, we get

The result is sharp.

In particular, by taking , we get

Theorem 7. Let the function be given by (1) belongs to the class . Then, for any real number , we have where for convenience If , then These results are sharp.

Proof. Between (23) and (24) and (31), we have where . Our result now follows by virtue of Lemma 3. To show that these bounds are sharp, we define the function by and the functions and by Clearly, the functions . Also, we write . If or , then the equality holds if and only if is or one of its rotations. When , then the equality holds if and only if is or one of its rotations. If , then the equality holds if and only if is or one of its rotations. If , then the equality holds if and only if is or one of its rotation.

2. Coefficient Estimates for the Function

Theorem 8. If and is the inverse function of with where is the greater than the radius of the Koebe domain of the class , then for any complex number , we have Also, for any complex number , we have The result is sharp. In particular,

Proof. Since is the inverse function of , we have From equations (23) to (24), we get Equating the coefficients of and on both sides of (45) and simplifying, we get By applying Lemma 2, we get For any complex number , consider where Taking modulus on both sides of (49) and applying Lemma 2, we get the estimate as stated in (41). This completes the proof of Theorem 8.

3. The Logarithmic Coefficients

The logarithmic coefficients of defined in are given by

Using series expansion of on the left hand side of (50) and equating various coefficients give

Theorem 9. Let with logarithmic coefficients given by (51) and (52). Then, and for any , then These inequalities are sharp. In particular, for , we get

Proof. Using (23) and (24) in (51) and (52) and after simplification, one may have To determine the bounds for , we express (57) in the form where then by applying Lemma 2, we get For any , from (56) to (57), we have where An application of Lemma 2 gives the desired estimate.

4. Coefficients Associated with

In this section, we determine the coefficient bounds and Fekete-Szegö problem associated with the function given by where .

Theorem 10. Let and are given by (63). Then The results are sharp.

Proof. By routine calculation, one may have Comparing the coefficients of and on both sides of (63) and (65), we get Using (23) and (24) in (66) and (67), we obtain By Lemma 2, we get Now, where Again by using Lemma 2, we get For any , between (68) and (70), we get That is, where The result follows by application of Lemma 2 and therefore completes the proof.

5. Second Hankel Inequality for

Theorem 11. Let the function be given by (1), then

Proof. Since , there exists an analytic function with and in such that, Therefore, between (23), (24), and (25), we get By writing and we have From (15) to (16), it follows that Replacing by and then substituting the values of , and from (81) yield Differentiating in (84) partially with respect to yields It is clear from (85) that ; thus, is an increasing function of for and for any fixed with . So, the maximum of occurs at and Note that Differentiating partially with respect to yields If then its root is Also, we have is negative for , which means that the function can take the maximum value at also which is

Data Availability

No data is used.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The author Marwan Amin Kutbi expresses his thanks and grateful to King AbdulAziz University (Jeddah, Saudi Arabia) for unlimited support during this research.