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

where a1 = 1. Hankel determinants are beneficial, for example, in viewing that whether the certain coefficient functionals related to functions are bounded in U 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 ∣H jðnÞ ∣ of univalent functions and p − valent functions as well as starlike functions. In [5], it is evidenced that the Hankel determinants of univalent functions satisfy

Let P be the class of functions with positive real part consisting of all analytic functions p : U ⟶ ℂ satisfying pð0Þ = 1 and RepðzÞ > 0.
Ma and Minda [30] amalgamated various subclasses of starlike and convex functions which are subordinate to a function ψ ∈ P with ψð0Þ = 1, ψ ′ ð0Þ > 0, ψ maps U onto a region starlike with respect to 1 and symmetric with respect to real axis and familiarized the classes as below: By choosing ψ satisfying Ma-Minda conditions and that maps U on to some precise regions like parabolas, cardioid, lemniscate of Bernoulli, and booth lemniscate in the righthalf of the complex plane, several fascinating subclasses of starlike and convex functions are familiarized and studied. Raina and Sokół [31] considered the class S * ðψÞ for ψðzÞ = ξ + ffiffiffiffiffiffiffiffiffiffiffi ffi 1 + ξ 2 p and established some remarkable inequalities (also see [32] and references cited therein). Gandhi in [33] considered a class S * ðψÞ with ψ = βe ξ + ð1 − βÞð1 + ξÞ, 0 ≤ β ≤ 1, 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 ST LðsÞ and CV LðsÞ of starlike and convex functions, respectively. Geometrically, they consist of functions f ∈ A such that ξf ′ðξÞ/f ðξÞ and ðξf ′ðξÞÞ′/f ′ðξÞ, 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.
We include the following results which are needed for the proofs of our main results. [38].

Lemma 2 see
and the outcome is sharp for the functions formulated by Lemma 3 see [30]. Suppose that Equality occurs when p 1 ðξÞ = ð1 + ξÞ/ð1 − ξÞ or one of its rotations.

Journal of Function Spaces
Theorem 6. Let the function f ∈ Mðϑ, φÞ be given by (1) and for any ϖ ∈ ℂ then Proof.
Let the function f ∈ Mðϑ, φÞ be given by (1), as in Theorem 5, from (23) to (24), we have where ðϑ + 1Þ 2 ÞÞ. Now by Lemma 2, we get The result is sharp. In particular, by taking ϖ = 1, we get Theorem 7. Let the function f ∈ A be given by (1) belongs to the class Mðϑ, φÞð0 ≤ ϑ ≤ 1Þ. Then, for any real number μ, we have where for convenience If δ 3 ≤ μ ≤ δ 2 , then These results are sharp.
Journal of Function Spaces > δ 2 , then the equality holds if and only if f is K ϕ or one of its rotations. When δ 1 < μ < δ 2 , then the equality holds if and only if f is K ϕ 3 = φðξ 2 Þ = 1 + ffiffi ffi 2 p ξ 2 + ð1/2Þξ 4 or one of its rotations. If μ = δ 1 , then the equality holds if and only if f is F η or one of its rotations. If μ = δ 2 , then the equality holds if and only if f is G η or one of its rotation.

Coefficient
Estimates for the Function f −1 Theorem 8. If f ∈ Mðϑ, φÞ and f −1 ðwÞ = w + ∑ ∞ n=2 d n w n is the inverse function of f with |w | <r 0 where r 0 is the greater than the radius of the Koebe domain of the class Mðϑ, φÞ, then for any complex number μ, we have Also, for any complex number μ, we have The result is sharp. In particular, Proof. Since is the inverse function of f , we have From equations (23) to (24), we get Equating the coefficients of ξ and ξ 2 on both sides of (45) and simplifying, we get By applying Lemma 2, we get For any complex number μ, 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.

The Logarithmic Coefficients
The logarithmic coefficients e n of f defined in U are given by Using series expansion of log ð1 + ξÞ on the left hand side of (50) and equating various coefficients give Theorem 9. Let f ∈ Mðϑ, φÞ with logarithmic coefficients given by (51) and (52). Then, and for any ν ∈ ℂ, then These inequalities are sharp. In particular, for ν = 1, we get Proof. Using (23) and (24) in (51) and (52) and after simplification, one may have To determine the bounds for e 2 , we express (57) in the form where An application of Lemma 2 gives the desired estimate.

Coefficients Associated with ξ/f ðξÞ
In this section, we determine the coefficient bounds and Fekete-Szegö problem associated with the function HðξÞ given by where f ∈ Mðϑ, φÞ.
Proof. By routine calculation, one may have Comparing the coefficients of ξ and ξ 2 on both sides of (63) and (65), we get