The Second Hankel Determinant of Logarithmic Coefficients for Starlike and Convex Functions Involving Four-Leaf-Shaped Domain

In this particular research article, we take an analytic function Q 4 L = 1 + 5/6 z + 1/6 z 5 , which makes a four-leaf-shaped image domain. Using this speci ﬁ c function, two subclasses, S ∗ 4 L and C 4 L , of starlike and convex functions will be de ﬁ ned. For these classes, our aim is to ﬁ nd some sharp bounds of inequalities that consist of logarithmic coe ﬃ cients. Among the inequalities to be studied here are Zalcman inequalities, the Fekete-Szegö inequality, and the second-order Hankel determinant.


Introduction and Definitions
To properly comprehend the findings provided in the paper, certain important literature on geometric function theory must first be discussed. In this regard, the letters S and A stand for the normalized univalent (or schlicht) functions class and the normalized holomorphic (or analytic) functions class, respectively. These primary notions are defined in the disc U d = fz ∈ ℂ : jzj < 1g by where GðU d Þ expresses holomorphic functions class, and This class S evolved as the foundational component of cutting-edge research in this area. In his paper [1], Koebe established the presence of a "covering constant" ζ, demonstrating that if F is holomorphic and Schlicht in U d with F ′ ð0Þ = 1 and Fð0Þ = 0, then FðU d Þ = fw : jwj < ζg. Many mathematicians were intrigued by this beautiful result. Within a few years, the wonderful article by Bieberbach [2], which gave rise to the renowned coefficient hypothesis, was published.
It is easy to deduce from equation (2) that, for F ∈ S, the logarithmic coefficients are computed by Currently, Lecko and Kowalczyk and Kowalczyk and Lecko [38,39] studied the following Hankel determinant H q,n ðJ F /2Þ of logarithmic coefficients It has been noted that By the virtue of the function Q 4L = 1 + 5/6z + 1/6z 5 , we define the following classes: 2 Journal of Function Spaces Alternatively, F ∈ S * 4L if and only if an analytic function q occurs that satisfies qðzÞ ≺ Q 4L in such that By taking qðzÞ = Q 4L in (18), we achieve the following function, which serves as an extremal in many of the class S * 4L problems.
The following Alexander-type connection-related two classes were mentioned above. The above two families are interlinked by the following Alexander-type relation From (19) and (20), we easily obtain the following extremal functions in various problems of the class C 4L Clearly, g 0 ðzÞ, g 0 ðz 2 Þ, g 0 ðz 3 Þ, and g 0 ðz 4 Þ belong to the class C 4L . That is, In the present paper, our core objective is to find the sharp coefficient type problems of logarithmic functions for the families S * 4L and C 4L : Among the inequalities to be studied here are Zalcman inequalities, the Fekete-Szegö inequality, and the second-order Hankel determinant H 2,1 ðJ F /2Þ.

A Set of Lemmas
We must first create the class P in the below set-builder form in order to declare the Lemmas that are employed in our primary findings.

4L
We start by establishing out the class S * 4L 's initial coefficient bounds.
3 Journal of Function Spaces Theorem 6. Let F be the series form (1) and if F ∈ S * 4L , then These bounds are sharp.
Proof. By utilizing (44) and (45) Implementation of (28) and triangle inequality, we get Equality is determined by using (10), (11), and This inequality is sharp and can be obtained by using (10), (11), and Theorem 9. Let F be the expansion (1) and if F ∈ S * 4L , then The above stated result is the best possible.
Proof. From (44)-(46), we easily attain By using Lemma 4 and triangle inequality, we obtain Equality is determined by using (10), (11), (12), and Theorem 10. Let F be the expansion (1) and if F ∈ S * 4L , then The last stated inequality is the finest.

Coefficient Inequalities for the Class C 4L
For the function of class C 4L , we start this portion by determining the absolute values of the first four initial logarithmic coefficients.
Theorem 12. Let F be given by (1) and if F ∈ C 4L , then , These bounds are sharp.

Theorem 13.
Let F ∈ C 4L be the series form (1). Then, This inequality is sharp.
For λ = 1, we get the below corollary.

Theorem 15.
Let F be the form (1) and if F ∈ C 4L , then This result is sharp.

Theorem 16.
Let F be the form (1) and F ∈ C 4L : Then, This result is sharp.

Theorem 17.
Let F be given the form (1) and F ∈ C 4L : Then, This result is sharp.

Data Availability
The numerical data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.