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In the current study, we investigate a subclass of convex functions, denoted by K n − 1 , L , associated with a domain bounded by an epicycloid with n − 1 cusps. The primary objective is to derive sharp bounds for the coefficient bounds, the Fekete–Szegö inequality, and the second Hankel determinant, as well as to establish an upper bound for the third Hankel determinant for this newly introduced class.

In our present investigation, a subclass of starlike function Sn−1,L* connected with a domain bounded by an epicycloid with n−1 cusps was considered. The main work is to investigate some coefficient inequalities, and second and third Hankel determinants for functions belonging to this class. In particular, we calculate the sharp bounds of the third Hankel determinant for f∈S4L* with zf′(z)f(z) bounded by a four-leaf shaped domain under the unit disk D.

Leaves are determinate organs produced by the shoot apical meristem. Land plants demonstrate a large range of variation in leaf form. Here we discuss evolution of leaf form in the context of our current understanding of leaf development, as this has emerged from molecular genetic studies in model organisms. We also discuss specific examples where parallel studies of development in different species have helped understanding how diversification of leaf form may occur in nature.

The significant characteristics of Associate Laguerre polynomials (ALPs) have noteworthy applications in the fields of complex analysis and mathematical physics. The present article mainly focuses on the inclusion relationships of ALPs and various analytic domains. Starting with the investigation of admissibility conditions of the analytic functions belonging to these domains, we obtained the conditions on the parameters of ALPs under which an ALP maps an open unit disc inside such analytical domains. The graphical demonstration enhances the outcomes and also proves the validity of our obtained results.

Bi-univalent functions associated with the leaf-like domain within open unit disks are investigated, and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with q-calculus. The class is proved to not be empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on |α2| and |α3| coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation.

Bi-univalent functions associated with the leaf-like domain within the open unit disk are investigated and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with q-calculus. The class is proved to be not empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on |α2| and |α3| coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation.

The purpose of this paper is to determine the estimates of some coefficient-related problems for the class B T 3 ℓ of bounded turning functions connected to a three-leaf shaped domain. We calculate the upper bounds of the second and third order Hankel determinants with the coefficients of the inverse functions. The bounds are proved to be sharp.

The purpose of this study was to obtain the sharp Hankel determinant H2,1Ff/2 and H2,2Ff/2 with a logarithmic coefficient as entry for the class BT3L of bounded turning functions connected with a three-leaf-shaped domain. In this study, we developed a novel method to prove the bound sharpness. Although the calculations are much easier using numerical analysis, all the proofs of our results can be checked with a basic knowledge of calculus.

First, by making use of the concept of basic (or q-) calculus, as well as the principle of subordination between analytic functions, generalization Rq(h) of the class R(h) of analytic functions, which are associated with the leaf-like domain in the open unit disk U, is given. Then, the coefficient estimates, the Fekete–Szegö problem, and the second-order Hankel determinant H2(1) for functions belonging to this class Rq(h) are investigated. Furthermore, similar results are examined and presented for the functions zf(z) and f−1(z). For the validity of our results, relevant connections with those in earlier works are also pointed out.

Let BT4l denote a subclass of bounded turning functions connected with a four-leaf-type domain. The goal of the study is to probe into the bounds of coefficients |b6|,|b7|,|b8|, the bounds of the logarithmic coefficients, and the third-order determinants |H3,1|,|H3,2|,|H3,3| for the functions in this class.