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A normalized analytic function defined on the open unit disk is a bounded turning function if its derivative has positive real part. Such functions are univalent, and therefore, we find sufficient conditions for a function to be a bounded turning function. In this paper, we prove a general differential subordination theorem in terms of the derivative, the pre-Schwarzian derivative, and the Schwarzian derivative, providing sufficient conditions for a function to be a bounded turning function. We then apply the result to obtain several simple sufficient conditions. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to introduce two new subclasses R sin m ( ℑ ) $\mathcal{R}_{\sin }^{m}(\Im )$ and R sin ( ℑ ) $\mathcal{R}_{\sin }(\Im )$ of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function ℑ ( v ) = 2 1 + e − v $\Im (v)=\frac{2}{1+e^{-v}}$ , v ≥ 0 $v\geq 0$ in the open unit disc E. Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results.

In the current article, we consider a class of bounded turning functions associated with the cosine hyperbolic function and give some results containing coefficient functionals using the familiar Carathéodory functions. An improvement on the bound of the third-order Hankel determinant for functions in this class is provided. Furthermore, we obtain sharp estimates of the Fekete-Szegö, Krushkal, and Zalcman functionals with logarithmic coefficients as entries. All the findings are proved to be sharp.

Let B T 4 l 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 | b 6 | , | b 7 | , | b 8 | , the bounds of the logarithmic coefficients, and the third-order determinants | H 3 , 1 | , | H 3 , 2 | , | H 3 , 3 | for the functions in this class. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce two new subclasses R sin m (ℑ) and R sin (ℑ) of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function ℑ (v) = 2 1 + e − v , v ≥ 0 in the open unit disc E. Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results. [ABSTRACT FROM AUTHOR]

In this research article, we introduce new family R G of holomorphic functions, which is related to the generalized bounded turning and generating functions of Gregory coefficients. Leveraging the concept of functions with positive real parts, we acquire the first five coefficients for the functions belonging to this newly defined family, demonstrating their sharpness. Furthermore, we find the third Hankel determinant for functions in the class R G . Moreover, the sharp bounds for logarithmic and inverse coefficients of functions belonging to the under-considered class R G are estimated. [ABSTRACT FROM AUTHOR]

The Hankel determinant H 2 , 1 F f - 1 / 2 of logarithmic coefficients is defined as H 2 , 1 F f - 1 / 2 : = Γ 1 Γ 2 Γ 2 Γ 3 = Γ 1 Γ 3 - Γ 2 2 , where Γ 1 , Γ 2 , and Γ 3 are the first, second, and third logarithmic coefficients of inverse functions belonging to the class S of normalized univalent functions. In this paper, we establish sharp inequalities H 2 , 1 F f - 1 / 2 ≤ 19 / 288 , H 2 , 1 F f - 1 / 2 ≤ 1 / 144 , and H 2 , 1 F f - 1 / 2 ≤ 1 / 36 for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively. [ABSTRACT FROM AUTHOR]

The objective of this paper is to estimate the sharp bound of the third Hankel determinant for the kth-root transformation to the class of functions whose derivative has a positive real part satisfying the normalized conditions f(0) = 0 and f0(0) = 1 in the open unit disk D:= {z ∊ C: |z| < 1. [ABSTRACT FROM AUTHOR]

In the last few years, numerous subfamilies of univalent functions, whether directly or indirectly associated with exponential functions, have been introduced and thoroughly investigated. Among these, the families S e * , C e and R e defined by subordination to e z have been intensively investigated. While the coefficient problem on the class S e * and C e has been solved in many cases, in this paper, we mainly intend to compute the sharp estimates of some initial coefficients, the Feketo–Szegö inequality, and the sharp bounds of second- and third-order Hankel determinants for functions belonging to the class R e . This work has the potential to significantly enrich and enhance the exploration of univalent functions in conjunction with exponential functions, making the field more comprehensive and robust. [ABSTRACT FROM AUTHOR]

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.

In this research article, we introduce new family RG of holomorphic functions, which is related to the generalized bounded turning and generating functions of Gregory coefficients. Leveraging the concept of functions with positive real parts, we acquire the first five coefficients for the functions belonging to this newly defined family, demonstrating their sharpness. Furthermore, we find the third Hankel determinant for functions in the class RG. Moreover, the sharp bounds for logarithmic and inverse coefficients of functions belonging to the under-considered class RG are estimated.

The present study introduces a new family of analytic functions by utilizing the q-derivative operator and the q-version of the hyperbolic tangent function. We find certain inequalities, including the coefficient bounds, second Hankel determinants, and Fekete–Szegö inequalities, for this novel family of bi-univalent functions. It is worthy of note that almost all the results are sharp, and their corresponding extremal functions are presented. In addition, some special cases are demonstrated to show the validity of our findings. [ABSTRACT FROM AUTHOR]

One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these coefficient-related problems, the problem of the third-order Hankel determinant sharp bound is the most difficult one. The aim of the present study is to determine the sharp bound of the Hankel determinant of third order by using the methodology of the aforementioned Carathéodory function family. Further, we also study some other coefficient-related problems, such as the Fekete–Szegő inequality and the second-order Hankel determinant. We examine these results for the family of bounded turning functions linked with a cardioid-shaped domain. [ABSTRACT FROM AUTHOR]

In this paper, a brief study of certain properties of bounded turning functions is carried out. Furthermore, bound to the famous Fekete - Szego functional H2 (1) = |a3 - ta22 |, with t real and the Second Hankel Determinant H2 (2) = |a2a4 - a32| for functions of bounded turning of order β associated with Salagean differential operator are obtained using a succinct mathematical approach. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the last few years, numerous subfamilies of univalent functions, whether directly or indirectly associated with exponential functions, have been introduced and thoroughly investigated. Among these, the families Se*, Ce and Re defined by subordination to ez have been intensively investigated. While the coefficient problem on the class Se* and Ce has been solved in many cases, in this paper, we mainly intend to compute the sharp estimates of some initial coefficients, the Feketo–Szegö inequality, and the sharp bounds of second- and third-order Hankel determinants for functions belonging to the class Re. This work has the potential to significantly enrich and enhance the exploration of univalent functions in conjunction with exponential functions, making the field more comprehensive and robust.

In this paper, we obtain the upper bounds of the Toeplitz determinants for the class of functions with bounded turning. We also present some consequences of our main results. Some estimates obtained on Toeplitz determinants are sharp. [ABSTRACT FROM AUTHOR]

The present study introduces a new family of analytic functions by utilizing the q-derivative operator and the q-version of the hyperbolic tangent function. We find certain inequalities, including the coefficient bounds, second Hankel determinants, and Fekete–Szegö inequalities, for this novel family of bi-univalent functions. It is worthy of note that almost all the results are sharp, and their corresponding extremal functions are presented. In addition, some special cases are demonstrated to show the validity of our findings.

One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these coefficient-related problems, the problem of the third-order Hankel determinant sharp bound is the most difficult one. The aim of the present study is to determine the sharp bound of the Hankel determinant of third order by using the methodology of the aforementioned Carathéodory function family. Further, we also study some other coefficient-related problems, such as the Fekete–Szegő inequality and the second-order Hankel determinant. We examine these results for the family of bounded turning functions linked with a cardioid-shaped domain.

The purpose of this study was to obtain the sharp Hankel determinant H 2 , 1 F f / 2 and H 2 , 2 F f / 2 with a logarithmic coefficient as entry for the class BT 3 L 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. [ABSTRACT FROM AUTHOR]

The purpose of this article is to obtain the sharp estimates of the first four initial logarithmic coefficients for the class BT s of bounded turning functions associated with a petal-shaped domain. Further, we investigate the sharp estimate of Fekete-Szegö inequality, Zalcman inequality on the logarithmic coefficients and the Hankel determinant H 2 , 1 F f / 2 and H 2 , 2 F f / 2 for the class BT s with the determinant entry of logarithmic coefficients. [ABSTRACT FROM AUTHOR]

We investigate the bounds of Hankel determinants H4,1(f), H4,2(f), H4,3(f) for the set of functions with bounded turning of order alpha. We also study these bounds for 2-fold symmetric and 3-fold symmetric functions in this class. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate the third and fourth Hankel determinants for the class of functions with bounded turning associated with Bernoulli's lemniscate. The fourth Hankel determinants for 2-fold symmetric and 3-fold symmetric functions are also studied. [ABSTRACT FROM AUTHOR]

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.

The purpose of this article is to obtain the sharp estimates of the first four initial logarithmic coefficients for the class BTs of bounded turning functions associated with a petal-shaped domain. Further, we investigate the sharp estimate of Fekete-Szegö inequality, Zalcman inequality on the logarithmic coefficients and the Hankel determinant H2,1Ff/2 and H2,2Ff/2 for the class BTs with the determinant entry of logarithmic coefficients.

The main aim of this paper is to study the fifth Hankel determinant for the class of functions with bounded turnings. The results are also investigated for 2-fold symmetric and 4-fold symmetric functions. [ABSTRACT FROM AUTHOR]

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.

First, by making use of the concept of basic (or q-) calculus, as well as the principle of subordination between analytic functions, generalization R q (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 H 2 (1) for functions belonging to this class R q (h) are investigated. Furthermore, similar results are examined and presented for the functions z f (z) and f − 1 (z) . For the validity of our results, relevant connections with those in earlier works are also pointed out. [ABSTRACT FROM AUTHOR]