Hermitian-Toeplitz determinants for a subclass of analytic functions

<abstract><p>In this paper, we introduce a new subclass of analytic and bi-univalent functions in the open unit disc $ U. $ For this subclass of functions, estimates of the initial coefficients $ \left\vert A_{2}\right\vert $ and $ \left\vert A_{3}\right\vert $ of the Taylor-Maclaurin series are given. An application of Legendre polynomials to this subclass of functions is presented. Furthermore, our study discusses several special cases.</p></abstract>

Let A \mathcal {A} be the class of normalized analytic functions in the open unit disc D ≔ { z ∈ C : | z | > 1 } \mathbb {D} ≔\{z \in \mathbb {C} : \lvert z \rvert >1 \} and S \mathcal {S} be a subset of A \mathcal {A} consisting of functions that are univalent in D \mathbb {D} . Let λ > 0 \lambda > 0 and μ ∈ C \mu \in \mathbb {C} such that | 1 − μ | > λ ≤ 1 \lvert 1-\mu \rvert > \lambda \leq 1 and λ + | 1 − μ | ≤ 1 \lambda + \lvert 1-\mu \rvert \leq 1 . A subclass of functions f f in S \mathcal {S} satisfying | f ′ ( z ) ( z / f ( z ) ) 2 − μ | > λ , z ∈ D \lvert f’(z) (z/f(z))^2 - \mu \rvert > \lambda , \,\, z \in \mathbb {D} , is considered in this article. By using the bound of the second coefficient of f f in this subclass, several coefficients bounds are being explored for this subclass and sharp results are obtained.

In the present investigation, our aim is to define a generalized subclass of analytic and bi-univalent functions associated with a certain $q$-integral operator in the open unit disk $\mathbb{U}$. We estimate bounds on the initial Taylor-Maclaurin coefficients $\left \vert a_{2}\right \vert$ and $\left \vert a_{3}\right \vert $ for normalized analytic functions $f$ in the open unit disk by considering the function $f$ and its inverse $g = f^{{-}{1}}$. Furthermore, we derive special consequences of the results presented here, which would apply to several (known or new) subclasses of analytic and bi-univalent functions.

In this research, we focused on presenting a novel subclass of multivalent analytic functions situated in the open unit disk, characterized by the use of Jackson’s derivative operator. Our investigation systematically establishes the requisite inclusion conditions in this class, offering detailed coefficient characterizations. The exploration encompasses an array of significant properties intrinsic to this subclass, encompassing coefficient estimates, growth and distortion theorems, identification of extreme points, and the determination of the radius of starlikeness and convexity for functions falling within this specialized category. Expanding the preliminary findings, this research extended the inquiry to delve deeper into the intriguing features and implications associated with this new subclass of multivalent analytic functions. The research concentrated the light on the nuanced intricacies of coefficient estimates, providing a comprehensive understanding of how these functions evolve within the open unit disk through exploring the growth and distortion theorems, unraveling the underlying mathematical principles governing the behavior of functions in this subclass as they extend beyond the unit disk. The findings of this research contribute to the broader understanding of multivalent analytic functions, paving the way for further exploration and applications in diverse mathematical contexts.

Examining the behavior of parametric convex operators on a certain set of analytical functions

Let Ω denote the class of functions f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ belonging to the normalized analytic function class A in the open unit disk U = z : z < 1 , which are bi-univalent in U , that is, both the function f and its inverse f − 1 are univalent in U . In this paper, we introduce and investigate two new subclasses of the function class Ω of bi-univalent functions defined in the open unit disc U , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

The primary object of this paper is to investigate sharp estimate to the Toeplitz determinants of third order for the class of bounded turning functions and fourth order for the class of starlike and convex functions in the open unit disk $\mathbb{D},$ which are the fundamental subclasses of univalent functions. The practical tools applied in the derivation of our main results are the coefficient inequalities for the analytic in $\mathbb{D}$ functions from the Carath\'{e}odory class. The problem of finding sharp estimates to the Toeplitz determinants for the function $f,$ when it is a member of certain subclass of univalent functions is technically difficult in the case when $q = 4$ and $s\in\{1, 2\}$, than that in the case when $q=3$ and $s\in\{1, 2\}.$ Here, in our present investigation, we have successfully derived the sharp bounds of third -order namely $T_{3,2}\big(f\big)$ for the class of Bounded turning functions and fourth-order Toeplitz determinants namely $T_{4,1}\big(f\big)$ and $T_{4,2}\big(f\big)$ for the class of starlike and convex functions. With the motivation of these results, researchers may obtain bounds (sharp) for other classes of analytic functions of higher order Toeplitz determinants.

The objective of this paper is to obtain the best possible sharp upper bound for the second Hankel functional associated with the kth root transform [f(zk)]1/k of normalized analytic function f(z) when it belongs to certain subclass of analytic functions, defined on the open unit disc in the complex plane using Toeplitz determinants.

The quasi-Hadamard products of certain subclasses of analytic p-valent functions with negative coefficients

In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions using the q-derivative operator Dq0<q<1 and the Gegenbauer polynomials in a symmetric domain, which is the open unit disc Λ=℘:℘∈Cand℘<1. For these subclasses of analytic and bi-univalent functions, the coefficient estimates and Fekete–Szegö inequalities are solved. Some special cases of the main results are also linked to those in several previous studies. The symmetric nature of quantum calculus itself motivates our investigation of the applications of such quantum (or q-) extensions in this paper.

The symmetric differential operator SDO is a simplification functioning of the recognized ordinary derivative. The purpose of this effort is to provide a study of SDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to deliver two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function, bounded turning function subclass and convolution structures. Consequently, we define a linear combination differential operator involving the Sàlàgean differential operator and the Ruscheweyh derivative. The new operator is a generalization of the Lupus differential operator. Moreover, we aim to solve some special complex boundary problems for differential equations, spatially the class of Briot-Bouquet differential equations. All solutions are symmetric under the suggested SDOs. Additionally, by using the SDOs, we introduce a generalized class of Briot-Bouquet differential equations to deliver, what is called the symmetric Briot-Bouquet differential equations. We shall show that the upper solution is symmetric in the open unit disk by considering a set of examples of univalent functions.KeywordsUnivalent functionAnalytic functionOpen unit diskFractional operatorFractional calculus

Fractional calculus is the prominent branch of applied mathematics, it deals with a lot of diverse possibility of finding differentiation as well as integration of function <i>f</i>(<i>z</i>) when the order of differentiation operator ‘D’ and integration operator ‘J’ is a real number or a complex number. The combination of fractional calculus with geometric function theory is the dynamic field of the current research scenario. It has many applications not only in the field of mathematics but also in the different fields like modern mathematical physics, electrochemistry, viscoelasticity, fluid dynamics, electromagnetic, the theory of partial differential equations systems, Mathematical modeling. Various new subclasses of univalent and multivalent functions defined by using different operators. In this research paper, we work on the formation of new subclass of analytic and multivalent functions defined under the open unit disk. By using Generalized Ruscheweyh derivative operator we define a new subclass of analytic and multivalent functions. The main aim of this research article is to derive interesting characteristics of new subclass of multivalent functions, which mainly include coefficient bound, growth and distortion bounds for function and its first derivative, extreme point and obtain unidirectional results for the multivalent functions which are belonging to this new subclass.

In this paper, we obtain the best possible upper bound to the second and third Hankel determinants for the functions belonging to a certain general subclass of analytic functions defined on the open unit disc in the complex plane using Toeplitz determinants. Relevant connections of the results presented here with those given in earlier works are also indicated.

In this effort, we present a new definition of the Steiner symmetrization by using special analytic functions in a complex domain (the open unit disk) with respect to the origin. This definition will be used to optimize the class of univalent analytic functions. Our method is based on the concept of differential subordination and the Carathéodory theory. Examples are illustrated in the sequel involving the modified Libera–Livingston–Bernardi integral operator over the open unit disk. The result gives that this integral satisfies the definition of bounded turning function (univalent analytic function).

Let A be a class of functions of the form f(z)=z+∑n=2∞anzn which are analytic in the open unit disc D={ z∈ℂ:| z |<1 } where an is a complex number. Also let S denotes a subclass of all functions in A which are univalent in D and let Σ denotes the class of bi-univalent functions in D. In this paper, we introduce two subclasses of Σ defined in the open unit disk D which are denoted by G∑s(α,β) and G*∑s(α,β) and we find the upper bounds for the second and S LS third coefficients for functions in these subclasses.