Third Hankel Determinant for a Subclass of Univalent Functions Associated with Lemniscate of Bernoulli

This paper deals with a new subclass of univalent function associated with the right half of the lemniscate of Bernoulli. We have find the upper bound of the Hankel determinant H3(1) for this subclass by applying the Carlson–Shaffer operator to it. The present work also deals with certain properties of this newly defined subclass, such as the upper bound of the Hankel determinant of order 3, the co-efficient estimate, etc.

Bounds for Two New Subclasses of Bi-Univalent Functions Associated with Legendre Polynomials

In this article, two new subclasses of the bi-univalent function class σ related with Legendre polynomials are presented. Additionally, the first two Taylor–Maclaurin coefficients a2 and a3 for the functions belonging to these new subclasses are estimated.

Applications of Borel Distribution for a New Family of Bi-Univalent Functions Defined by Horadam Polynomials

In this paper, by making use of Borel distribution we introduce a new family GΣ(δ, γ, λ, τ, r) of normalized analytic and bi-univalent functions in the open unit disk U, which are associated with Horadam polynomials. We establish upper bounds for the initial Taylor-Maclaurin coefficients |a2| and |a3| of functions belonging to the analytic and bi-univalent function family which we have introduced here. Furthermore, we establish the Fekete-Szego problem of functions in this new family.

On Some Subclasses of $m$-fold Symmetric Bi-univalent Functions associated with the Sakaguchi Type Functions

In the present investigation, we introduce the subclasses $\Lambda_{\Sigma_m}^{\rightthreetimes}(\sigma,\phi,\upsilon)$ and $\Lambda_{\Sigma_m}^{\rightthreetimes}(\sigma,\gamma,\upsilon)$ of $m$-fold symmetric bi-univalent function class $\Sigma_m$, which are associated with the Sakaguchi type of functions and defined in the open unit disk. Further, we obtain estimates on the initial coefficients $b_{m+1}$ and $b_{2m+1}$ for the functions of these subclasses and find out connections with some of the familiar classes.

Some Geometric Properties of a Hyperbolic Univalent Function

In this paper, we analyze several aspects of a hyperbolic univalent function related to convexity properties, by assuming to be the univalent holomorphic function maps of the unit disk onto the hyperbolic convex region ( is an open connected subset of). This assumption leads to the coverage of some of the findings that are started by seeking a convex univalent function distortion property to provide an approximation of the inequality and confirm the form of the lower bound for . A further result was reached by combining the distortion and growth properties for increasing inequality . From the last result, we wanted to demonstrate the effect of the unit disk image on the condition of convexity estimation by proving the two inequalities of , and .

Coefficient Estimates for Bi-univalent Functions Defined By (P, Q) Analogue of the Salagean Differential Operator Related to the Chebyshev Polynomials

In the present investigation we use the Jackson (p,q)-differential operator to introduce the extended Salagean operator denoted by Rkp,q. Certain bi-univalent function classes based on operator Rkp,q related to the Chebyshev polynomials are introduced. First, two coefficient bounds and Fekete-Szego inequalities for the function classes are established. A number of corollaries are developed by varying parameters involved.

Radius of starlikeness for some classes containing non-univalent functions

A starlike univalent function [Formula: see text] is characterized by [Formula: see text]; several subclasses of starlike functions were studied in the past by restricting [Formula: see text] to take values in a region [Formula: see text] on the right-half plane, or, equivalently, by requiring [Formula: see text] to be subordinate to the corresponding mapping of the unit disk [Formula: see text] to the region [Formula: see text]. The mappings [Formula: see text], [Formula: see text], defined by [Formula: see text] and [Formula: see text] map the unit disk [Formula: see text] to certain nice regions in the right-half plane. For normalized analytic functions [Formula: see text] with [Formula: see text] and [Formula: see text] are subordinate to the function [Formula: see text] for some analytic functions [Formula: see text] and [Formula: see text], we determine the sharp radius for them to belong to various subclasses of starlike functions.

Coefficient Estimates for Some Subclasses of m-Fold Symmetric Bi-univalent Functions Defined by Linear Operator

The articles introduces and investigates "two new subclasses of the bi-univalent functions ." These are analytical functions related to the m-fold symmetric function and . We calculate the initial coefficients for all the functions that belong to them, as well as the coefficients for the functions that belong to a field where finding these coefficients requires a complicated method. Between the remaining results, the upper bounds for "the initial coefficients "are found in our study as well as several examples. We also provide a general formula for the function and its inverse in the m-field. A function is called analytical if it does not take the same values twice . It is called a univalent function if it is analytical at all its points, and the function is called a bi-univalent if it and its inverse are univalent functions together. We also discuss other concepts and important terms. .

On the Ninth Coefficient of the Inverse of a Convex Function

We consider the inverse function z=g(w)=w+b2w2+⋯ of a normalized convex univalent function w=f(z)=z+a2z2+⋯ on the unit disk in the complex plane. So far, it is known that |bn|≤1 for n=2,3,⋯,8. On the other hand, the inequality |bn|≤1 is not valid for n=10. It is conjectured that |b9|≤1. The present paper offers the estimate |b9|<1.617.