Coefficient functionals and radius problems associated with an evolute of a nephroid curve

The most essential and the needed concept used by the theory of complex function is the Quasi subordination. The subordination along with the majorization concepts are getting collaborated with the help of this Quasi subordination concept. In this article, a novel subclass consisting of univalent analytic functions are investigated, analysed and reviewed. The Chebyshev polynomials that is associated with the open unit disk is defined. Further the estimates are found in these classes having the coefficients of functions by utilizing the benefits of Chebyshev polynomials. Then the Fekete-Szego inequalities are obtained which provides the results representing the ¨ associated new classes thereby briefly making the quasi-subordination to involve along with majorization results.

In this work, we introduce a new subclas of bi-univalent functions which is defined by Hadamard product andsubordination in the open unit disk. and find upper bounds for the second and third coefficients for functions in this new subclass. Further, we generalize and improve some of the previously published results.

In this paper, we introduce and investigate a new subclass of bi-prestarlike functions defined in the open unit disk, associated with Chebyshev Polynomials. Furthermore, we find estimates of first two coefficients of functions in these classes, making use of the Chebyshev polynomials. Also, we obtain the Fekete-Szegö inequalities for function in these classes. Several consequences of the results are also pointed out as corollaries.

A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to H", the linear invariant family of locally univalent functions of finite order or.If f(z) is subordinate to F(z) in the open unit disc, /'(0) > 0, and F(z) is in U" 1.65 < a < oo, then f'(z) is majorized by F'(z) in \z\ < (a + 1) -(a2 + 2a)vi.The result is sharp.I. Introduction.Let <S denote the set of all normalized analytic univalent functions in the open unit disc D. Let/(z), F(z) and <p(z) be analytic in |z| < r.We say that/(z) is majorized byLet tla be the set of all locally univalent (f'(z) = 0) analytic functions in D with order < a which are of the form f(z) = z A-.The family U" is known as the universal linear invariant family of order a [4].A concise summary and introduction to properties of linear invariant families which relate to the following material is contained in [1].The present paper concludes the proof of results announced in [1].Majorization-subordination theory begins with Biernacki who showed in 1936 that if/(z) is subordinate in D to F(z) (F(z) G <S), then/(z) is majorized by F(z) in |z| < 1/4.In the succeeding years Goluzin, Tao Shah, Lewandowski and MacGregor examined various related problems but always under the stipulation that the dominant function F(z) is in <S (for greater detail see [1]).In 1951 Goluzin showed that if f(z) is majorized by a univalent function F(z), then f'(z) would be majorized by F'(z) in |z| < 0.12.He conjectured that majorization would always occur for jz| < 3 -y and this was proved by Tao Shah in 1958.In this paper we show that the result is actually true for functions in 11. and obtain the sharp radius of majorization as a + 1 -(a2 + 2a)^2 for 1.65 < o < oo.This yields 3 -y for the case a = 2.Our investigation shows that the important datum for majorization-subordination theory is not univalence, but the order of a linear invariant family.In particular, many classically derived estimates for univalent functions are true for functions of infinite valence.

In this paper, we present a new subclass $${\mathcal {T}}_{\varSigma }(\mu )$$ of bi univalent functions belong to $$\varSigma $$ in the open unit disc $${\mathcal {U}} =\left\{ z\, :\,\,z\in {\mathcal {C}}\,\,and \,\, |z| <1\right\} $$ . Then, we use the concepts of Faber polynomial expansions to find upper bound for the general coefficient of such functions belongs to the defined class. Further, for the functions in this subclass we obtain bound on first three coefficients $$|a_{2}|$$ , $$|a_{3}|$$ and $$|a_{4}|$$ . We hope that this paper will inspire future researchers in applying our approach to other related problems.

The goal of this paper is to investigate some of the features of differential subordination of analytic univalent functions in an open unit disc. In addition, it has shed light on geometric features such as coefficient inequality, Hadamard product qualities, and the Komatu integral operator. Some intriguing results for third-order differential subordination and superordination of analytic univalent functions have been installed. Then, using the convolution of two linear operators, certain results of third order differential subordination involving linear operators were reported. As a result, we use features of the Komatu integral operator to analyze and study third-order subordinations and superordinations in relation to the convolution. Finally, several results for third order differential subordination in the open unit disk using generalized hypergeometric function have been addressed using the convolution operator.

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 &lt; 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.

By using the most generalized gamma function (parametric gamma function, or p-gamma function), we present the most generalized Rabotnov function, called the p-Rabotnov function. Consequently, new fractal–fractional differential and integral operators of a complex variable in an open unit disk are defined and investigated analytically and geometrically. We address some inequalities involving the generalized fractal–fractional integral operator in some spaces of analytic functions. A novel complex fractal–fractional integral transform (CFFIT) is presented. A normalization of the proposed CFFIT is observed in the open unit disk. Examples are illustrated for power series of analytic functions.

It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO 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 impose 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 SCDOs 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 and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk.

For ??(?,?], let Ra(?) denote the class of all normalized analytic functions in the open unit disk U satisfying the following differential subordination: f'(z)+1/2(1+ei?)z f''(z)&lt;?(z) z ? U), where the function ?(z) is analytic in the open unit disk U such that ?(0)=1. In this paper, various integral and convolution characterizations, coefficient estimates and differential subordination results for functions belonging to the class R?(?) are investigated. The Fekete-Szeg? coefficient functional associated with the kth root transform [f(zk)]1/k of functions in R?(?) is obtained. A similar problem for a corresponding class R?,?(?) of bi-univalent functions is also considered. Connections with previous known results are pointed out.

Recently, the generalized Mittag-Leffler (EGML) function is extended by utilizing the extended Beta function. Based on this type of function, we provide a new integral operator in the open unit disk. The present article discusses several applications of differential subordination for certain normalized analytic functions in the open unit disk, which are acted upon by Salagean type integral operator.

It is well known that there are two important classes of analytic functions of Ma-Minda type (MMT): Ma-Minda starlike and Ma-Minda convex functions. In this work, we suggest a new class of analytic functions, which is normalized in the open unit disk. The suggested class is generated by a roulette curve formula, which satisfies the symmetric behavior in the open unit disk. A roulette curve is shaped as the path outlined by the sum of two complex numbers, each affecting at a uniform rapidity in a circle. Special cases are illustrated involving special functions. Graphics of the curve are illustrated by using Mathematica 13.3.

In Geometric Function Theory, it is well known that the familiar Koebe function f(z) = z/(1 − z)2 is the extremal function for the class 𝒮* of starlike functions in the open unit disk 𝕌 and also that the function g(z) = z/(1 − z) is the extremal function for the class 𝒦 of convex functions in the open unit disk 𝕌. However, the partial sum fn (z) of f(z) is not starlike in 𝕌 and the partial sum gn (z) of g(z) is not convex in 𝕌. The aim of the present paper is to investigate the starlikeness and convexity of these partial sums fn (z) and gn (z). Computational and graphical usages of Mathematica (Version 4.0) as well as geometrical descriptions of the image domains in several illustrative examples are also presented. E-mail: owa@math.kindai.ac.jp

We introduce and investigate new subclasses of analytic and bi-univalent functions defined by modified Tremblay operator in the open unit disk. Also we obtain upper bounds for the coefficients of functions belonging to these classes.

In the present investigation, we use the Horadam Polynomials to establish upper bounds for the second and third coefficients of functions belongs to a new subclass of analytic and Bazilevic bi-univalent functions defined in the open unit disk ∪. Also, we discuss Fekete-Szegä problem for functions belongs to this subclass.2010 Mathematics Subject Classification: 30C45, 30C50.