New subclasses of bi-univalent Ma-Minda starlike function involving Fox-Wright function

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.

Let <TEX>$\mathcal{A}$</TEX> be the class of analytic functions f in the open unit disk <TEX>$\mathbb{U}$</TEX>={z : <TEX>${\mid}z{\mid}$</TEX> < 1} with the normalization conditions <TEX>$f(0)=f^{\prime}(0)-1=0$</TEX>. If <TEX>$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$</TEX> and <TEX>${\delta}$</TEX> > 0 are given, then the <TEX>$T_{\delta}$</TEX>-neighborhood of the function f is defined as <TEX>$$TN_{\delta}(f)\{g(z)=z+\sum_{n=2}^{\infty}b_nz^n{\in}\mathcal{A}:\sum_{n=2}^{\infty}T_n{\mid}a_n-b_n{\mid}{\leq}{\delta}\}$$</TEX>, where <TEX>$T=\{T_n\}_{n=2}^{\infty}$</TEX> is a sequence of positive numbers. In the present paper we investigate some problems concerning <TEX>$T_{\delta}$</TEX>-neighborhoods of function in various classes of analytic functions with <TEX>$T=\{2^{-n}/n^2\}_{n=2}^{\infty}$</TEX>. We also find bounds for <TEX>$^{\delta}^*_T(A,B)$</TEX> defined by <TEX>$$^{\delta}^*_T(A,B)=jnf\{{\delta}</TEX><TEX>&</TEX><TEX>gt;0:B{\subset}TN_{\delta}(f)\;for\;all\;f{\in}A\}$$</TEX> where A, B are given subsets of <TEX>$\mathcal{A}$</TEX>.

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

Let S l * denote the class of analytic functions f in the open unit disk D = { z : | z | &lt; 1 } normalized by f ( 0 ) = f ′ ( 0 ) − 1 = 0 , which is subordinate to exponential function, z f ′ ( z ) f ( z ) ≺ e z ( z ∈ D ) . In this paper, we aim to investigate the third-order Hankel determinant H 3 ( 1 ) for this function class S l * associated with exponential function and obtain the upper bound of the determinant H 3 ( 1 ) . Meanwhile, we give two examples to illustrate the results obtained.

In this paper, using Hadmard product, we introduce certain new classes of analytic functions in the open unit disk. Such results as inclusion and subordination relationships, characterization and coecient estimates, growth and distortion theorems, extreme points, closure theorems and radius of starlikeness and convexity belonging to the class TS(f;g;;;;A;B ) are obtained. Further subordination results for the class S(f;g;;;;A;B ) are derived:

Let T n ( A , B , α ) denote the class of functions of the form: f ( z ) = z + ∑ k = n + 1 ∞ a k z k ( n ∈ N = { 1 , 2 , 3 , … } ) , which are analytic in the open unit disk U and satisfy the following subordination condition: f ′ ( z ) + α zf ″ ( z ) ≺ 1 + Az 1 + Bz ( z ∈ U ; - 1 ⩽ B < 1 ; B < A ; α > 0 ) . In this paper, we find the sharp lower bound on | z | = r < 1 for the functional Re { f ′ ( z ) + α zf ″ ( z ) } over the class: T n ( A , B , 0 ) = f ( z ) = z + ∑ k = n + 1 ∞ a k z k : f ′ ( z ) ≺ 1 + Az 1 + Bz ( n ∈ N = { 1 , 2 , 3 , … } ; z ∈ U ) . By applying this result, several interesting consequences are given.

We discuss the best linear approximation methods in the Hardy spaceH q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained.

Majorization properties for certain classes of analytic functions using the Sălăgean operator

Let 𝒜 be the class of analytic functions in the open unit disk 𝕌. We define Θα,β : 𝒜 → 𝒜 by (Θα,βf)(α, β ≠ 2, 3, 4…), where is the fractional derivative of f of order γ. If α, β ∈ [0, 1], then a function f in 𝒜 is said to be in the class SPα,β if Θα,βf is a parabolic starlike function. In this paper, several properties and characteristics of the class SPα,β are investigated. These include subordination, characterization and inclusions, growth theorems, distortion theorems, and class‐preserving operators. Furthermore, sandwich theorem related to the fractional derivative is proved.

Making use of a convolution structure, we introduce a new class of analytic functions PTg(λ,α,β,γ) defined in the open unit disc and investigate its various characteristics. Further we obtained distortion bounds , extreme points and radii of close-to-convexity, starlikeness and convexity for functions belonging to the class PTg(λ,α,β,γ).

In this article, the present study employs the utilization of the concepts pertaining to (x,y)-symmetrical functions, Janowski type functions, and q-calculus in order to establish a novel subclass within the open unit disk. Specifically, we delve into the examination of convolution properties, which serve as a tool for investigating and inferring adequate and equivalent conditions. Moreover, we also explore specific characteristics of the class S˜qx,y(α,β,λ), thereby further scrutinizing the convolution properties of these newly defined classes.

A characterization theorem for a certain class of analytic functions

Introduction: : In this work, we introduce and investigate a new class of analytic functions in the open unit disc with negative coefficients. The object of the present paper is to determine coefficient estimates, neighborhoods and partial sums for functions belonging to this class.

The study on fractional integrals of confluent hypergeometric functions provides interesting subordination and superordination results and inspired the idea of using this operator to introduce a new class of analytic functions. Given the class of functions An=f∈HU:fz=z+an+1zn+1+…,z∈U written simply A when n=1, the newly introduced class involves functions f∈A considered in the study due to their special properties. The aim of this paper is to present the outcomes of the study performed on the new class, which include a coefficient inequality, a distortion theorem and extreme points of the class. The starlikeness and convexity properties of this class are also discussed, and partial sums of functions from the class are evaluated in order to obtain class boundary properties.

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.