Subclass of Analytic Functions Associated with Differential 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.

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.

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:

The notion of strong differential subordination was introduced in 1994 and the theory related to it was developed in 2009. The dual notion of strong differential superordination was also introduced in 2009. In a paper published in 2012, the notion of strong differential subordination was given a new approach by defining new classes of analytic functions on U×U¯ having as coefficients holomorphic functions in U¯. Using those new classes, extended Sălăgean and Ruscheweyh operators were introduced and a new extended operator was defined as Lαm:Anζ*→Anζ*,Lαmf(z,ζ)=(1−α)Rmf(z,ζ)+αSmf(z,ζ),z∈U,ζ∈U¯, where Rmf(z,ζ) is the extended Ruscheweyh derivative, Smf(z,ζ) is the extended Sălăgean operator and Anζ*={f∈H(U×U¯),f(z,ζ)=z+an+1ζzn+1+⋯,z∈U,ζ∈U¯}. This operator was previously studied using the new approach on strong differential subordinations. In the present paper, the operator is studied by applying means of strong differential superordination theory using the same new classes of analytic functions on U×U¯. Several strong differential superordinations concerning the operator Lαm are established and the best subordinant is given for each strong differential superordination.

Inequality study is a magnificent field for investigating the geometric behaviors of analytic functions in the open unit disk calling the subordination and superordination. In this work, we aim to formulate a generalized differential-difference operator. We introduce a new class of analytic functions having the generalized operator. Some subordination results are included in the sequel.

We introduce a new class of analytic functions on the unit disc of the complex plane by modifying the F(p, q, s) spaces introduced by the last author by adding a measure factor. We prove some basic properties and relationships with other classical function spaces

In this paper, we introduce and study some new subclasses of analytic functions defined in the open unit disc using the convolution technique. Inclusion results, radius problems and several other properties of these classes are discussed. Key words: Convolution integral operator, functions with positive real part, alpha-starlike, bounded Mocanu variation, univalent.

<abstract><p>The present investigation focus on applying the theories of differential subordination, differential superordination and related sandwich-type results for the study of some subclasses of symmetric functions connected through a linear extended multiplier operator, which was previously defined by involving the $ q $-Choi-Saigo-Srivastava operator. The aim of the paper is to define a new class of analytic functions using the aforementioned linear extended multiplier operator and to obtain sharp differential subordinations and superordinations using functions from the new class. Certain subclasses are highlighted by specializing the parameters involved in the class definition, and corollaries are obtained as implementations of those new results using particular values for the parameters of the new subclasses. In order to show how the results apply to the functions from the recently introduced subclasses, numerical examples are also provided.</p></abstract>

The aim of the present paper is to introduce a new class of analytic functions by using a q-integral operator in the conic region. It is worth mentioning that these regions are symmetric along the real axis. We find the coefficient estimates, the Fekete–Szegö inequality, the sufficiency criteria, the distortion result, and the Hankel determinant problem for functions in this class. Furthermore, we study the inverse coefficient estimates for functions in this class.

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.

Problem statement: We introduced a new bijective convolution linear operator defined on the class of normalized analytic functions. This operator was motivated by many researchers namely Srivastava, Owa, Ruscheweyh and many others. The operator was essential to obtain new classes of analytic functions. Approach: Simple technique of Ruscheweyh was used in our preliminary approach to create new bijective convolution linear operator. The preliminary concept of Hadamard products was mentioned and the concept of subordination was given to give sharp proofs for certain sufficient conditions of the linear operator aforementioned. In fact, the subordinating factor sequence was used to derive different types of subordination results. Results: Having the linear operator, subordination theorems were established by using standard concept of subordination. The results reduced to well-known results studied by various researchers. Coefficient bounds and inclusion properties, growth and closure theorems for some subclasses were also obtained. Conclusion: Therefore, many interesting results could be obtained and some applications could be gathered.

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

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.

Consider a family of integral operators and a related family of differential operators, both defined on a class of analytic functions holomorphic in the unit disk, distortion properties of the real part are derived from a general aspect. 1. Integral and differential operators. Let F denote the class of analytic functions which are holomorphic in the unit disk E = {| z |< 1}. Let F nad G be its subclasses consisting of f ∈ F normalized by f(0) = f ′(0) − 1 = 0 and f(0) = 1, respectively. In previous papers [3, 6] we have observed an integral operator L(a) defined on F , which is represented by L(a)f(z) = a ∫