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The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and Hurwitz–Lerch zeta functions to formulate a new special function, namely, the logarithm-Hurwitz–Lerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the Hurwitz–Lerch zeta function behave under different transformations. [ABSTRACT FROM AUTHOR]

In this investigation, we study a class of analytic functions of type Carathéodory style in the open unit disk connected with some fractal domains. This class of analytic functions is formulated based on a kind of Langevin differential equations (LDEs). We aim to study the analytic solvability of LDEs in the advantage of geometric function theory consuming the geometric properties of the Julia fractal (JF) and other fractal connected with the logarithmic function. The analytic solutions of the LDEs are obtainable by employing the subordination theory. [ABSTRACT FROM AUTHOR]

Univalent functions with asymptotically conformal extension to the boundary form a subclass of functions with quasiconformal extension with rather special features. Such functions arise in various questions of geometric function theory and Teichmüller space theory and have important applications involving conformal and quasiconformal maps. The paper provides an approximative characterization of local conformality and its connection with univalent polynomials. Also, some other quantitative applications of this connection are given. [ABSTRACT FROM AUTHOR]

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases. [ABSTRACT FROM AUTHOR]

In the present paper, we investigated some conditions to be in the class of Carathéodory functions by using the concept of the first-order differential subordinations. Moreover, various interesting special cases were considered in the geometric function theory as applications of main results presented here. [ABSTRACT FROM AUTHOR]

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ , convexity of order δ , k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature. [ABSTRACT FROM AUTHOR]

The research presented in this paper concerns the notion of geometric function theory called fuzzy differential subordination. Using the technique associated with fuzzy differential subordination, a new subclass of analytic functions related with the strongly Janowski-type functions is defined. The class is introduced by using a new operator defined by the convolution of the generalized Sălăgean differential operator and Choi integral linear operator. Certain inclusion relations are proved for this class by using the notion of fuzzy differential subordination. In addition, new fuzzy differential subordinations are obtained that characterize this class. [ABSTRACT FROM AUTHOR]

The study presented in this paper follows a line of research familiar for Geometric Function Theory, which consists in defining new integral operators and conducting studies for revealing certain geometric properties of those integral operators such as univalence, starlikness, or convexity. The present research focuses on the Bessel function of the first kind and order ν unveiling the conditions for this function to be univalent and further using its univalent form in order to define a new integral operator on the space of holomorphic functions. For particular values of the parameters implicated in the definition of the new integral operator involving the Bessel function of the first kind, the well-known Alexander, Libera, and Bernardi integral operators can be obtained. In the first part of the study, necessary and sufficient conditions are obtained for the Bessel function of the first kind and order ν to be a starlike function or starlike of order α ∈ [ 0 , 1) . The renowned prolific method of differential subordination due to Sanford S. Miller and Petru T. Mocanu is employed in the reasoning. In the second part of the study, the outcome of the first part is applied in order to introduce the new integral operator involving the form of the Bessel function of the first kind, which is starlike. Further investigations disclose the necessary and sufficient conditions for this new integral operator to be starlike or starlike of order 1 2 . [ABSTRACT FROM AUTHOR]

In this paper, we employ the concept of the $ q $-derivative to derive certain differential and integral operators, $ D_{q, \lambda}^{n} $ and $ I_{q, \lambda}^{n} $, resp., to generalize the class of Salagean operators over the set of univalent functions. By means of the new operators, we establish the subclasses $ M^n_{q, \lambda} $ and $ D^n_{q, \lambda} $ of analytic functions on an open unit disc. Further, we study coefficient inequalities for each function in the given classes. Over and above, we derive some properties and characteristics of the set of differential subordinations by following specific techniques. In addition, we study the general properties of $ D_{q, \lambda}^{n} $ and $ I_{q, \lambda}^{n} $ and obtain some interesting differential subordination results. Several results are also derived in some details.

In this paper, we employ the concept of the q -derivative to derive certain differential and integral operators, D q , λ n and I q , λ n , resp., to generalize the class of Salagean operators over the set of univalent functions. By means of the new operators, we establish the subclasses M q , λ n and D q , λ n of analytic functions on an open unit disc. Further, we study coefficient inequalities for each function in the given classes. Over and above, we derive some properties and characteristics of the set of differential subordinations by following specific techniques. In addition, we study the general properties of D q , λ n and I q , λ n and obtain some interesting differential subordination results. Several results are also derived in some details. [ABSTRACT FROM AUTHOR]

Gegenbauer polynomials constitute a numerical tool that has attracted the interest of many function theorists in recent times mainly due to their real-life applications in many areas of the sciences and engineering. Their applications in geometric function theory (GFT) have also been considered by many researchers. In this paper, this powerful tool is associated with the prolific concepts of convolution and subordination. The main purpose of the research contained in this paper is to introduce and study a new subclass of analytic functions. This subclass is presented using an operator defined as the convolution of the generalized distribution and the error function and applying the principle of subordination. Investigations into this subclass are considered in connection to Carathéodory functions, the modified sigmoid function and Bell numbers to obtain coefficient estimates for the contained functions. [ABSTRACT FROM AUTHOR]

Geometric function theory combines geometric tools and their applications for information and communication analysis. It is also successfully used in the field of advanced signals, image processing, machine learning, speech and sound recognition. Various new subclasses of analytic functions have been defined using quantum calculus to investigate many interesting properties of these subclasses. This article defines a new class of q-starlike functions in the open symmetric unit disc ∇ using symmetric quantum calculus. Extreme points for this class as well as coefficient estimates and closure theorems have been investigated. By fixing several coefficients finitely, all results were generalized into families of analytic functions. [ABSTRACT FROM AUTHOR]

The topics studied in the geometric function theory of one variable functions are connected with the concept of Symmetry because for some special cases the analytic functions map the open unit disk onto a symmetric domain. Thus, if all the coefficients of the Taylor expansion at the origin are real numbers, then the image of the open unit disk is a symmetric domain with respect to the real axis. In this paper, we formulate the q-differential operator associated with the q-Raina function using quantum calculus, that is the so-called Jacksons' calculus. We establish a new subclass of analytic functions in the unit disk by using this newly developed operator. The theory of differential subordination inspired our approach; therefore, we geometrically explore the most popular properties of this new operator: subordination properties, coefficient bounds, and the Fekete-Szegő problem. As special cases, we highlight certain well-known corollaries of our primary findings. [ABSTRACT FROM AUTHOR]

Ever since Lotfi A. Zadeh published the paper "Fuzzy Sets" in 1965 setting the basis of a new theory named fuzzy sets theory, many scientists have developed this theory and its applications. Mathematicians were especially interested in extending classical mathematical results in the fuzzy context. Such an extension was also done relating fuzzy sets theory and geometric theory of analytic functions. The study begun in 2011 has many interesting published outcomes and the present paper follows the line of the previous research in the field. The aim of the paper is to give some references related to the connections already made between fuzzy sets theory and geometric theory of analytic functions and to present some new results that might prove interesting for mathematicians willing to enlarge their views on certain aspects of the merge between the two theories. Using the notions of fuzzy differential subordination and the classical notion of differential subordination for analytic functions, two criteria for the univalence of the analytic functions are stated in this work. [ABSTRACT FROM AUTHOR]

The central purpose of this effort is to investigate analytic and geometric properties of a class of normalized analytic functions in the open unit disk involving Bernoulli's formula. As a consequence, some solutions are indicated by the well-known hypergeometric function. The class of starlike functions is investigated containing the suggested class. [ABSTRACT FROM AUTHOR]

In this note, we formulate a new linear operator given by Airy functions of the first type in a complex domain. We aim to study the operator in view of geometric function theory based on the subordination and superordination concepts. The new operator is suggested to define a class of normalized functions (the class of univalent functions) calling the Airy difference formula. As a result, the suggested difference formula joining the linear operator is modified to different classes of analytic functions in the open unit disk. [ABSTRACT FROM AUTHOR]

Quantum calculus (the calculus without limit) appeared for the first time in fluid mechanics, noncommutative geometry and combinatorics studies. Recently, it has been included into the field of geometric function theory to extend differential operators, integral operators, and classes of analytic functions, especially the classes that are generated by convolution product (Hadamard product). In this effort, we aim to introduce a quantum symmetric conformable differential operator (Q-SCDO). This operator generalized some well-know differential operators such as Sàlàgean differential operator. By employing the Q-SCDO, we present subclasses of analytic functions to study some of its geometric solutions of q-Painlevé differential equation (type III). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let A be the class of analytic functions in the unit disk that are normalized with f(0) = f′(0) - 1 = 0: In this paper we study the class Gλ = { f ϵ A : ∣1+z f″(z)/f′(z)/zf′(z)/f(z)=1∣<λ, z ϵ u}, and give sharp sufficient conditions that embed it into the class S*[A, B, b] = {f ϵ A : +1/b (z f′(z)/f(z)-1) ≺ 1+Az/1+Bz}where -1 ≤ B < A ≤ 1; b ≠ 0 and " ≺ " denotes the usual subordination. [ABSTRACT FROM AUTHOR]

This paper is related to notions adapted from fuzzy set theory to the field of complex analysis, namely fuzzy differential subordinations. Using the ideas specific to geometric function theory from the field of complex analysis, fuzzy differential subordination results are obtained using a new integral operator introduced in this paper using the well-known confluent hypergeometric function, also known as the Kummer hypergeometric function. The new hypergeometric integral operator is defined by choosing particular parameters, having as inspiration the operator studied by Miller, Mocanu and Reade in 1978. Theorems are stated and proved, which give corollary conditions such that the newly-defined integral operator is starlike, convex and close-to-convex, respectively. The example given at the end of the paper proves the applicability of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider convex subordination chains (c.s.c.) and alpha-prestarlike subordination chains (α-p.s.c.) over on the unit disc in the complex plane. We obtain sufficient conditions for a mapping to be an α-p.s.c. over , which generalize a well-known result of Ruscheweyh [St. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l''Université de Montreal, 1982]. We also obtain sufficient conditions for injectivity for nonanalytic mappings on the unit disc, and give certain examples of α-c.s.c. over on the unit disc. [Copyright &y& Elsevier]

Asymptotic analysis is a branch of mathematical analysis that describes the limiting behavior of the function. This behavior appears when we study the solution of differential equations analytically. The recent work deals with a special class of third type of Painlevé differential equation (PV). Our aim is to find asymptotic, symmetric univalent solution of this class in a symmetric domain with respect to the real axis. As a result that the most important problem in the asymptotic expansion is the connections bound (coefficients bound), we introduce a study of this problem. [ABSTRACT FROM AUTHOR]

In the z- domain, differential subordination is a complex technique of geometric function theory based on the idea of differential inequality. It has formulas in terms of the first, second and third derivatives. In this study, we introduce some applications of the third-order differential subordination for a newly defined linear operator that includes ξ -Generalized-Hurwitz–Lerch Zeta functions (GHLZF). These outcomes are derived by investigating the appropriate classes of admissible functions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The symmetric differential operator is a generalization operating of the well-known ordinary derivative. These operators have advantages in boundary value problems, statistical studies and spectral theory. In this effort, we introduce a new symmetric differential operator (SDO) and its integral in the open unit disk. This operator is a generalization of the Sàlàgean differential operator. Our study is based on geometric function theory and its applications in the open unit disk. We formulate new classes of analytic functions using SDO depending on the symmetry properties. Moreover, we define a linear combination operator containing SDO and the Ruscheweyh derivative. We illustrate some inclusion properties and other inequalities involving SDO and its integral. [ABSTRACT FROM AUTHOR]