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This document is a summary of a special issue of the journal "Mathematics" that focuses on new trends in complex analysis research. The issue includes 14 papers that cover various aspects of complex-valued functions of one or several complex variables. The papers explore topics such as coefficient estimates, starlikeness and convexity of analytic functions, holomorphic and bi-univalent functions, and integral operators. The research presented in the papers aims to contribute to the development of complex analysis and inspire further studies in the field. The document also acknowledges the authors and reviewers who contributed to the special issue's success. [Extracted from the article]

Orthogonal polynomials have been widely employed by renowned authors within the context of geometric function theory. This study is driven by prior research and aims to address the —Fekete-Szegö problem. Additionally, we provide bound estimates for the coefficients and an upper bound estimate for the second Hankel determinant for functions belonging to the category of analytical and bi-univalent functions. This investigation incorporates the utilization of Euler polynomials. [ABSTRACT FROM AUTHOR]

This document titled "Fractional Calculus and Hypergeometric Functions in Complex Analysis" explores the impact of fractional calculus on various scientific and engineering disciplines. It emphasizes the significance of fractional operators in the study of fractional calculus and their applications in complex analysis research, specifically in the theory of univalent functions. The document also introduces hypergeometric functions and their connection to the theory of univalent functions. It compiles 12 research papers that cover topics such as geometric properties of fractional differential operators, logarithmic-related problems of univalent functions, and the study of generalized bi-subordinate functions. This document serves as a valuable resource for researchers interested in these subjects and their applications in complex analysis. Additionally, it provides a summary of three articles published in the Special Issue on "Fractional Calculus and Hypergeometric Functions in Complex Analysis." The first article explores the use of the Sălăgean q-differential operator for meromorphic multivalent functions, introducing new subclasses of functions. The second article presents three general double-series identities using Whipple transformations for terminating generalized hypergeometric functions, which can be used to derive additional identities. The third article defines a new generalized domain based on the quotient of two analytic functions and investigates the upper bounds of certain coefficients and determinants. The authors anticipate that these findings will inspire further research in the field. [Extracted from the article]

This document is a summary of a special issue of the journal Axioms titled "New Developments in Geometric Function Theory II." The special issue contains 14 research papers that explore various topics related to complex-valued functions in the field of Geometric Function Theory. The papers cover subjects such as coefficient estimates, subordination theories, hypergeometric functions, and differential operators. Each paper presents new findings and results that contribute to the development of Geometric Function Theory. The special issue is recommended for researchers and scholars interested in this field of study. The document also acknowledges the authors, reviewers, and editors involved in the creation of the special issue. [Extracted from the article]

This work presents a novel investigation that utilizes the integral operator I p , λ n in the field of geometric function theory, with a specific focus on sandwich theorems. We obtained findings about the differential subordination and superordination of a novel formula for a generalized integral operator. Additionally, certain sandwich theorems were discovered. [ABSTRACT FROM AUTHOR]

This study presents a subclass S (β) of bi-univalent functions within the open unit disk region D . The objective of this class is to determine the bounds of the Hankel determinant of order 3, ( Ⱨ 3 (1) ). In this study, new constraints for the estimates of the third Hankel determinant for the class S (β) are presented, which are of considerable interest in various fields of mathematics, including complex analysis and geometric function theory. Here, we define these bi-univalent functions as S (β) and impose constraints on the coefficients │ a n │ . Our investigation provides the upper bounds for the bi-univalent functions in this newly developed subclass, specifically for n = 2, 3, 4, and 5. We then derive the third Hankel determinant for this particular class, which reveals several intriguing scenarios. These findings contribute to the broader understanding of bi-univalent functions and their potential applications in diverse mathematical contexts. Notably, the results obtained may serve as a foundation for future investigations into the properties and applications of bi-univalent functions and their subclasses. [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]

Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical biology, plasma physics, electromagnetic theory, and many more. This article explores some mathematical applications of the fractional q-differential and integral operator in the field of geometric function theory. By using the linear multiplier fractional q-differintegral operator D q , λ m ρ , σ and subordination, we define and develop a collection of q-starlike functions that are linked to the cardioid domain. This study also investigates sharp inequality problems like initial coefficient bounds, the Fekete–Szego problems, and the coefficient inequalities for a new class of q-starlike functions in the open unit disc U. Furthermore, we analyze novel findings with respect to the inverse function (μ − 1) within the class of q-starlike functions in U. The findings in this paper are easy to understand and show a connection between present and past studies. [ABSTRACT FROM AUTHOR]

In this research, we combine ideas from geometric function theory and fuzzy set theory. We define a new operator D τ − λ L α , ζ m : A → A of analytic functions in the open unit disc Δ with the help of the Riemann–Liouville fractional integral operator, the linear combination of the Noor integral operator, and the generalized Sălăgean differential operator. Further, we use this newly defined operator D τ − λ L α , ζ m together with a fuzzy set, and we next define a new class of analytic functions denoted by R ϝ ζ (m , α , δ). Several innovative results are found using the concept of fuzzy differential subordination for the functions belonging to this newly defined class, R ϝ ζ (m , α , δ). The study includes examples that demonstrate the application of the fundamental theorems and corollaries. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel class of analytic functions in the form h (z) = z p + ∑ k = p + 1 ∞ a k z k in the unit disk. These functions establish a connection between the extended Mittag–Leffler function and the quantum operator presented in this paper, which is denoted by ℵ q , p n (L , a , b) and is also an extension of the Raina function that combines with the Jackson derivative. Through the application of differential subordination methods, essential properties like bounds of coefficients and the Fekete–Szegő problem for this class are derived. Additionally, some results of special cases to this study that were previously studied were also highlighted. [ABSTRACT FROM AUTHOR]

Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since ℜ (1 + sinh (z)) ≯ 0 , it implies that the class S sinh * introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter λ with the restriction 0 ≤ λ ≤ ln (1 + 2) , and by doing that, ℜ (1 + sinh (λ z)) > 0. The present research intends to provide a novel subclass of starlike functions in the open unit disk U , denoted as S sinh λ * , and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients a n for n = 2 , 3 , 4 , 5. Then, we prove a lemma, in which the largest disk contained in the image domain of q 0 (z) = 1 + sinh (λ z) and the smallest disk containing q 0 (U) are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including S * (β) and K (β) of starlike functions of order β and convex functions of order β. Investigating S sinh λ * radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding S sinh λ * radii of different subclasses is the calculation of that value of the radius r < 1 for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of λ , is also obtained. [ABSTRACT FROM AUTHOR]

One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions X defined in the new domain, including the sharp estimates for the coefficients a 2 , a 3 , and a 4 , and for three second-order and third-order Hankel determinants, H 2 , 1 X , H 2 , 2 X , and H 3 , 1 X . The optimality of each obtained estimate is given as well. [ABSTRACT FROM AUTHOR]

A branch of complex analysis with a rich history is geometric function theory, which first appeared in the early 20th century. The function theory deals with a variety of analytical tools to study the geometric features of complex-valued functions. The main purpose of this paper is to estimate more accurate bounds for the coefficient | a n | of the functions that belong to a class of bi-univalent functions with missing coefficients that are defined by using the subordination. The significance of our present results consists of improvements to some previous results concerning different recent subclasses of bi-univalent functions, and the aim of this paper is to improve the results of previous outcomes. In addition, important examples of some classes of such functions are provided, which can help to understand the issues related to these functions. [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]

In this study, we investigate a possible relationship between fuzzy differential subordination and the theory of geometric functions. First, using the Al-Oboudi differential operator and the Babalola convolution operator, we establish the new operator BS α , λ m , t : A n → A n in the open unit disc U. The second step is to develop fuzzy differential subordination for the operator BS α , λ m , t . By considering linear transformations of the operator BS α , λ m , t , we define a new fuzzy class of analytic functions in U which we denote by T ϝ λ , t (m , α , δ) . Several innovative results are found using the concept of fuzzy differential subordination and the operator BS α , λ m , t for the function f in the class T ϝ λ , t (m , α , δ) . In addition, we explore a number of examples and corollaries to illustrate the implications of our key findings. Finally, we highlight several established results to demonstrate the connections between our work and existing studies. [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]

The Theory of Complex Functions has been studied by many scientists and its application area has become a very wide subject. Harmonic functions play a crucial role in various fields of mathematics, physics, engineering, and other scientific disciplines. Of course, the main reason for maintaining this popularity is that it has an interdisciplinary field of application. This makes this subject important not only for those who work in pure mathematics, but also in fields with a deep-rooted history, such as engineering, physics, and software development. In this study, we will examine a subclass of Harmonic functions in the Theory of Geometric Functions. We will give some definitions necessary for this. Then, we will define a new subclass of complex-valued harmonic functions, and their coefficient relations, growth estimates, radius of univalency, radius of starlikeness and radius of convexity of this class are investigated. In addition, it is shown that this class is closed under convolution of its members. [ABSTRACT FROM AUTHOR]

This document is an editorial introducing a special issue of the journal Axioms titled "Theory of Functions and Applications." The special issue contains 15 articles that explore various topics in the theory of functions, real and complex variables, and their applications. The articles cover fields such as function approximation, functional analysis, complex analysis, differential equations, numerical methods, and mathematical modeling. The aim of the special issue is to share scholars' theories, methods, and findings in function theory and provide solutions for applied problems in related scientific fields. [Extracted from the article]

The idea of fuzzy differential subordination is a generalisation of the traditional idea of differential subordination that evolved in recent years as a result of incorporating the idea of fuzzy set into the field of geometric function theory. In this investigation, we define some general classes of p-valent analytic functions defined by the fuzzy subordination and generalizes the various classical results of the multivalent functions. Our main focus is to define fuzzy multivalent functions and discuss some interesting inclusion results and various other useful properties of some subclasses of fuzzy p-valent functions, which are defined here by means of a certain generalized Srivastava-Attiya operator. Additionally, links between the significant findings of this study and preceding ones are also pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we first modify one of the most famous theorems on the principle of differential subordination to hold true for normalized analytic functions with a fixed initial Taylor-Maclaurin coefficient. By using this modified form, we generalize and improve several results, which appeared recently in the literature on the geometric function theory of complex analysis. We also prove some simple conditions for starlikeness, convexity, and the strong starlikeness of several one-parameter families of integral operators, including (for example) a certain μ -convex integral operator and the familiar Bernardi integral operator. [ABSTRACT FROM AUTHOR]

This work is concerned with the branch of complex analysis known as geometric function theory, which has been modified for use in the study of fuzzy sets. We develop a novel operator L α , λ m : A n → A n in the open unit disc Δ using the Noor integral operator and the generalized Sălăgean differential operator. First, we develop fuzzy differential subordination for the operator L α , λ m and then, taking into account this operator, we define a particular fuzzy class of analytic functions in the open unit disc Δ , represented by R ϝ λ (m , α , δ) . Using the idea of fuzzy differential subordination, several new results are discovered that are relevant to this class. The fundamental theorems and corollaries are presented, and then examples are provided to illustrate their practical use. [ABSTRACT FROM AUTHOR]

The Gould–Hopper–Laguerre–Sheffer matrix polynomials were initially studied using operational methods, but in this paper, we investigate them using matrix techniques. By leveraging properties of Pascal functionals and Wronskian matrices, we derive several identities for these polynomials, including recurrence relations. It is highlighted that these identities, acquired via matrix techniques, are distinct from the ones obtained when using operational methods. [ABSTRACT FROM AUTHOR]

Many researchers have defined the q-analogous of differential and integral operators for analytic functions using the concept of quantum calculus in the geometric function theory. In this study, we conduct a comprehensive investigation to identify the uses of the Sălăgean q-differential operator for meromorphic multivalent functions. Many features of functions that belong to geometrically defined classes have been extensively studied using differential operators based on q-calculus operator theory. In this research, we extended the idea of the q-analogous of the Sălăgean differential operator for meromorphic multivalent functions using the fundamental ideas of q-calculus. With the help of this operator, we extend the family of Janowski functions by adding two new subclasses of meromorphic q-starlike and meromorphic multivalent q-starlike functions. We discover significant findings for these new classes, including the radius of starlikeness, partial sums, distortion theorems, and coefficient estimates. [ABSTRACT FROM AUTHOR]

The notion of competing HT ht -Laplacians with weights is considered for the first time. They are driven by a degenerated HT ht -Laplacian with weights and a competing HT ht -Laplacian with weights, respectively. Finally, the authors indicate how the result can be extended to HT ht -equations ( HT ht ). This research aims to present a linear operator HT ht by utilizing the I q i -Mittag-Leffler function and to introduce the subclass of harmonic HT ht -convex functions HT ht related to the Janowski function. [Extracted from the article]

The study of the q -analogue of the classical results of geometric function theory is currently of great interest to scholars. In this article, we define generalized classes of close-to-convex functions and quasi-convex functions with the help of the q -difference operator. Moreover, by using the q -analogues of a certain family of linear operators, the classes K q , b s h , K ˜ q , s b h , Q q , b s h , and Q ˜ q , s b h are introduced. Several interesting inclusion relationships between these newly defined classes are discussed, and the invariance of these classes under the q -Bernadi integral operator was examined. Furthermore, some special cases and useful consequences of these investigations were taken into consideration. [ABSTRACT FROM AUTHOR]

Pre-functions are functions that possess a sequence {fn(z, β)} which tends to one of the elementary functions as n tends to infinity and β tends to 0. The main objective of this paper is to broaden the scope of pre-functions from functions of a real variable to functions of a complex variable by introducing pre-functions of a complex variable. We have analyzed the pre-functions of a complex variable for their properties. The pre-Laguerre, pre-Bessel and pre-Legendre polynomials of a complex variable have been obtained as special cases. Graphs have been used to visualize complex pre-functions. [ABSTRACT FROM AUTHOR]

In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of differential equations. Many topics, including complex analysis, hypergeometric series, and particle physics, have been generalized in q-calculus. In this study, first of all, we define the q-analogues of a differential operator ( D R λ , q m , n ) by using the basic idea of q-calculus and the definition of convolution. Additionally, using the newly constructed operator ( D R λ , q m , n ), we establish the q-analogues of two new integral operators ( F λ , γ 1 , γ 2 , ... γ l m , n , q and G λ , γ 1 , γ 2 , ... γ l m , n , q ), and by employing these operators, new subclasses of the q-starlike and q-convex functions are defined. Sufficient conditions for the functions (f) that belong to the newly defined classes are investigated. Additionally, certain subordination findings for the differential operator ( D R λ , q m , n ) and novel geometric characteristics of the q-analogues of the integral operators in these classes are also obtained. Our results are generalizations of results that were previously proven in the literature. [ABSTRACT FROM AUTHOR]

Determining the sharp bounds for coefficient-related problems that appear in the Taylor–Maclaurin series of univalent functions is one of the most difficult aspects of studying geometric function theory. The purpose of this article is to establish the sharp bounds for a variety of problems, such as the first three initial coefficient problems, the Zalcman inequalities, the Fekete–Szegö type results, and the second-order Hankel determinant for families of Sakaguchi-type functions related to the cardioid-shaped domain. Further, we study the logarithmic coefficients for both of these classes. [ABSTRACT FROM AUTHOR]

Quantum calculus plays a significant role in many different branches such as quantum physics, hypergeometric series theory, and other physical phenomena. In our paper and using quantitative calculus, we introduce a new family of normalized analytic functions in the open unit disk, which relates to both the generalized Mittag–Leffler function and the Jackson differential operator. By using a differential subordination virtue, we obtain some important properties such as coefficient bounds and the Fekete–Szegő problem. Some results that represent special cases of this family that have been studied before are also highlighted. [ABSTRACT FROM AUTHOR]

Zadeh's fuzzy set theory offers a logical, adaptable solution to the challenge of defining, assessing and contrasting various sustainability scenarios. The results presented in this paper use the fuzzy set concept embedded into the theories of differential subordination and superordination established and developed in geometric function theory. As an extension of the classical concept of differential subordination, fuzzy differential subordination was first introduced in geometric function theory in 2011. In order to generalize the idea of fuzzy differential superordination, the dual notion of fuzzy differential superordination was developed later, in 2017. The two dual concepts are applied in this article making use of the previously introduced operator defined as the convolution product of the generalized Sălgean operator and the Ruscheweyh derivative. Using this operator, a new subclass of functions, normalized analytic in U, is defined and investigated. It is proved that this class is convex, and new fuzzy differential subordinations are established by applying known lemmas and using the functions from the new class and the aforementioned operator. When possible, the fuzzy best dominants are also indicated for the fuzzy differential subordinations. Furthermore, dual results involving the theory of fuzzy differential superordinations and the convolution operator are established for which the best subordinants are also given. Certain corollaries obtained by using particular convex functions as fuzzy best dominants or fuzzy best subordinants in the proved theorems and the numerous examples constructed both for the fuzzy differential subordinations and for the fuzzy differential superordinations prove the applicability of the new theoretical results presented in this study. [ABSTRACT FROM AUTHOR]

In this paper, the author combines the geometric theory of analytic function regarding differential superordination and subordination with fuzzy theory for the convolution product of Ruscheweyh derivative and multiplier transformation. Interesting fuzzy inequalities are obtained by the author. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study the behavior of the modules of the first two coefficients for the classes N Σ (γ , λ , δ , μ ; α) and N Σ * (γ , λ , δ , μ ; β) of normalized holomorphic and bi-univalent functions that are connected with the prestarlike functions. We determine the upper bounds for the initial Taylor–Maclaurin coefficients | a 2 | and | a 3 | for the functions of each of these families, and we also point out some special cases and consequences of our main results. The study of these classes is closely connected with those of Ruscheweyh who in 1977 introduced the classes of prestarlike functions of order μ using a convolution operator and the proofs of our results are based on the well-known Carathédory's inequality for the functions with real positive part in the open unit disk. Our results generalize a few of the earlier ones obtained by Li and Wang, Murugusundaramoorthy et al., Brannan and Taha, and could be useful for those that work with the geometric function theory of one-variable functions. [ABSTRACT FROM AUTHOR]

Some well-known authors have extensively used orthogonal polynomials in the framework of geometric function theory. We are motivated by the previous research that has been conducted and, in this study, we solve the Fekete–Szegö problem as well as give bound estimates for the coefficients and an upper bound estimate for the second Hankel determinant for functions in the class  G Σ (v , σ)  of analytical and bi-univalent functions, implicating the Euler polynomials. [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]

One of the most important problems in the study of geometric function theory is knowing how to obtain the sharp bounds of the coefficients that appear in the Taylor–Maclaurin series of univalent functions. In the present investigation, our aim is to calculate some sharp estimates of problems involving coefficients for the family of convex functions with respect to symmetric points and associated with a hyperbolic tangent function. These problems include the first four initial coefficients, the Fekete–Szegö and Zalcman inequalities, and the second-order Hankel determinant. Additionally, the inverse and logarithmic coefficients of the functions belonging to the defined class are also studied in relation to the current problems. [ABSTRACT FROM AUTHOR]

This article defines a new operator called the q-Babalola convolution operator by using quantum calculus and the convolution of normalized analytic functions in the open unit disk. We then study a new class of analytic and bi-univalent functions defined in the open unit disk associated with the q-Babalola convolution operator. The main results of the investigation include some upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szego inequalities for the functions in the new class. Many applications of the finds are highlighted in the corollaries based on the various unique choices of the parameters, improving the existing results in Geometric Function Theory. [ABSTRACT FROM AUTHOR]

At the present time, the study of various classical properties of the geometric function theory using the concept of a fuzzy subset remains limited. In this article, our main aim is to introduce the subclasses of spiral-like functions of complex order in terms of the fuzzy notion and we generalize these subclasses by applying a family of linear operators. The relationships between the newly defined subclasses are examined. In addition, we show that these subclasses are preserved under the well-known Bernardi integral operator. [ABSTRACT FROM AUTHOR]

In this paper, we define two classes of meromorphic multivalent functions in the punctured disc U * = w ∈ C : 0 < | w | < 1 by using the principle of subordination. We investigate a number of useful results including subordination results, some connections with a certain integral operator, sandwich properties, an inclusion relationship, and Fekete-Szegö inequalities for the functions belonging these classes. Our results are connected with those in several earlier works, which are related to this field of Geometric Function Theory (GFT) of Complex Analysis. [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]

In this paper, we discuss and introduce a new study using an integral operator w k , μ m in geometric function theory, especially sandwich theorems. We obtained some conclusions for differential subordination and superordination for a new formula generalized integral operator. In addition, certain sandwich theorems were found. The differential subordination theory's features and outcomes are symmetric to those derived using the differential subordination theory. [ABSTRACT FROM AUTHOR]

The results contained in this paper are the result of a study regarding fractional calculus combined with the classical theory of differential subordination established for analytic complex valued functions. A new operator is introduced by applying the Libera integral operator and fractional integral of order λ for analytic functions. Many subordination properties are obtained for this newly defined operator by using famous lemmas proved by important scientists concerned with geometric function theory, such as Eenigenburg, Hallenbeck, Miller, Mocanu, Nunokawa, Reade, Ruscheweyh and Suffridge. Results regarding strong starlikeness and convexity of order α are also discussed, and an example shows how the outcome of the research can be applied. [ABSTRACT FROM AUTHOR]

A previously introduced operator defined by applying the Riemann-Liouville fractional integral to the convex combination of well-known Ruscheweyh and Salagean differential operators is used for defining a new fuzzy subclass. The authors suggest that the operator introduced here can be utilized to define other classes of analytic functions or to generalize other types of differential operators. The new operator defined in this paper can be used to introduce other specific subclasses of analytic functions, and quantum calculus can be also investigated in future studies. The fractional differential operator and the Mittag-Leffler functions are combined to formulate and arrange a new operator of fractional calculus. [Extracted from the article]

In the geometric theory of functions of a complex variable, the well-known direction is related to the estimates of the products of the inner radii of pairwise non-overlapping domains. This direction is called extreme problems in classes of pairwise non-overlapping domains. One of the problems of this type is considered in the present work. [ABSTRACT FROM AUTHOR]

The theory of geometric functions was first introduced by Bernard Riemann in 1851. In 1916, with the concept of normalized function revealed by Bieberbach, univalent function concept has found application area. Assume f(z)=z+Σn>2(anzn converges for all complex numbers z with 1,|z|, and f(z)is one-to-one on the set of such z. Convex and starlike functions f(z) and g(z) are discussed with the help of subordination. The f(z) and g(z) are analytic in unit disc and f(0)0,f'(0)=1, and g(0)=0,g'(0)-1=0. A single valued function f(z) is said to be univalent (or schlict or one-to-one) in domain DCC never gets the same value twice; that is, if f(x1)-fz2 for all z1 and z2 with z1= z2. Let A be the class of analytic functions in the unit disk U={z:|z|1} that are normalized with f(0)=0,f'(0)=1. In this paper we give the some necessary conditions for f(z)E S*[a,a²] and 0< a²

We have previously shown that a -fold holomorphic covering of a domain in the complex plane by another domain is extremal in the majorization principle for -valent functions and quadratic forms associated with Green's functions of these domains. In this paper, dual majorization principles involving both Green's and Neumann functions are obtained, in which -fold coverings are also extremal. The results are exemplified by applications of these principles to geometric function theory. [ABSTRACT FROM AUTHOR]

The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on differential subordination, one of the newest techniques used in the field, also known as the technique of admissible functions. For that, the appropriate classes of admissible functions are first defined. Based on these classes, we obtain some differential subordination and superordination results for multivalent meromorphic functions, analytic in the punctured unit disc, related to a linear operator ℑ ρ , τ p (ν) , for τ > 0 , ν , ρ ∈ C , such that R e (ρ − ν) ≧ 0 , R e (ν) > τ p , (p ∈ N) . Moreover, taking into account both subordination and superordination results, we derive a sandwich-type theorem. The connection with some other known results and an example are also provided. [ABSTRACT FROM AUTHOR]

The concepts of fuzzy differential subordination and superordination were introduced in the geometric function theory as generalizations of the classical notions of differential subordination and superordination. Fractional calculus is combined in the present paper with quantum calculus aspects for obtaining new fuzzy differential subordinations and superordinations. For the investigated fuzzy differential subordinations and superordinations, fuzzy best subordinates and fuzzy best dominants were obtained, respectively. Furthermore, interesting corollaries emerge when using particular functions, frequently involved in research studies due to their geometric properties, as fuzzy best subordinates and fuzzy best dominants. The study is finalized by stating the sandwich-type results connecting the previously proven results. [ABSTRACT FROM AUTHOR]

We define a new class of exponential starlike functions constructed by a linear operator involving normalized form of the generalized Struve function. Making use of a technique of differential subordination introduced by Miller and Mocanu, we investigate several new results related to the Briot–Bouquet differential subordinations for the linear operator involving the normalized form of the generalized Struve function. We also obtain univalent solutions to the Briot–Bouquet differential equations and observe that these solutions are the best dominant of the Briot–Bouquet differential subordinations for the exponential starlike function class. Moreover, we give an application of fractional integral operator for a complex-valued function associated with the generalized Struve function. The significance of this paper is due to the technique employed in proving the results and novelty of these results for the Struve functions. The approach used in this paper can lead to several new problems in geometric function theory associated with special functions. [ABSTRACT FROM AUTHOR]

In this article we explore several applications of q-calculus in geometric function theory. Using the method of differential subordination, we obtain interesting univalence properties for the q-Sălăgean differential operator. Sharp subordination results are obtained by using functions with remarkable geometric properties as subordinating functions and considering the conditions of expressions involving the q-Sălăgean differential operator and a convex combination using it. [ABSTRACT FROM AUTHOR]

In this paper, we discuss and introduce a new study on the connection between geometric function theory, especially sandwich theorems, and Viete's theorem in elementary algebra. We obtain some conclusions for differential subordination and superordination for a new formula of complete homogeneous symmetric functions class involving an ordered cyclic operator. In addition, certain sandwich theorems are found. [ABSTRACT FROM AUTHOR]