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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 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]

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]

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]

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]

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]

In the past few years, many scholars gave much attention to the use of q-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric q-calculus in geometric function theory, very little work has been published so far. In this research, with the help of fundamental concepts of symmetric q-calculus and the symmetric q-Salagean differential operator for harmonic functions, we define a new class of harmonic functions connected with Janowski functions S H 0 ˜ m , q , A , B . First, we illustrate the necessary and sufficient convolution condition for S H 0 ˜ m , q , A , B and then prove that this sufficient condition is a sense preserving and univalent, and it is necessary for its subclass TS H 0 ˜ m , q , A , B . Furthermore, by using this necessary and sufficient coefficient condition, we establish some novel results, particularly convexity, compactness, radii of q-starlike and q-convex functions of order α , and extreme points for this newly defined class of harmonic functions. Our results are the generalizations of some previous known results. [ABSTRACT FROM AUTHOR]

The word "symmetry" is a Greek word that originated from "symmetria". It means an agreement in dimensions, due proportion, and arrangement; however, in complex analysis, it means objects remaining invariant under some transformation. This idea has now been recently used in geometric function theory to modify the earlier classical q-derivative introduced by Ismail et al. due to its better convergence properties. Consequently, we introduce a new class of analytic functions by using the notion of q-symmetric derivative. The investigation in this paper obtains a number of the latest important results in q-theory, including coefficient inequalities and convolution characterization of q-symmetric starlike functions related to Janowski mappings. [ABSTRACT FROM AUTHOR]

Aspects related to applications in the geometric function theory of q-calculus are presented in this paper. The study concerns the investigation of certain q-analogue differential operators in order to obtain their geometrical properties, which could be further developed in studies. Several interesting properties of the q-analogue of the Sălăgean differential operator are obtained here by using the differential subordination method. [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]

The Atangana–Baleanu fractional integral and multiplier transformations are two functions successfully used separately in many recently published studies. They were previously combined and the resulting function was applied for obtaining interesting new results concerning the theories of differential subordination and fuzzy differential subordination. In the present investigation, a new approach is taken by using the operator previously introduced by applying the Atangana–Baleanu fractional integral to a multiplier transformation for introducing a new subclass of analytic functions. Using the methods familiar to geometric function theory, certain geometrical properties of the newly introduced class are obtained such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and the radii of starlikeness, convexity, and close-to-convexity of functions belonging to the class. This class may have symmetric or assymetric properties. The results could prove interesting for future studies due to the new applications of the operator and because the univalence properties of the new subclass of functions could inspire further investigations having it as the main focus. [ABSTRACT FROM AUTHOR]

Applications of the Atangana–Baleanu fractional integral were considered in recent studies related to geometric function theory to obtain interesting differential subordinations. Additionally, the multiplier transformation was used in many studies, providing elegant results. In this paper, a new operator is defined by combining those two prolific functions. The newly defined operator is applied for introducing a new subclass of analytic functions, which is investigated concerning certain properties, such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and radii of starlikeness, convexity and close-to-convexity. This class may have symmetric or asymmetric properties. The results could prove interesting due to the new applications of the Atangana–Baleanu fractional integral and of the multiplier transformation. Additionally, the univalence properties of the new subclass of functions could inspire researchers to conduct further investigations related to this newly defined class. [ABSTRACT FROM AUTHOR]

This Special Issue contains 14 articles from the MDPI journal Symmetry on the general subject area of "Special Functions and Polynomials", written by scholars belonging to different countries of the world. A similar number of submitted articles was not accepted for publication. Several successful Special Issues on the same or closely related topics have already appeared in MDPI's Symmetry, Mathematics and Axioms journals, in particular those edited by illustrious colleagues such as Hari Mohan Srivastava, Charles F. Dunkl, Junesang Choi, Taekyun Kim, Gradimir Milovanović, and many others, who testify to the importance of this matter for its applications in every field of mathematical, physical, chemical, engineering and statistical sciences. The subjects treated in this Special Issue include, in particular, the following Keywords. [ABSTRACT FROM AUTHOR]

Theorems giving the best dominants for some fuzzy differential subordinations are proven, and interesting corollaries are provided with the use of particular functions as fuzzy best dominants. By means of the newly obtained operator, the subclass Sn( , , ) of analytic functions in the unit disc is introduced, and various properties and characteristics of this class are derived by applying techniques specific to the differential subordination concept. A sandwich-type theorem is stated combining the results given in two theorems proven in this paper, using the two dual theories of fuzzy differential subordination and fuzzy differential superordination. Owning to the importance and great interest of differential operators, two generalized differential operators, which may be symmetric or assymetric, are newly introduced in paper [[8]]. [Extracted from the article]

The motivation of the present article is to define the (p − q) -Wanas operator in geometric function theory by the symmetric nature of quantum calculus. We also initiate and explore certain new families of holormorphic and bi-univalent functions A E (λ , σ , δ , s , t , p , q ; ϑ) and S E (μ , γ , σ , δ , s , t , p , q ; ϑ) which are defined in the unit disk U associated with the (p − q) -Wanas operator. The upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö-type inequalities for the functions in these families are obtained. Furthermore, several consequences of our results are pointed out based on the various special choices of the involved parameters. [ABSTRACT FROM AUTHOR]

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class A p , where class A p is invariant (or symmetric) under rotations. The well-known class of Janowski functions are used with the help of the principle of subordination between analytic functions in order to define this subclass of analytic and p-valent functions. This function class generalizes various other subclasses of analytic functions, not only in classical Geometric Function Theory setting, but also some q-analogue of analytic multivalent function classes. We study and investigate some interesting properties such as sufficiency criteria, coefficient bounds, distortion problem, growth theorem, radii of starlikeness and convexity for this newly-defined class. Other properties such as those involving convex combination are also discussed for these functions. In the concluding part of the article, we have finally given the well-demonstrated fact that the results presented in this article can be obtained for the (p , q) -variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter p is obviously unnecessary. [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]

This article presents certain families of analytic functions regarding q-starlikeness and q-convexity of complex order γ (γ ∈ C \ 0) . This introduced a q-integral operator and certain subclasses of the newly introduced classes are defined by using this q-integral operator. Coefficient bounds for these subclasses are obtained. Furthermore, the (δ , q) -neighborhood of analytic functions are introduced and the inclusion relations between the (δ , q) -neighborhood and these subclasses of analytic functions are established. Moreover, the generalized hyper-Bessel function is defined, and application of main results are discussed. [ABSTRACT FROM AUTHOR]