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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 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]

In this note, we present the state-of-the-art theory of bi-Sobolev mappings. We recall that f is a Sobolev homeomorphism if f belongs to W loc 1 , 1 ∩ Hom (Ω ; Ω ′) , and f is a bi-Sobolev map if and only if f and f - 1 are Sobolev homeomorphisms. This concept, introduced in Hencl et al. (J. Math. Anal. Appl. 355, 22–32 2009), plays a central role in Geometric Function Theory. For instance, we just mention here that maps of bi-Sobolev type are strictly related to the notion of mappings of finite distortion; see, among others, the papers (Hencl and Koskela 2014) Pratelli (Nonlinear Anal. 154, 258–268 2017). [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros be 0. In this paper, we investigate the relative growth of a polynomial p(z) with respect to two circles z=r and z = R and obtain inequalities about the dependence of p(rz) on p(Rz), where z = 1, for 0 while taking into account the placement of the zeros of the underlying polynomial. Our results improve as well as generalize certain well-known polynomial inequalities. Some numerical examples are also given in order to illustrate and compare graphically the obtained inequalities with some recent results. [ABSTRACT FROM AUTHOR]

In this article, we define a generalized q-integral operator on multivalent functions. It generalizes many known linear operators in Geometric Function Theory (GFT). Inclusions results, convolution properties and q-Bernardi integral preservation of the subclasses of analytic functions are discussed. [ABSTRACT FROM AUTHOR]

The results of this work have a connection with the geometric function theory and they were obtained using methods based on subordination along with information on q -calculus operators. We defined the q -analogue of multiplier- Ruscheweyh operator of a certain family of linear operators I q , μ s (λ , ℓ) f (ς) (s ∈ N 0 = N ∪ { 0 } , N = { 1 , 2 , 3 ,.. } ; ℓ , λ , μ ≥ 0 , 0 < q < 1) . Our major goal was to build some analytic function subclasses using I q , μ s (λ , ℓ) f (ς) and to look into various inclusion relationships that have integral preservation features. The results of this work have a connection with the geometric function theory and they were obtained using methods based on subordination along with information on -calculus operators. We defined the -analogue of multiplier- Ruscheweyh operator of a certain family of linear operators . Our major goal was to build some analytic function subclasses using and to look into various inclusion relationships that have integral preservation features. [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]

This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order λ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular. [ABSTRACT FROM AUTHOR]

In Geometric function theory, the Ma-Minda class of starlike functions has a unique place as it unifies various subclasses of starlike functions. There has been an vivid interplay between special functions and their geometric properties, like starlikeness. In this article, we establish certain special function's radius of Ma-Minda starlikness. As an application, we obtain conditions on parameters for these special functions to be in the Ma-Minda class. Further, we focus on certain convolution properties for the Ma-Minda class that are not done so far, and study their applications in radius problem. Finally, we prove a variational problem of Goluzin, namely, the region of variability for the Ma-Minda class. Our results simplify and generalize the already-known ones. [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 recent years, the problem of finding the sharp radii bounds for certain properties in geometric function theory has attracted several researchers. However, there are several instances where only lower bounds for the radius problems have been established. In this paper, we have worked in a similar direction to compute the upper bounds in these cases which coincides with the conjectured values. Moreover, explicit functions are provided which yield that these bounds are attainable. [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]

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]

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]

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]

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]

In light of the Alexander transformation, the class of spirallike functions is significant. The characteristics of special functions also appear very frequently in Geometric function theory. In this paper, we find the radii of γ -spirallike and convex γ -spirallike of order α of certain special functions. [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]

In the complex field, special functions are closely related to geometric holomorphic functions. Koebe function is a notable contribution to the study of the geometric function theory (GFT), which is a univalent function. This sequel introduces a new class that includes a more general Koebe function which is holomorphic in a complex domain. The purpose of this work is to present a new operator correlated with GFT. A new generalized Koebe operator is proposed in terms of the convolution principle. This Koebe operator refers to the generality of a prominent differential operator, namely the Ruscheweyh operator. Theoretical investigations in this effort lead to a number of implementations in the subordination function theory. The tight upper and lower bounds are discussed in the sense of subordinate structure. Consequently, the subordinate sandwich is acquired. Moreover, certain relevant specific cases are examined. [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]

In this paper, trajectory of time changing along a helical line is represented by parametric equations in Cartesian coordinates of Euclidean space. On the basis of a cycloidal sweep of a cylindrical surface onto a plane, analytical form of a helix is determined. On its basis, integral surface is determined, which is called the periodic characteristic of the diagonal differentiation operator and its connection with its linear characteristic is established. a) elements of new approach related to the periodic characteristic of diagonal differentiation operator are proposed, b) method for reducing integral along the periodic characteristic to an integral with linear characteristic, c) conditions establishing structure of the integral as sum of linear and multiperiodic functions. Some consequences of these results and recommendations of an algorithmic nature for further expansion of research in this direction are given. [ABSTRACT FROM AUTHOR]

Over the last few years, Geometric Function Theory (GFT) as one of the most prime branch of complex analysis has gained a considerable and an impressive attention from many researchers, largely because it deals with the study of the geometric properties of analytic functions and their numerous applications in various fields of mathematics such as in special functions, probability distributions, and fractional calculus. The investigations in this paper are on two new classes of analytic functions defined in the unit disk ε = {z ∈ C : |z| < 1} and denoted by XSq (b,K) and XTq (b,K). Function f in the classes satisfy the conditions f(0) = f ′(0) -- 1 = 0, hence can be of series type f(z) = z + a2z² + a3z³ + …, z ∈ ε. The definition of the two new classes of analytic functions embed some well-known special functions such as the Galuê-type Struve function, modified error function and a starlike function whose coefficients are Bell's numbers while some involving mathematical principles are the q-derivative, inequalities, convolution and subordination. The main results from these classes are however, the upper estimates of some initial bounds such as |an| (n = 2, 3, 4) and the Fekete-Szegö functional |a3 - øa2/2| (ø ∈ C) of functions f ∈ XSq (b,K) and f ∈ XTq (b,K). [ABSTRACT FROM AUTHOR]

The research into theory for analytic univalent as well as multivalent functions is an ancient subject for mathematics, especially in complex analysis, which has attracted a great number for scholars due to utter elegance of the its geometrical characteristics as well as numerous research opportunities. The study of univalent functions is one of most important areas of complex analysis for only one and many variables. Researchers have been interested in the traditional study of this subject since at least 1907. During this time until now many researchers in the field of complex analysis, including as Euler, Gauss, Riemann, Cauchy, and many others, have developed. Geometric function theory is a combination or interplay of geometry and analysis. The main goal of this article is to investigate the principle for dependence as well as add an additional subset for polyvalent functions with a different operator that is related to derivatives of higher order. As a result, the findings were important in terms of various geometric properties, including coefficient estimation, distortion as well as growth borders, radii for starlikeness, convexity, as well as close-to-convexity. [ABSTRACT FROM AUTHOR]

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]

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]

The paper contains new results in the field of Geometric Function Theory of one variable functions, specially connected with the concepts of differential subordinations and superordinations, and that could be used for further investigation in this area. We defined a new subclasses of analytic multivalent functions in the open unit disk D with the aid of the generalized well-known Srivastava-Attiya operator obtained by a convolution product with the general Hurwitz-Lerch Zeta function. For the functions belonging to these subclasses we obtain sharp subordination and superordination results, that generalizes some previous well-known subordination properties obtained by different authors. The main results are followed by some particular cases obtained for special choices of the parameters, some of them being connected with the Janowski type functions. The technique used in the proofs is based on the general theory of differential subordinations and superordination initiated and developed by S.S. Miller and P. T. Mocanu. We emphasize that these results are sharp in the sense that there are the best possible under the given assumptions of our theorems and corollaries, that is the dominants cannot be improved. These new results generalizes some previous well-known subordination properties obtained by different authors. [ABSTRACT FROM AUTHOR]

The numerical tools that have outshinning many others in the history of Geometric Function Theory (GFT) are the Chebyshev and Gegenbauer polynomials in the present time. Recently, Gegenbauer polynomials have been used to define several subclasses of an analytic functions and their yielded results are in the public domain. In this work, analytic univalent functions defined by Gegenbauer polynomials is considered using close-to-convex approach of starlike function. Some early few coefficient bounds obtained are used to establish the famous Fekete-Szego inequalities. [ABSTRACT FROM AUTHOR]

We investigate on some subclasses of analytic fuctions defined by subordination. Also, we give estimates of... In this work, we find the sharp norm estimate for the functions f in the extended classes of starlike functions. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions for neutrosophic Poisson probability distribution series to be in the classes Mnγ (θ) were derived by means of coefficient inequalities. The results obtained further strengthening the relationships between geometric function theory and statistics and by extension, fuzzy logic. [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]

This note presents a calculus proof of the famous Pythagorean theorem, which has been proved in more than 350 different ways by using geometric arguments since the first proof given by Pythagoras (565-495 BC). The proof presented here also involves function concepts in such a way that it provides a rather straightforward argument. [ABSTRACT FROM AUTHOR]

The sigmoid function increases the size of the hypothesis space that the network can represent. Neural networks can be used for complex learning tasks. It is therefore necessary to investigate the role of sigmoid function in geometric function theory. In this study, a new subclass of bi-starlike functions involving Sigmoid function and Bernol li Lemniscate was defined. Some coefficient bounds belonging to this newly defined subclass were also obtained by using subordination principle. The key tools in the proof of our main results are the coefficient Fekete-Szegö inequalities for this subclass. The results obtained agree and extend some earlier results. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel subclass of bi-univalent functions, which are of considerable interest in various fields of mathematics, including complex analysis and geometric function theory. By employing the property of subordination, we define these bi-univalent functions as R(t, γ, λ) and impose constraints on the coefficients | an |. 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]

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 article we continue our investigation of the Paatero space. We prove that the intersection of every Paatero class V(k) with every Hardy space Hp is closed in that Hp and associate singular continuous measures to elements of V(k). There have been no examples in the literature of functions in V(k) with zeros in the unit disk other than the one at the origin. We close this gap in the literature. We derive a representation of the measure associated to a function in V(k) for functions whose derivatives are rational, or algebraic, or transcendental functions in the unit disk.Finally, we consider the notion of regulated domains, introduced by Pommerenke and show that there are regulated domains whose boundary is not of bounded boundary rotation.

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]