452 results on '"Geometric function theory"'
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2. Second Hankel Determinant and Fekete–Szegö Problem for a New Class of Bi-Univalent Functions Involving Euler Polynomials.
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Gebur, Semh Kadhim and Atshan, Waggas Galib
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EULER polynomials , *GEOMETRIC function theory , *HANKEL functions , *ORTHOGONAL polynomials - Abstract
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]
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- 2024
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3. Fractional Calculus and Hypergeometric Functions in Complex Analysis.
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Oros, Gheorghe and Oros, Georgia Irina
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FRACTIONAL calculus , *HYPERGEOMETRIC functions , *ANALYTIC functions , *GEOMETRIC function theory , *HANKEL functions , *MEROMORPHIC functions , *SPECIAL functions - Abstract
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]
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- 2024
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4. New Developments in Geometric Function Theory II.
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Oros, Georgia Irina
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GEOMETRIC function theory , *UNIVALENT functions , *ANALYTIC functions , *MEROMORPHIC functions , *SYMMETRIC functions , *HYPERGEOMETRIC functions , *INVERSE functions - Abstract
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]
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- 2024
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5. Differential Subordination and Superordination Using an Integral Operator for Certain Subclasses of p -Valent Functions.
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Almutairi, Norah Saud, Shahen, Awatef, and Darwish, Hanan
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GEOMETRIC function theory , *GENERALIZED integrals , *ANALYTIC functions , *INTEGRAL operators - Abstract
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]
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- 2024
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6. Issue Information.
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QUANTUM groups , *COMPUTATIONAL number theory , *GEOMETRIC function theory , *ANALYTIC number theory , *HARMONIC analysis (Mathematics) , *ALGEBRAIC number theory , *DISCRETE mathematics - Published
- 2024
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7. New differential operator for analytic univalent functions associated with binomial series.
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Rossdy, Munirah, Omar, Rashidah, and Soh, Shaharuddin Cik
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ANALYTIC functions , *UNIVALENT functions , *GEOMETRIC function theory , *DIFFERENTIAL operators , *STAR-like functions , *MATHEMATICIANS - Abstract
The investigation of operators is significant in mathematics, particularly in geometric function theory. Operators have generated a considerable amount of attention in the theory of geometric functions because of their significance in generalising and preserving subclasses of analytic univalent functions. Many mathematicians have discussed operators and the new generalisation in various articles. Research on operators has been conducted since 1915. Since then, mathematicians are still very interested in discussing operators with many properties of analytic univalent functions. Hence, this paper introduces a new generalised operator D λ , α s , m , k f (z) of analytic functions in the open unit disc using the generalised Srivastava-Attiya operator, generalised Al-Oboudi operator, and binomial series. Based on this new operator, some subclasses are defined using differential subordinations method. Furthermore, the inclusion properties are obtained for functions in these subclasses. [ABSTRACT FROM AUTHOR]
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- 2024
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8. SOME REMARKS ON SOBOLEV AND BI-SOBOLEV MAPPINGS.
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D'Onofrio, Luigi and Molica Bisci, Giovanni
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GEOMETRIC function theory - Abstract
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]
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- 2024
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9. On Third Hankel Determinant for Certain Subclass of Bi-Univalent Functions.
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Shakir, Qasim Ali and Atshan, Waggas Galib
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UNIVALENT functions , *GEOMETRIC function theory , *HANKEL functions , *GEOMETRIC analysis , *ANALYTIC functions , *MATHEMATICS - Abstract
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]
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- 2024
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10. Geometric Properties of Normalized Galué Type Struve Function.
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Sarkar, Samanway, Das, Sourav, and Mondal, Saiful R.
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GEOMETRIC function theory , *HARDY spaces , *SYMMETRIC functions , *STAR-like functions , *CONVEX functions , *UNIVALENT functions - Abstract
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]
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- 2024
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11. Certain radii problems for 퓢∗(ψ) and special functions.
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Gangania, Kamaljeet and Kumar, S. Sivaprasad
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GEOMETRIC function theory , *STAR-like functions , *RADIUS (Geometry) , *SPECIAL functions - Abstract
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]
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- 2024
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12. New Applications of Fractional q -Calculus Operator for a New Subclass of q -Starlike Functions Related with the Cardioid Domain.
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Khan, Mohammad Faisal and AbaOud, Mohammed
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FRACTIONAL calculus , *GEOMETRIC function theory , *STAR-like functions , *INTEGRAL operators , *PLASMA physics , *INVERSE functions - Abstract
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]
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- 2024
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13. Applications of Fuzzy Differential Subordination to the Subclass of Analytic Functions Involving Riemann–Liouville Fractional Integral Operator.
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Breaz, Daniel, Khan, Shahid, Tawfiq, Ferdous M. O., and Tchier, Fairouz
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FRACTIONAL integrals , *GEOMETRIC function theory , *INTEGRAL operators , *DIFFERENTIAL operators , *ANALYTIC functions , *UNIVALENT functions , *MERGERS & acquisitions , *FUZZY sets - Abstract
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]
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- 2023
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14. Certain Quantum Operator Related to Generalized Mittag–Leffler Function.
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Yassen, Mansour F. and Attiya, Adel A.
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QUANTUM operators , *GEOMETRIC function theory , *ANALYTIC functions , *OPERATOR functions , *DIFFERENTIAL operators - Abstract
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]
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- 2023
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15. On the Study of Starlike Functions Associated with the Generalized Sine Hyperbolic Function.
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Gul, Baseer, Arif, Muhammad, Alhefthi, Reem K., Breaz, Daniel, Cotîrlă, Luminiţa-Ioana, and Rapeanu, Eleonora
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HYPERBOLIC functions , *STAR-like functions , *GEOMETRIC function theory , *SINE function , *ANALYTIC functions , *UNIVALENT functions , *CONVEX functions - Abstract
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]
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- 2023
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16. Analytic Functions Related to a Balloon-Shaped Domain.
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Ahmad, Adeel, Gong, Jianhua, Al-Shbeil, Isra, Rasheed, Akhter, Ali, Asad, and Hussain, Saqib
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UNIVALENT functions , *GEOMETRIC function theory , *HANKEL functions , *ANALYTIC functions - Abstract
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]
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- 2023
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17. Coefficient Bounds for Some Families of Bi-Univalent Functions with Missing Coefficients †.
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Analouei Adegani, Ebrahim, Jafari, Mostafa, Bulboacă, Teodor, and Zaprawa, Paweł
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GEOMETRIC function theory , *UNIVALENT functions , *COEFFICIENTS (Statistics) , *ANALYTIC functions , *TWENTIETH century - Abstract
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]
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- 2023
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18. Fuzzy differential subordination related to strongly Janowski functions.
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Kanwal, Bushra, Hussain, Saqib, and Saliu, Afis
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ANALYTIC functions , *LINEAR operators , *INTEGRAL operators , *DIFFERENTIAL operators , *UNIVALENT functions , *GEOMETRIC function theory - Abstract
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]
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- 2023
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19. Fuzzy Differential Subordination Associated with a General Linear Transformation.
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Malik, Sarfraz Nawaz, Khan, Nazar, Tawfiq, Ferdous M. O., Khan, Mohammad Faisal, Ahmad, Qazi Zahoor, and Xin, Qin
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ANALYTIC functions , *GEOMETRIC function theory , *DIFFERENTIAL operators , *LINEAR operators - Abstract
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]
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- 2023
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20. Study on the Criteria for Starlikeness in Integral Operators Involving Bessel Functions.
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Oros, Georgia Irina, Oros, Gheorghe, and Bardac-Vlada, Daniela Andrada
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INTEGRAL operators , *STAR-like functions , *GEOMETRIC function theory , *BESSEL functions , *UNIVALENT functions , *HOLOMORPHIC functions , *FUNCTION spaces - Abstract
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]
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- 2023
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21. Certain Properties of Harmonic Functions Defined by a Second-Order Differential Inequality.
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Breaz, Daniel, Durmuş, Abdullah, Yalçın, Sibel, Cotirla, Luminita-Ioana, and Bayram, Hasan
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HARMONIC functions , *UNIVALENT functions , *DIFFERENTIAL inequalities , *GEOMETRIC function theory , *COMPUTER software development - Abstract
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]
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- 2023
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22. Sandwich Subordinations Imposed by New Generalized Koebe-Type Operator on Holomorphic Function Class.
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Moureh, Anwar H. and Al-Janaby, Hiba F.
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HOLOMORPHIC functions , *OPERATOR functions , *GEOMETRIC function theory , *SPECIAL functions , *DIFFERENTIAL operators , *UNIVALENT functions - Abstract
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]
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- 2023
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23. Theory of Functions and Applications.
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Kal'chuk, Inna
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COMPOSITION operators , *MATHEMATICAL complex analysis , *QUASILINEARIZATION , *GEOMETRIC function theory - Abstract
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]
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- 2024
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24. Certain Results on Fuzzy p -Valent Functions Involving the Linear Operator.
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Ali, Ekram Elsayed, Vivas-Cortez, Miguel, Ali Shah, Shujaat, and Albalahi, Abeer M.
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GEOMETRIC function theory , *ANALYTIC functions , *LINEAR operators , *FUZZY sets - Abstract
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]
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- 2023
- Full Text
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25. New Criteria for Starlikness and Convexity of a Certain Family of Integral Operators.
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Srivastava, Hari M., Alavi, Rogayeh, Shams, Saeid, Aghalary, Rasoul, and Joshi, Santosh B.
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INTEGRAL operators , *GEOMETRIC function theory , *ANALYTIC functions , *UNIVALENT functions , *STAR-like functions - Abstract
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]
- Published
- 2023
- Full Text
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26. Applications of Fuzzy Differential Subordination for a New Subclass of Analytic Functions.
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Khan, Shahid, Ro, Jong-Suk, Tchier, Fairouz, and Khan, Nazar
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GEOMETRIC function theory , *ANALYTIC functions , *DIFFERENTIAL operators , *FUZZY sets , *INTEGRAL operators , *GEOMETRIC analysis , *GENERALIZED integrals - Abstract
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]
- Published
- 2023
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27. Matrix Approaches for Gould–Hopper–Laguerre–Sheffer Matrix Polynomial Identities.
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Nahid, Tabinda, Alam, Parvez, and Choi, Junesang
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POLYNOMIALS , *GEOMETRIC function theory , *MATRICES (Mathematics) - Abstract
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]
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- 2023
- Full Text
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28. The application of quasi-subordination for non-bazilević functions.
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Al-Khafaji, Saba N., Al-Fayadh, Ali, and Abbood, Mohaimen Muhammed
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EXPONENTIAL functions , *GEOMETRIC function theory - Abstract
The inequality of finding the upper bounds for the nonlinear functional |a3 - µa22| of the Taylor-Mclaurin series is popularly famous as the Fekete-Szegö inequality. This inequality has a rich history in the geometric function theory. Its source by Fekete-Szegö in 1933. In this paper, through the advantage of applying concepts quasi-subordinate, Sakaguchi functions and exponential functions, the authors construct a new subclass Ne,q(s, b, λ) of Non-Bazilević functions. As well as, obtained the first and two Taylor-Maclaurin coefficient estimates, sharp upper bounds of the Fekete-Szegö inequality and majorization for functions which belong to this subclass. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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29. The Properties of Meromorphic Multivalent q -Starlike Functions in the Janowski Domain.
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Al-Shbeil, Isra, Gong, Jianhua, Ray, Samrat, Khan, Shahid, Khan, Nazar, and Alaqad, Hala
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GEOMETRIC function theory , *STAR-like functions , *DIFFERENTIAL operators , *OPERATOR theory , *ANALYTIC functions , *INTEGRAL operators , *MEROMORPHIC functions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
30. A calculus-assisted proof of the Pythagorean theorem.
- Author
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Villaseñor, José A.
- Subjects
- *
PYTHAGOREAN theorem , *CALCULUS , *PROOF theory , *GEOMETRIC function theory , *TRIGONOMETRY - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
31. Editorial: Overview and Some New Directions.
- Author
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Zeng, Shengda, Migórski, Stanislaw, and Liu, Yongjian
- Subjects
- *
HOPF bifurcations , *GEOMETRIC function theory , *SYSTEMS theory , *QUANTUM field theory , *NONLINEAR dynamical systems , *BOUNDARY value problems - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
32. On Generalizations of the Close-to-Convex Functions Associated with q -Srivastava–Attiya Operator.
- Author
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Breaz, Daniel, Alahmari, Abdullah A., Cotîrlă, Luminiţa-Ioana, and Ali Shah, Shujaat
- Subjects
- *
GEOMETRIC function theory , *INTEGRAL operators , *GENERALIZATION , *ANALYTIC functions , *DIFFERENTIAL inclusions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
33. Estimate of Third and Fourth Hankel Determinants for Certain Subclasses of Analytic Functions.
- Author
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Singh, Gurmeet, Singh, Gagandeep, and Singh, Gurcharanjit
- Subjects
- *
HANKEL functions , *ANALYTIC functions , *UNIVALENT functions , *GEOMETRIC function theory - Abstract
In geometric function theory, the estimation of upper bound of the Hankel determinants for various subclasses of analytic functions is an interesting topic of current research. Till now, extensive work has been done on the estimation of second and third Hankel determinants. The present investigation deals with the estimate of fourth Hankel determinant for certain subclasses of analytic functions in the open unit disc E = {z: |z| < 1}. This work will set the stage in the direction of investigation of fourth Hankel determinant for several other classes. [ABSTRACT FROM AUTHOR]
- Published
- 2023
34. Pre-functions and Extended pre-functions of Complex Variables.
- Author
-
Thirumalai, A., Muthunagai, K., and Agarwal, Ritu
- Subjects
- *
POLYNOMIALS , *COMPLEX variables , *GEOMETRIC function theory , *HYPERBOLIC functions , *TRIGONOMETRIC functions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
35. Some New Applications of the q -Analogous of Differential and Integral Operators for New Subclasses of q -Starlike and q -Convex Functions.
- Author
-
Al-Shaikh, Suha B., Abubaker, Ahmad A., Matarneh, Khaled, and Khan, Mohammad Faisal
- Subjects
- *
INTEGRAL operators , *DIFFERENTIAL operators , *STAR-like functions , *GEOMETRIC function theory , *NUMERICAL solutions to differential equations , *INTEGRAL inequalities , *ANALYTIC functions , *HYPERGEOMETRIC series - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
36. Investigation of the Second-Order Hankel Determinant for Sakaguchi-Type Functions Involving the Symmetric Cardioid-Shaped Domain.
- Author
-
Ullah, Khalil, Arif, Muhammad, Aldawish, Ibtisam Mohammed, and El-Deeb, Sheza M.
- Subjects
- *
SYMMETRIC domains , *SYMMETRIC functions , *UNIVALENT functions , *HANKEL functions , *STAR-like functions , *CONVEX functions , *GEOMETRIC function theory - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
37. Jackson Differential Operator Associated with Generalized Mittag–Leffler Function.
- Author
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Attiya, Adel A., Yassen, Mansour F., and Albaid, Abdelhamid
- Subjects
- *
DIFFERENTIAL operators , *UNIVALENT functions , *ANALYTIC functions , *GEOMETRIC function theory , *HYPERGEOMETRIC series , *CALCULUS - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
38. New Applications of Fuzzy Set Concept in the Geometric Theory of Analytic Functions.
- Author
-
Alb Lupaş, Alina
- Subjects
- *
GEOMETRIC function theory , *FUZZY sets , *UNIVALENT functions , *SET theory , *CONVEX functions , *ANALYTIC functions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
39. Fuzzy Differential Inequalities for Convolution Product of Ruscheweyh Derivative and Multiplier Transformation.
- Author
-
Alb Lupaş, Alina
- Subjects
- *
DIFFERENTIAL inequalities , *GEOMETRIC function theory , *DIFFERENTIAL operators - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
40. Initial Coefficients Upper Bounds for Certain Subclasses of Bi-Prestarlike Functions.
- Author
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Hamadneh, Tareq, Abu Falahah, Ibraheem, AL-Khassawneh, Yazan Alaya, Al-Husban, Abdallah, Wanas, Abbas Kareem, and Bulboacă, Teodor
- Subjects
- *
UNIVALENT functions , *STAR-like functions , *GEOMETRIC function theory , *HOLOMORPHIC functions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
41. Fekete–Szegö Problem and Second Hankel Determinant for a Class of Bi-Univalent Functions Involving Euler Polynomials.
- Author
-
Riaz, Sadia, Shaba, Timilehin Gideon, Xin, Qin, Tchier, Fairouz, Khan, Bilal, and Malik, Sarfraz Nawaz
- Subjects
- *
EULER polynomials , *HANKEL functions , *GEOMETRIC function theory , *ORTHOGONAL polynomials - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
42. Coefficient Results concerning a New Class of Functions Associated with Gegenbauer Polynomials and Convolution in Terms of Subordination.
- Author
-
Olatunji, Sunday Olufemi, Oluwayemi, Matthew Olanrewaju, and Oros, Georgia Irina
- Subjects
- *
GEGENBAUER polynomials , *GEOMETRIC function theory , *ANALYTIC functions , *UNIVALENT functions , *ERROR functions , *STAR-like functions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
43. Results on Second-Order Hankel Determinants for Convex Functions with Symmetric Points.
- Author
-
Ullah, Khalil, Al-Shbeil, Isra, Faisal, Muhammad Imran, Arif, Muhammad, and Alsaud, Huda
- Subjects
- *
SYMMETRIC functions , *CONVEX functions , *HYPERBOLIC functions , *UNIVALENT functions , *LOGARITHMIC functions , *GEOMETRIC function theory - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
44. Coefficients Inequalities for the Bi-Univalent Functions Related to q -Babalola Convolution Operator.
- Author
-
Al-shbeil, Isra, Gong, Jianhua, and Shaba, Timilehin Gideon
- Subjects
- *
GEOMETRIC function theory , *ANALYTIC functions , *MATHEMATICAL convolutions , *QUANTUM operators - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
45. On Fuzzy Spiral-like Functions Associated with the Family of Linear Operators.
- Author
-
Azzam, Abdel Fatah, Shah, Shujaat Ali, Cătaș, Adriana, and Cotîrlă, Luminiţa-Ioana
- Subjects
- *
GEOMETRIC function theory , *INTEGRAL operators , *STAR-like functions , *ANALYTIC functions , *CONVEX functions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
46. Some Properties of Certain Classes of Meromorphic Multivalent Functions Defined by Subordination.
- Author
-
Seoudy, Tamer M. and Shammaky, Amnah E.
- Subjects
- *
GEOMETRIC function theory , *MEROMORPHIC functions , *INTEGRAL operators , *ANALYTIC functions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
47. On a New Subclass of q -Starlike Functions Defined in q -Symmetric Calculus.
- Author
-
Razzaque, Asima, Noor, Saima, and Hussain, Saqib
- Subjects
- *
STAR-like functions , *CALCULUS , *ANALYTIC functions , *UNIVALENT functions , *GEOMETRIC function theory , *SYMMETRIC functions , *IMAGE processing , *CONVEX functions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
48. New Results on Integral Operator for a Subclass of Analytic Functions Using Differential Subordinations and Superordinations.
- Author
-
Salman, Fatima Obaid and Atshan, Waggas Galib
- Subjects
- *
ANALYTIC functions , *GENERALIZED integrals , *STAR-like functions , *GEOMETRIC function theory , *INTEGRAL operators - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
49. Subordination Properties of Certain Operators Concerning Fractional Integral and Libera Integral Operator.
- Author
-
Oros, Georgia Irina, Oros, Gheorghe, and Owa, Shigeyoshi
- Subjects
- *
GEOMETRIC function theory , *INTEGRAL operators , *FRACTIONAL calculus , *FRACTIONAL integrals , *ANALYTIC functions - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
50. New Developments in Geometric Function Theory.
- Author
-
Oros, Georgia Irina
- Subjects
- *
GEOMETRIC function theory , *UNIVALENT functions , *ANALYTIC functions , *MEROMORPHIC functions , *CONVEX functions , *FRACTIONAL calculus - Abstract
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]
- Published
- 2023
- Full Text
- View/download PDF
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