Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions

1. Introduction

The aim of this Special Issue was to attract valuable developments of researchers interested in introducing new operators that involve complex valued functions of one or several variables, studying their properties, and then using the newly defined operators in various ways.

Fractional calculus operators have also developed significant applications in the field of analytic functions theory. The classical definition of these operators and their generalizations have provided notable applications in the development and investigation of new function classes with notable geometric features. In addition to fractional calculus tools and certain hypergeometric functions, components of quantum calculus have also been incorporated in investigations regarding different types of operators. The theories of differential subordination and superordination, along with their newer forms of strong and fuzzy differential subordination and superordination, have also constantly generated interesting outcomes involving different types of operators.

Academic contributions have been welcomed on applications of various operators, such as differential, integral, fractional, or quantum calculus operators, for introducing new classes of functions, for studies involving all types of differential subordination and superordination techniques, or for any other investigations in different areas concerning complex analysis. Hopefully, the published material will inspire new developments regarding complex analysis operators, special classes of analytic functions, and any other aspects associated with geometric function theory.

2. Overview of the Published Papers

Of the many submissions received, 20 high-quality papers were accepted for publication in this issue after a thorough and conscientious review.

The study submitted by Ali Hamzah Alibrahim and Saptarshi Das (Contribution 1) concerns the extension of the p-trigonometric functions $sinp$ and $cosp$ family in complex domains and p-hyperbolic functions $sinhp$ and the $coshp$ family in hyperbolic complex domains. This paper explores the essential duality between the p-trigonometric and p-hyperbolic functions. Although standard hyperbolic and trigonometric functions have long been used to solve various differential equations, this work extends those boundaries to the realm of p-complex numbers. The authors demonstrate that by first uncovering the deep connection between these p-variants with complex arguments, they can finally develop the orthogonality of their basis functions. It is this specific pursuit—moving from a theoretical duality to a practical tool for solving complex ODEs—that forms the heart and primary motivation of their study. The authors propose future studies on the inverse p-trigonometric functions and the inverse p-hyperbolic functions and also on ordinary differential equations (ODEs) involving complex numbers and their solutions in the generalized p-trigonometric and hyperbolic function basis.

The research by Abdullah Alsoboh and Georgia Irina Oros (Contribution 2) introduces a new subclass of bi-univalent functions associated with the leaf-like domain in open unit disk. The subordination principle for analytic functions is applied in conjunction with q-calculus. Coefficient bounds are examined, and Fekete–Szegö inequalities are investigated for the functions in the new class after the authors prove that it is not empty. Illustrative examples are generated using Geogebra, and they showcase the practical value of our results.

Sheza M. El-Deeb and Luminita-Ioana Cotîrlă (Contribution 3) initiate research on a new subclass of convex functions defined by using tangent functions and by applying the combination of Babalola operators and Binomial series. Sharp coefficient bounds are determined and sharp Fekete–Szegö inequality is proved. Radius of convexity and growth and distortion approximation are also investigated for the new class.

Tariq Al-Hawary, Basem Frasin and Ibtisam Aldawish (Contribution 4) provide certain sufficient criteria for the generalized hypergeometric distribution series to be in two families of analytic functions defined in the open unit disk. An integral operator is investigated for the generalized distribution series and inclusion connections between the integral operator and the new families are proved.

The study presented by Georgia Irina Oros, Simona Dzitac and Daniela Andrada Bardac-Vlada (Contribution 5) involves the special case of the differential superordination theory, namely fuzzy differential superordiantion. Following the introduction of the concept of third-order fuzzy differential subordination, the paper introduces the dual concept of third-order fuzzy differential superordination by generalizing the concepts already established for the second-order fuzzy differential superordination. Necessary and sufficient conditions for determining subordinants of a third-order fuzzy differential superordination are established, and the method for finding the best subordinant for such fuzzy differential superordiantion is presented when it can be obtained. The study is concluded by the presentation of a numerical example in the hope that it would suggest further uses of the new results.

The study presented by Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi and Marwa Ennaceur (Contribution 6) further develops the idea of the fuzzy set concept embedded into differential subordiantion theory. A new integral operator is introduced involving the class of meromorphic functions, the notion of convolution, and the Hurwitz–Lerch Zeta function. New fuzzy differential subordinations are investigated for which the best fuzzy dominants are provided. The new results include starlikeness and convexity properties for the new operator, established by applying the means of the fuzzy differential subordination theory.

Sondekola Rudra Swamy, Basem Aref Frasin, Daniel Breaz and Luminița-Ioana Cotîrlă (Contribution 7) address the topic of introducing new families of holomorphic and bi-univalent functions by using generalized bivariate Fibonacci polynomials and a the $(p,q)$-derivative operator. Estimates for the coefficients $|d_2|$ and $|d_3|$ are found, where $d_2$, $d_3$ are the Taylor–Maclaurin coefficients, and the Fekete-Szegő inequality is investigated. Relevant connections to the existing results and consequences are also highlighted.

Prathviraj Sharma, Srikandan Sivasubramanian, Gangadharan Murugusundaramoorthy and Nak Eun Cho (Contribution 8) introduce a new subclass of concave bi-univalent functions associated with bounded boundary rotation defined in the open unit disk. The initial Taylor–Maclaurin coefficients are estimated for functions belonging to the new class and certain corollaries show the connection of the new results to previously established ones. Futhermore, Fekete–Szegö inequality is also investigated for the newly introduced class.

In their research, Sarem H. Hadi, Yahea Hashem Saleem, Alina Alb Lupaş, Khalid M. K. Alshammari and Abdullah Alatawi (Contribution 9) apply estimates bounds for Carathéodory functions in the complex domain in order to obtain sharp limits for the inverse of analytic functions. The authors find estimations for the second and third-order Toeplitz determinants with coefficients depending on inverse analytic functions that contain the families of pre-starlike, starlike, convex, and symmetric-starlike functions. The authors suggest further developments for a variety of holomorphic function families to determine upper estimates for the first inverse coefficients following the methods presented in this research.

Narjes Alabkary and Saiful R. Mondal determine in their study (Contribution 10) the radius of $\alpha$-spiral-likeness of order ${\displaystyle \frac{cos\left(\alpha \right)}{2}}$ for entire functions represented as infinite products of their positive zeros. The focus is on three classes of functions represented by a convergent infinite product of factors involving the positive zeros of the function. Special functions such as Gamma functions, Bessel functions, Struve functions, Wright functions, Ramanujan-type entire functions, and q-Bessel functions are also integrated into the investigation. The research is enhanced by numerical computations of the radius for some functions, done using Mathematica 12 software.

Approaches involving subordiantion techniques and q-calculus aspects are taken by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi and Wael W. Mohammed (Contribution 11) to investigate generalized classes of quasi-convex and close-to-convex functions using a new q-analogue operator. This study examines the invariance of the newly defined classes under the q-Bernardi integral operator and establishes several interesting inclusion relationships between them.

Research by Lateef Ahmad Wani and Saiful R. Mondal (Contribution 12) investigates extremal problems for a classical class of univalent functions defined in the unit disk, extending and refining certain classical results in the theory of univalent functions, with new insights into the geometric and analytic properties of the class. Estimates for $|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}|$ are established and the length of the boundary curve of the image domain $f\left(\mathbb{D}\right)$ is computed. The radius of univalence for the partial sums $f_n$ is also determined.

In their endeavor, Mohammad El-Ityan, Qasim Ali Shakir, Tariq Al-Hawary, Rafid Buti, Daniel Breaz and Luminita-Ioana Cotîrlă (Contribution 13), define a new class of bi-univalent functions by applying a generalized $(p,q)$-derivative operator. Insights into the geometrical and analytical properties of this class of functions are provided by establishing estimates for important coefficients, the Fekete–Szegö functional, as well as for the second and third Hankel determinants.

The research presented by Ekram E. Ali, Rabha M. El-Ashwah, Teodor Bulboacă and Abeer M. Albalahi (Contribution 14) examines a new class of meromorphic functions in the punctured unit disk $\dot{U}$ defined using the generalized q-Sălăgean operator. The recently established q-Schwarz–Pick lemma and the generalization of Nehari’s lemma for the Jackson’s q-difference operator are the main tools of investigation for a majorization problem associated with this class.

Through their investigation, Saddaf Noreen, Saiful R. Mondal, Muhey U. Din, Saima Mushtaq, Zhang Wei and Adil Murtaza (Contribution 15) derive new sufficient conditions for q-close-to-convexity with respect to certain starlike functions involving three different normalizations of q-Bessel–Struve functions.

The study presented by Pengfei Bai, Adeel Ahmad, Akhter Rasheed, Saqib Hussain, Huo Tang and Saima Noor (Contribution 16) introduces a new q-starlike function subclass based on the application of the q-analogues of Janowski-type functions and q-hyperbolic secant function. The geometric interpretation of hyperbolic secant function’s domain is also given by adjusting the q-parameter. The research provides results like coefficient estimates, growth and distortion bounds, and the radius of starlikeness for the new class by utilizing the convolution operator techniques.

The work of Ji Eun Kim (Contribution 17) addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. The results demonstrate concrete applications related to boundary value problems and potential theoretic formulations that involve extended hypercomplex fields.

In their research, Ekram E. Ali, Rabha M. El-Ashwah, Wafaa Y. Kota and Abeer M. Albalahi (Contribution 18) define a new linear operator which maps analytic functions to new functions with transformed coefficients derived from the Mittag–Leffler function. Sufficient conditions are established to ensure that functions related to the generalized Mittag–Leffler function are members of certain analytic function subclasses defined in this work. Examples are constructed to show the applicability of the theoretical results.

A new generalized class of functions is defined by Adel Salim Tayyah, Sibel Yalçın and Hasan Bayram (Contribution 19), and sharp bounds are found for various coefficient-related problems within this class, including Hankel determinants. The most significant result obtained is the use of the solution of a linear differential equation to construct the general formula of functions belonging to this class, which offers a systematic and clear method for generating precise examples and analyzing their properties. The Mathematica TM program (version 13.2) is used both as a computational tool and to enhance the applied aspect of this research.

Ningegowda Ravikumar, Basem Aref Frasin, Hari Mohan Srivastava, Haladasanahalli Shivanna Roopa and Ibtisam Aldawish (Contribution 20) use a linear multiplier q-differintegral operator with a generalized binomial series to define two new classes of analytic functions. The Hankel and Toeplitz determinant boundary values are established and a Fekete–Szegö-type inequality is investigated for the newly defined classes. The results enhance the understanding of coefficient estimates and analytic behavior of functions belonging to these subclasses.

3. Conclusions

This Special Issue features 20 papers exploring diverse methodologies in the study of complex functions of one or several variables. These findings are meant to inspire researchers and serve as a valuable resource within the field. Furthermore, this project continues with a second edition titled “Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions, 2nd Edition” Together, these two collections aim to provide scholars with innovative ideas and new directions for study in geometric function theory.