New Trends in Complex Analysis Research

1. Introduction

This Special Issue aims to present some of the newest results obtained from the study of complex-valued functions of one or several complex variables. It was also intended to be a tribute to the memory of the great researcher, Professor Gabriela Kohr from Babeș-Bolyai University, Cluj-Napoca, Romania. She will be missed, but her brilliant ideas will live on and inspire generations to come.

Contributions of scholars studying different aspects regarding complex-valued functions were expected concerning original results on holomorphic functions of both one and several variables. For complex-valued functions of one variable, classical differential subordination and superordination theories could be considered, as well as special cases of strong differential subordination and superordination and fuzzy differential subordination and superordination theories. Interesting outcomes were likely to appear regarding the addition of special functions to studies. Quantum calculus, which has been proven to offer tremendous applications in geometric function theory, was also expected to provide lines of research. Classical aspects of the starlikeness and convexity of special classes of analytic functions continue to generate successful studies, and such studies were thus also welcome. Any aspects related to the study of holomorphic functions of several variables were considered valuable to the success of this Special Issue.

This Special Issue was intended to contain research regarding complex-valued functions of one and several variables and hence push forward the development of both branches, also inspiring parallels between them.

2. Overview of the Published Papers

The present Special Issue contains 14 papers accepted for publication after a conscientious review process.

The study submitted by Georgia Irina Oros and Liminița-Ioana Cotîrlă (Contribution 1) concerns the classical, but still prevalent, problem of introducing new classes of m-fold symmetric bi-univalent functions and studying the properties related to coefficient estimates. Quantum calculus aspects are also considered in this study in order to enhance its novelty and to obtain more interesting results. Three new classes of bi-univalent functions are defined, generalizing certain previously studied classes. For these new classes, coefficient estimates are given regarding the Taylor–Maclaurin coefficients $|a_{m+1}|$ and $|a_{2m+1}|$, and the Fekete–Szegő problem is investigated. The bounds of coefficient estimates obtained here are not sharp, and thus further investigation is required in order to improve these estimates.

The research by Matthew Olanrewaju Oluwayemi, Kaliappan Vijaya and Adriana Cătaş (Contribution 2) introduces a new subclass of analytic functions involving a generalized differential operator and investigates certain properties including the radius of starlikeness, closure properties and integral mean results for the class of analytic functions with negative coefficients. Further, the relationship between the newly established results and some known results in the literature is also highlighted. The study can be extended for results on Hölder inequalities, partial sums and subordination involving the functions from the newly defined class.

Abbas Kareem Wanas and Liminița-Ioana Cotîrlă (Contribution 3) initiate and explore a certain family of holomorphic and bi-univalent functions in the open unit disk, making use of a certain operator also using the $(M,N)$-Lucas polynomials. Upper bounds for the initial Taylor–Maclaurin coefficients and the Fekete–Szegő-type inequality for functions in this family are provided.

Daniel Breaz, Kadhavoor R. Karthikeyan and Elangho Umadevi (Contribution 4) reported a systematic study to discover the properties of a subclass of meromorphic starlike functions defined using the Mittag–Leffler three-parameter function. The Prabhakar function (or a three-parameter Mittag–Leffler function) is used for the study, since it has several applications in science and engineering problems. The restrictions to obtain sufficient conditions for meromorphic starlikeness involving quasi-subordination are given. The solution to the Fekete–Szegő problem and inclusion relationships for functions belonging to the defined function classes are also provided. Several new and classical results can be obtained as a special case of the main results presented in the paper.

The study presented by Alina Alb Lupaş and Georgia Irina Oros (Contribution 5) involves the fractional integral of the confluent hypergeometric function and presents its new applications for introducing a certain subclass of analytic functions. Conditions for functions to belong to this class are determined and the class is investigated considering aspects regarding coefficient bounds as well as partial sums of these functions. Distortion properties of the functions belonging to the class are proven and radii estimates are established for starlikeness and convexity properties of the class. For future investigations, the class could be transformed considering aspects of fuzzy differential subordination and superordination and quantum calculus.

A new operator is defined by Georgia Irina Oros, Gheorghe Oros and Shigeyoshi Owa (Contribution 6), considering functions that belong to the known class of p-valent analytic functions in the open unit disk U. Applying this operator, a new subclass of p-valent analytic functions is introduced and some interesting subordination- and coefficient-related properties of the functions in this class are discussed. It is also shown that for certain values given to the parameters involved in the definition of the class, p-valent starlike and p-valent convex functions of certain orders can be obtained, respectively. Examples are also given as applications of the newly proven results. A future approach to the operator introduced in this paper is suggested by the investigations of q-type operators related to multivalent functions cited in the paper. Also, further investigations on p-valent functions involving special functions are suggested by other published works mentioned in the paper.

The topic regarding new families of holomorphic and bi-univalent functions defined in the unit disk is addressed by Abbas Kareem Wanas and Liminița-Ioana Cotîrlă (Contribution 7). A new such family is defined using the q-Srivastava–Attiya operator and Gegenbauer polynomials. For the new family of functions, the bounds for $|a_2|$ and $|a_3|$ are found, where $a_2$ and $a_3$ are the initial Taylor–Maclaurin coefficients and the Fekete–Szegő inequality, and special cases and consequences are investigated.

Mohsan Raza, Sarfraz Nawaz Malik, Qin Xin, Muhey U. Din and Liminița-Ioana Cotîrlă (Contribution 8) studied the necessary conditions for the univalence of integral operators that involve two functions: the generalized Bessel function and a function from the well-known class of normalized analytic functions in the open unit disk. For particular parameters, the univalence of the integral operators that are defined by Bessel, modified Bessel and spherical Bessel functions is obtained. The main tools for the discussions are the Kudriasov conditions for the univalency of functions, as well as functional inequalities for the generalized Bessel functions. Sufficient conditions for the univalency of the integral operators that involve trigonometric, as well as hyperbolic, functions are provided as an application of the new results.

In their research, the fractional integral of the Gaussian hypergeometric function is defined by Georgia Irina Oros and Simona Dzitac (Contribution 9). Using this operator and certain properties of the subordination chains, new fuzzy differential subordinations are investigated for which the best dominants are provided. Considering particular functions as such dominants, interesting geometric properties interpreted as inclusion relations of certain subsets of the complex plane are presented. An example is constructed in order to show the applicability of the results obtained in the study. As future directions of study, the fractional integral of the Gaussian hypergeometric function can be used in similar studies involving the dual theory of fuzzy differential superordination. Sandwich-type results are likely to be obtained when combining the findings of the present study and the ones obtained using dual theory. Then, the fractional integral of the Gaussian hypergeometric function could be applied for introducing fuzzy classes of analytic functions. A fractional derivative could also be used combined with the Gaussian hypergeometric function.

The Bessel–Struve kernel function defined in the unit disc is used by Najla M. Alarifi and Saiful R. Mondal in their study (Contribution 10). The Bessel–Struve kernel functions are generalized in this article, and differential equations are derived. The conditions under which the generalized Bessel–Struve function is lemniscate convex are obtained by using a subordination technique. The relation between the Janowski class and exponential class is also derived. This study explores a range of possible geometric features, including lemniscate and exponential Carathéodory properties and lemniscate convexity. Further theoretical concepts or different approaches require studying the exponential or Janowski convexity.

Lei Shi, Hari M. Srivastava, Ayesha Rafiq, Muhammad Arif and Muhammad Ihsan (Contribution 11) aimed to discuss certain coefficient-related problems for the inverse functions associated with a bounded turning function class subordinated with the exponential function. The bounds of some initial coefficients, the Fekete–Szegő-type inequality, and the estimation of Hankel determinants of second and third order are investigated. All of these bounds were proven to be sharp. The results established in this paper are useful for understanding the geometric properties of this function class. By improving present methods, one may be able to obtain more outcomes on the known various subclasses of univalent functions.

The research by Muhammad Arif, Safa Marwa, Qin Xin, Fairouz Tchier, Muhammad Ayaz and Sarfraz Nawaz Malik (Contribution 12) deals with analytic functions with bounded turnings defined in the unit disk. A new class is defined with the condition that these functions are subordinated by the sigmoid function. Sharp coefficient inequalities, including the third Hankel determinant, are provided for this new class. Similar results are also included for the logarithmic coefficients related to functions of this class. The results presented here could help in finding the fourth-order Hankel determinants for the same types of analytic functions that have been considered in this study.

In the first part of the research authored by Hidetaka Hamada, Gabriela Kohr and Mirela Kohr (Contribution 13), generalizations of the Fekete–Szegő inequalities for quasiconvex mappings F of type B are presented in addition to the first elements F of g-Loewner chains on the unit ball of a complex Banach space, recently obtained by H. Hamada, G. Kohr and M. Kohr. The Fekete–Szegő inequalities are established by using the norm under the restrictions on the second- and third-order terms of the homogeneous polynomial expansions of the mapping F. In the second part of this paper, an estimation of the difference in the moduli of successive coefficients is given for the first elements of g-Loewner chains on a unit disc. An estimation of the difference in the moduli of successive coefficients is also given for the first elements F of g-Loewner chains on the unit ball of a complex Banach space under restrictions on the second- and third-order terms of the homogeneous polynomial expansions of the mapping F.

The research presented by Dorina Răducanu (Contribution 14) examines a new class of analytic functions normalized by a certain condition. Upper bounds for a number of coefficient estimates are given, among which are the initial coefficients, the second Hankel determinant, and the Zalcman functional. Upper estimates for higher-order Schwarzian derivatives are also obtained for the new class. The results obtained in this note could be the subject of further investigations related to the Fekete–Szegő-type functional.

3. Conclusions

The 14 papers published as part of this Special Issue cover a variety of topics related to areas of research in complex functions of one or several variables. Researchers interested in the field should hopefully find some interesting and inspiring results so that the papers are cited and the Special Issue can reach a large audience. There is a follow-up to this Special Issue entitled “New Trends in Complex Analysis Research II”. Hopefully, by reading the papers in these two projects, scholars will find interesting ideas for new directions of study in geometric function theory.