New Developments in Geometric Function Theory II

1. Introduction

This Special Issue is a sequel to the successful first volume entitled “New Developments in Geometric Function Theory”. Following the same idea as the previous Special Issue, this project aimed to gather the latest developments in research concerning complex-valued functions in the Geometric Function Theory field.

Scholars’ contributions were accepted on topics which included but were not limited to:

• New classes of univalent and bi-univalent functions;
• Studies regarding coefficient estimates including the Fekete–Szegő functional, Hankel determinants and Toeplitz matrices;
• Applications of different types of operators in Geometric Function Theory, including differential, integral, fractional or quantum calculus operators;
• Differential subordination and superordination theories in their classical form, also concerning their recent extensions, and strong and fuzzy differential subordination and superordination theories;
• Applications of different hypergeometric functions and orthogonal polynomials in Geometric Function Theory.

New results obtained by using any other techniques which can be applied in the field of complex analysis and its applications were also welcome. Hopefully, new lines of research associated with Geometric Function Theory have been highlighted and will provide a boost in the development of this field.

2. Overview of the Published Papers

Following a comprehensive review process, 14 articles were accepted for publication in this Special Issue.

Research by Sunday Olufemi Olatunji, Matthew Olanrewaju Oluwayemi and Georgia Irina Oros (Contribution 1) associates the powerful numerical tool provided by Gegenbauer polynomials with the prolific concepts of convolution and subordination. The investigation presented in this paper concerns a new subclass of functions introduced using an operator defined as the convolution of the generalized distribution and the error function using the concept of subordination. The research presented here targets a current topic of interest in Geometric Function Theory, namely coefficient-related studies. 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. The initial results regarding the coefficient estimates obtained by the authors can be used for further specific investigations regarding the coefficients of the functions from this class, such as estimations of Hankel determinants of different orders, Toeplitz determinants or the Fekete–Szegö problem.

Ibtisam Aldawish, Basem Frasin and Ala Amourah (Contribution 2) introduce a new family of normalized bi-univalent functions in the open unit disk associated with the Horadam polynomials using the concept of subordination and they estimate the second and the third coefficients in the Taylor–Maclaurin expansions of functions belonging to this class. Furthermore, the Fekete–Szegö inequality is evaluated for the functions in the newly defined family. Making use of the Bell distribution series could inspire researchers to derive the estimates of the Taylor–Maclaurin coefficients and Fekete–Szegö functional problems for functions belonging to new subclasses of bi-univalent functions defined by means of the Horadam polynomials associated with this distribution series.

Rasoul Aghalary, Ali Ebadian, Nak Eun Cho and Mehri Alizadeh (Contribution 3) present a new method of studying harmonic functions in Geometric Function Theory. In this paper, a specified class of new log-harmonic functions is constructed taking the convex-exponent product combination of two elements. Sufficient conditions for this class to be starlike log-harmonic are given as a result of this study. Earlier work in the literature is proven to be generalized by the outcome of this research, and examples connected to the new results are presented in order to encourage future investigations.

In their research (Contribution 4), Maryam Al-Towailb and Zeinab S. I. Mansour use quantum calculus aspects in order to introduce a q-analog of the class of completely convex functions. This class of functions is a generalization of the class of completely convex functions. Specific properties, including the convergence of q-Lidstone series expansions of q-completely convex functions, are proven in the study, and it also provides a sufficient and necessary condition for a real function to have an absolutely convergent q-Lidstone series expansion.

The main aim of the study conducted by Abbas Kareem Wanas, Fethiye Müge Sakar and Alina Alb Lupaş (Contribution 5) was to investigate two new classes of bi-univalent functions described through a generalized q-calculus operator using the generator function for the Laguerre polynomial. Initial Taylor–Maclaurin coefficient estimates for functions of these newly introduced bi-univalent function classes are obtained, and the well-known Fekete–Szegö inequalities are examined for each of these classes.

The study performed by Abdullah Alsoboh, Ala Amourah, Fethiye Müge Sakar, Osama Ogilat, Gharib Mousa Gharib and Nasser Zomot (Contribution 6) provides deeper insights into the theory and applications of bi-univalent functions. A new family of analytic bi-univalent functions that are injective and possess analytic inverses is introduced by employing a q-analogue of the derivative operator and the concept of subordination. Moreover, the upper bounds of the Taylor–Maclaurin coefficients of these functions are established, which can aid in approximating the accuracy of approximations using a finite number of terms. The upper bounds are obtained by approximating analytic functions using Faber polynomial expansions. The results obtained in this article can be generalized in the future using post-quantum calculus and other q-analogs of the fractional derivative operator.

The primary objective of the study published by Sercan Kazımoğlu, Erhan Deniz and Luminița-Ioana Cotîrlă (Contribution 7) is to investigate the criteria for univalence and convexity of the integral operators that employ Miller–Ross functions. Differential inequalities related to the Miller–Ross functions and well-known lemmas are employed in the proofs of the new results. By using Mathematica (version 8.0), some graphics are generated that support the main results. The original results presented here could stimulate and inspire researchers, just as all the operators introduced before in studies related to functions of a complex variable have done. Other geometric properties related to these operators could be investigated and they could also prove useful in introducing special classes of functions based on these properties.

By utilizing the concept of the q-fractional derivative operator and bi-close-to-convex functions, Hari Mohan Srivastava, Isra Al-Shbeil, Qin Xin, Fairouz Tchier, Shahid Khan and Sarfraz Nawaz Malik (Contribution 8) define and investigate a new subclass of normalized analytic functions in an open unit disk employing a novel fractional differential operator. By using the Faber polynomial expansion technique, the l-th coefficient bound for the functions contained within this class is provided, and a further explanation for the first few coefficients of bi-close-to-convex functions defined by the q-fractional derivative is also given. The Fekete–Szegö problem is also considered for this class and some examples are provided. It is also demonstrated how some previously published results could be improved and generalized as a result of the primary findings of this study, as well as their corollaries and consequences.

In their research (Contribution 9), Abeer A. Al-Dohiman, Basem Aref Frasin, Naci Taşar and Fethiye Müge Sakar discover some inclusion relations of a certain harmonic class with other classes of harmonic analytic functions defined in an open disk by applying a convolution operator associated with the Mittag–Leffler function. Several special cases of the main results are also obtained as corollaries of the main results. Following this study, one can find new inclusion relations for new harmonic classes of analytic functions using the convolution operator presented in this study.

Mohsan Raza, Mehak Tariq, Jong-Suk Ro, Fairouz Tchier and Sarfraz Nawaz Malik (Contribution 10) aim to introduce a class of starlike functions that are related to Bernoulli’s numbers of the second kind using the concept of subordination. Coefficient bounds, several radii problems, structural formulas, and inclusion relations are established, and sharp Hankel determinant problems of this class are presented. The newly defined class can be further investigated for determining the bounds of higher-order Hankel and Toeplitz determinants, and the same estimates can also be derived for logarithmic coefficients and for the coefficients of inverse functions.

Hasan Bayram, Kaliappan Vijaya, Gangadharan Murugusundaramoorthy and Sibel Yalçın (Contribution 11) introduce two novel subclasses of bi-univalent functions by leveraging generalized telephone numbers and binomial series through convolution. The analysis of the initial Taylor–Maclaurin coefficients is performed and Fekete–Szegö inequalities are established for these functions.

In the research presented by Sondekola Rudra Swamy and Luminita-Ioana Cotîrlă (Contribution 12), a new pseudo-type $\kappa$-fold symmetric bi-univalent function class that meets certain subordination conditions is introduced and studied with regard to coefficient bounds. For functions in the newly defined class, the upper bounds are obtained for certain coefficients that are further used for the evaluation of the Fekete–Szegö problem. In addition, pertinent links to previous results are highlighted and a few observations are given.

The goal of the study presented by Ebrahim Analouei Adegani, Mostafa Jafari, Teodor Bulboacă and Paweł Zaprawa (Contribution 13) is to estimate the upper bounds of the coefficients of the functions that belong to a set of bi-univalent functions with missing coefficients defined by using subordination. The results improve some previous results concerning different subclasses of bi-univalent functions that have been recently studied. In addition, important examples of some classes of such functions are provided, which can aid in the understanding of issues related to these functions. The authors expect that this method can be applied to the classes of harmonic and meromorphic functions in future work.

The notion of third-order strong differential subordination is investigated by Madan Mohan Soren, Abbas Kareem Wanas and Luminiţa-Ioana Cotîrlǎ (Contribution 14), who propose a new line of investigation for third-order strong differential subordination. Several intriguing properties are given within the context of specific classes of admissible functions. Certain definitions are extended to fit the third-order strong differential subordination theory, presenting new and interesting results. Several properties of the results of third-order strong differential subordinations for analytic functions associated with the Srivastava–Attiya operator are given. Studies of the dual theory of third-order strong differential superordination could be inspired by the results presented in this paper.

3. Conclusions

A book published under the same title, “New Developments in Geometric Function Theory II”, is available and contains the 14 papers that were published in this Special Issue. In the papers released as part of this initiative, a wide range of topics are discussed. Therefore, this Special Issue should be of interest to researchers studying Geometric Function Theory and its related fields.