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Fractal Fract
Special Issues
Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis
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Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: 31 October 2025 | Viewed by 6701

Special Issue Editors

Prof. Dr. Gheorghe Oros

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Faculty of Informatics and Sciences, University of Oradea, 410087 Oradea, Romania
Interests: special classes of univalent functions; differential subordinations and superordinations; differential operators; integral operators; differential-integral operators
Special Issues, Collections and Topics in MDPI journals
Dr. Georgia Irina Oros

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Faculty of Informatics and Sciences, University of Oradea, 410087 Oradea, Romania
Interests: special classes of univalent functions; differential subordinations and superordinations; differential operators; integral operators; differential–integral operators
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue is a follow-up to the first volume, entitled "Fractional Calculus and Hypergeometric Functions in Complex Analysis", which was well received. This new initiative, which builds upon the initial idea of the previous Special Issue by enlarging the focus of the targeted research, attempts to collect the most recent advancements in research regarding fractional calculus or/and quantum calculus combined with special functions in studies related to complex analysis.

Fractional calculus is a known and prolific tool in various scientific and engineering domains, as well as in theoretical studies regarding different branches of mathematics. In particular, comprehensive research has developed within the domain of geometric function theory, with the inclusion of fractional calculus. Furthermore, notable results have been obtained through enhancing investigative tools with quantum calculus aspects and through the impressive characteristics of special functions, among which hypergeometric functions are the most notable type. 

Scholars with an interest in any of these topics or in combining them with applications in other domains related to complex analysis are encouraged to submit their research in order to further the success of this Special Issue.

The topics to be covered include, but are not restricted to, the following:

  • New definitions and applications in fractional calculus and quantum calculus operators;
  • Applications of fractional calculus involving various special functions in complex analysis topics;
  • Applications of quantum calculus involving various special functions in complex analysis topics;
  • Orthogonal polynomials, including Jacobi polynomials and their special cases, Legendre polynomials, Chebyshev polynomials and Gegenbauer polynomials;
  • Applications of logarithmic, exponential and trigonometric functions regarding univalent functions’ theory;
  • Applications of gamma, beta and digamma functions;
  • Applications of fractional calculus and special functions in differential subordinations and superordiantions and their special forms of strong differential subordination and superordination and fuzzy differential subordiantion and superordination;
  • Different applications of quantum calculus combined with fractional calculus and/or special functions in geometric function theory.

Prof. Dr. Gheorghe Oros
Dr. Georgia Irina Oros
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • univalent functions
  • special functions
  • fractional operator
  • q–operator
  • differential subordination
  • differential superordination
  • quantum calculus

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (6 papers)

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Research

22 pages, 378 KiB  
Article
A Novel Family of Starlike Functions Involving Quantum Calculus and a Special Function
by Baseer Gul, Daniele Ritelli, Reem K. Alhefthi and Muhammad Arif
Fractal Fract. 2025, 9(3), 179; https://doi.org/10.3390/fractalfract9030179 - 14 Mar 2025
Viewed by 442
Abstract
The intent of quantum calculus, briefly q-calculus, is to find q-analogues of mathematical entities so that the original object is achieved when a certain limit is taken. In the case of q-analogue of the ordinary derivative, the limit is [...] Read more.
The intent of quantum calculus, briefly q-calculus, is to find q-analogues of mathematical entities so that the original object is achieved when a certain limit is taken. In the case of q-analogue of the ordinary derivative, the limit is q→1−. Also, the study of integral as well as differential operators has remained a significant field of inquiry from the early developments of function theory. In the present article, a subclass Sscμ,q of functions being analytic in D=z∈C:z<1 is introduced. The definition of Sscμ,q involves the concepts of subordination, that of q-derivative and q-Ruscheweyh operators. Since coefficient estimates and coefficient functionals provide insights into different geometric properties of analytic functions, for this newly defined subclass, we investigate coefficient estimates up to a4, in which both bounds for |a2| and |a3| are sharp, while that of |a4| is sharp in one case. We also discuss the sharp Fekete–Szegö functional for the said class. In addition, Toeplitz determinant bounds up to T32 (sharp in some cases) and sufficient condition are obtained. Several consequences derived from our above-mentioned findings are also part of the discussion. Full article
(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)
16 pages, 352 KiB  
Article
Sandwich-Type Results and Existence Results of Analytic Functions Associated with the Fractional q-Calculus Operator
by Sudhansu Palei, Madan Mohan Soren, Daniel Breaz and Luminiţa-Ioana Cotîrlǎ
Fractal Fract. 2025, 9(1), 4; https://doi.org/10.3390/fractalfract9010004 - 25 Dec 2024
Viewed by 519
Abstract
In the present investigation, we present certain subordination and superordination results for the q-integral operator of a fractional order associated with analytic functions in the open unit disk U. Using this q-integral operator, we obtain sandwich-type results. Furthermore, we employ [...] Read more.
In the present investigation, we present certain subordination and superordination results for the q-integral operator of a fractional order associated with analytic functions in the open unit disk U. Using this q-integral operator, we obtain sandwich-type results. Furthermore, we employ the existence of univalent solutions to a q-differential equation connected with a fractional q-integral operator of fractional order. We use these results to demonstrate the significance of our findings for particular functions. We also derive some examples and corollaries that are pertinent to our results. Full article
(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)
9 pages, 778 KiB  
Communication
Applications of Mittag–Leffler Functions on a Subclass of Meromorphic Functions Influenced by the Definition of a Non-Newtonian Derivative
by Daniel Breaz, Kadhavoor R. Karthikeyan and Gangadharan Murugusundaramoorthy
Fractal Fract. 2024, 8(9), 509; https://doi.org/10.3390/fractalfract8090509 - 29 Aug 2024
Cited by 2 | Viewed by 1012
Abstract
In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, [...] Read more.
In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, major deviation or adaptation was required in defining a class of meromorphic functions influenced by the multiplicative derivative. In addition, we redefined the subclass of meromorphic functions analogous to the class of the functions with respect to symmetric points. Initial coefficient estimates and Fekete–Szegö inequalities were obtained for the defined function classes. Some examples along with graphs have been used to establish the inclusion and closure properties. Full article
(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)
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21 pages, 346 KiB  
Article
Extensions of Bicomplex Hypergeometric Functions and Riemann–Liouville Fractional Calculus in Bicomplex Numbers
by Ahmed Bakhet, Mohamed Fathi, Mohammed Zakarya, Ghada AlNemer and Mohammed A. Saleem
Fractal Fract. 2024, 8(9), 508; https://doi.org/10.3390/fractalfract8090508 - 28 Aug 2024
Cited by 2 | Viewed by 1313
Abstract
In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville [...] Read more.
In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field. Full article
(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)
13 pages, 341 KiB  
Article
Applications of Caputo-Type Fractional Derivatives for Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation
by Kholood M. Alsager, Gangadharan Murugusundaramoorthy, Adriana Catas and Sheza M. El-Deeb
Fractal Fract. 2024, 8(9), 501; https://doi.org/10.3390/fractalfract8090501 - 26 Aug 2024
Cited by 1 | Viewed by 891
Abstract
In this article, for the first time by using Caputo-type fractional derivatives, we introduce three new subclasses of bi-univalent functions associated with bounded boundary rotation in an open unit disk to obtain non-sharp estimates of the first two Taylor–Maclaurin coefficients, [...] Read more.
In this article, for the first time by using Caputo-type fractional derivatives, we introduce three new subclasses of bi-univalent functions associated with bounded boundary rotation in an open unit disk to obtain non-sharp estimates of the first two Taylor–Maclaurin coefficients, |a2| and |a3|. Furthermore, the famous Fekete–Szegö inequality is obtained for the newly defined subclasses of bi-univalent functions. Several consequences of our results are pointed out which are new and not yet discussed in association with bounded boundary rotation. Some improved results when compared with those already available in the literature are also stated as corollaries. Full article
(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)
14 pages, 321 KiB  
Article
Ozaki-Type Bi-Close-to-Convex and Bi-Concave Functions Involving a Modified Caputo’s Fractional Operator Linked with a Three-Leaf Function
by Kaliappan Vijaya, Gangadharan Murugusundaramoorthy, Daniel Breaz, Georgia Irina Oros and Sheza M. El-Deeb
Fractal Fract. 2024, 8(4), 220; https://doi.org/10.3390/fractalfract8040220 - 10 Apr 2024
Cited by 6 | Viewed by 1625
Abstract
The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo’s fractional operator that is associated with [...] Read more.
The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo’s fractional operator that is associated with three-leaf functions in the open unit disc. The classes are defined using the notion of subordination based on the previously established fractional integral operators and classes of starlike functions associated with a three-leaf function. For functions in these classes, the Fekete-Szegö inequalities and the initial coefficients, |a2| and |a3|, are discussed. Several new implications of the findings are also highlighted as corollaries. Full article
(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)
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