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Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis

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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 March 2026 | Viewed by 13743

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Prof. Dr. Gheorghe Oros

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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

Prof. Dr. Georgia Irina Oros

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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

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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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Published Papers (14 papers)

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Research

21 pages, 1881 KB

Open AccessArticle

Applications of the Generalized Marcum Q-Function to Janowski Subclasses of Harmonic Functions

by Mohammad Faisal Khan and Mohammed AbaOud

Fractal Fract. 2026, 10(3), 209; https://doi.org/10.3390/fractalfract10030209 - 23 Mar 2026

Viewed by 52

Abstract

In this work, we provide a convolution type operator

Λ_{ν, b}

that is produced by the generalized Marcum Q-function and examine how it maps to various Janowski-type subclasses of harmonic univalent functions. Since the Marcum Q-function has an integral [...] Read more.

In this work, we provide a convolution type operator

Λ_{ν, b}

Λ_{ν, b}

can be understood as a fractional type integral operator operating on the coefficients of harmonic mappings. Applying

Λ_{ν, b}

to harmonic mappings

f = h + \bar{g}

in the unit disk

D

, we derive coefficient inequalities, and inclusion relations for various subclasses of harmonic and analytic univalent functions. In particular, we give sufficient conditions for

Λ_{ν, b} (f)

to belong to Janowski-starlike families such as

S_{H}^{*} (F, G)

K_{H}^{0}

, and

R_{H} (F, G)

. Closure properties of the Janowski class under the proposed operator are also established. Numerical tables and examples confirm the inclusion results, and graphical plots illustrate how the operator reshapes the image domains for different parameter pairs

(ν, b)

. Numerical illustrations are provided to visualize the geometric steering effect induced by the Marcum Q-function and its fractional-order damping behavior. 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, 365 KB

Open AccessArticle

Sandwich Results for Holomorphic Functions Related to an Integral Operator

by Amal Mohammed Darweesh, Adel Salim Tayyah, Sarem H. Hadi and Alina Alb Lupaş

Fractal Fract. 2026, 10(3), 171; https://doi.org/10.3390/fractalfract10030171 - 4 Mar 2026

Viewed by 202

Abstract

In this paper, we introduce a new logarithmic integral operator that unifies differentiation and fractional integration within the complex domain. The present work addresses this gap by applying the proposed operator to analytic functions represented by alternating power series. The method demonstrates that the coefficients can be reorganized in a controlled manner without affecting convergence or analytic behavior. Using this framework, we derive third-order differential subordination and superordination results, which naturally lead to corresponding sandwich-type results. The findings confirm that the introduced operator offers an effective analytical tool for studying distortion, growth, and mapping properties of analytic functions, with promising potential for future applications in fluid mechanics. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

28 pages, 400 KB

Open AccessArticle

New Certain Results of a Linear Multiplier Fractional q-Differintegral Operator for Fuzzy Differential Subordination and Superordination

by Ningegwoda Ravikumar, Basem Aref Frasin, Rmsen Abdulbari Ali Ahmed and Ibtisam Aldawish

Fractal Fract. 2026, 10(3), 170; https://doi.org/10.3390/fractalfract10030170 - 4 Mar 2026

Viewed by 180

Abstract

The concept of fuzzy differential subordination was introduced in 2011 as a natural generalization of classical differential subordination, reflecting the contemporary trend of incorporating fuzzy set theory into well-established mathematical frameworks. This work aims to explore multiple fuzzy differential subordinations (FDS) and fuzzy differential superordinations (FDSs) associated with the linear multiplier fractional q-differintegral operator. Utilizing the linear multiplier fractional q-differintegral operator, we introduce a novel fuzzy subclass of analytic functions, denoted by

S D_{F}^{σ, m} (q, λ, γ)

. Using the concept of FDS and FDSs, we identify important characteristics and analytical aspects of the class

S D_{F}^{σ, m} (q, λ, γ)

. Furthermore, we derive a collection of FDS and FDSs results specifically related to the linear multiplier fractional q-differintegral operator. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

21 pages, 730 KB

Open AccessArticle

Certain Geometric Properties of Normalized Euler Polynomial

by Suha B. Al-Shaikh, Mohammad Faisal Khan and Naeem Ahmad

Fractal Fract. 2026, 10(3), 136; https://doi.org/10.3390/fractalfract10030136 - 24 Feb 2026

Viewed by 280

Abstract

In this paper, we introduce and investigate a new class of analytic functions generated by Euler polynomials through a suitable normalization. Using classical tools from geometric function theory, including coefficient monotonicity, Fejér-type inequalities, MacGregor’s criteria, and Ozaki’s close-to-convexity condition, we establish sufficient conditions for the univalence, starlikeness, convexity, and close-to-convexity of the proposed Euler-polynomial-based normalized function. Sharp radius results for starlikeness, convexity, and close-to-convexity in the disk

D_{1 / 2}

are derived by exploiting refined coefficient bounds involving higher-order Euler polynomial terms. Several illustrative examples and graphical demonstrations are provided to verify the theoretical findings. The results obtained extend the known geometric properties of special function-based analytic classes and offer a new perspective on the geometric behavior of Euler polynomials in the unit disk. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

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15 pages, 375 KB

Open AccessArticle

Third Order Differential Subordination Results Associated with Tremblay Fractional Derivative Operator for p-Valent Analytic Functions

by Mohammad El-Ityan, Adriana Cătaş, Suha Hammad and Sheza El-Deeb

Fractal Fract. 2026, 10(2), 103; https://doi.org/10.3390/fractalfract10020103 - 2 Feb 2026

Cited by 1 | Viewed by 339

Abstract

This paper studies a class of p-valent analytic functions in the open unit disk using a modified Tremblay operator based on the Riemann–Liouville fractional integral and derivative. We derive a series representation of the operator, which serves as a useful tool to explore its analytic and geometric properties. New third-order differential subordination results are obtained by defining suitable classes of admissible functions. We provide sufficient conditions to ensure subordination relations with a given dominant function, leading to inclusion results in general complex domains. In addition, several applications are given for specific choices of the target domain, showing how the framework is flexible and able to produce unified results in the theory of differential subordination for multivalent analytic functions. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

16 pages, 315 KB

Open AccessArticle

New Mock Theta Function Identities via Fractional q-Calculus and Bilateral ₂ψ₂ Series

by Qiuxia Hu and Bilal Khan

Fractal Fract. 2026, 10(2), 86; https://doi.org/10.3390/fractalfract10020086 - 26 Jan 2026

Viewed by 536

Abstract

Mock theta functions, introduced by Ramanujan in his last letter to Hardy, play a significant role in q-series theory and have natural connections to fractional q-calculus. In this paper, we study bilateral hypergeometric series of the form

{}_{2}ψ_{2}

= [...] Read more.

{}_{2}ψ_{2}

\sum_{n = - \infty}^{\infty} \frac{{(a, b; q)}_{n}}{{(c, d; q)}_{n}} z^{n},

where

{(a; q)}_{n}

denotes the q-shifted factorial. Using Slater’s three-term transformation formula for bilateral

{}_{2}ψ_{2}

series, we derive new identities for Ramanujan’s mock theta functions of orders 2, 3, 6, and 8. These transformations reveal previously unknown relationships between different q-series representations and extend the classical theory of mock theta functions within the framework of q-special functions. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

13 pages, 319 KB

Open AccessArticle

Inclusive Subfamilies of Complex Order Generated by Liouville–Caputo-Type Fractional Derivatives and Horadam Polynomials

by Feras Yousef, Tariq Al-Hawary, Basem Frasin and Amerah Alameer

Fractal Fract. 2025, 9(11), 698; https://doi.org/10.3390/fractalfract9110698 - 30 Oct 2025

Cited by 2 | Viewed by 595

Abstract

In this paper, we introduce the inclusive subfamilies of complex order

E (δ_{1}, δ_{2}, δ_{3}, δ_{4}, a, b)

and [...] Read more.

In this paper, we introduce the inclusive subfamilies of complex order

E (δ_{1}, δ_{2}, δ_{3}, δ_{4}, a, b)

and

C (δ_{1}, δ_{2}, δ_{3}, δ_{4}, a, b)

, defined by means of the Liouville–Caputo-type derivative operator and subordination to the Horadam polynomials. For these subfamilies, we derive estimates for the initial coefficients

| q_{2} |

and

| q_{3} |

, as well as results concerning the Fekete–Szegö functional

| q_{3} - ϱ q_{2}^{2} |

. In addition, several related results are established as corollaries, accompanied by a concluding remark. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

23 pages, 551 KB

Open AccessArticle

Sharp Bounds on Hankel Determinant of q-Starlike and q-Convex Functions Subordinate to Secant Hyperbolic Functions

by Lifen Zhang, Zhigang Wang and Lei Shi

Fractal Fract. 2025, 9(6), 346; https://doi.org/10.3390/fractalfract9060346 - 26 May 2025

Cited by 2 | Viewed by 905

Abstract

In the present paper, using the q-difference operator, we introduce two classes of q-starlike functions and q-convex functions subordinate to secant hyperbolic functions. As functions in these classes have unique characteristic of missing coefficients on the second term in their analytic expansions, we define a new functional to unify the Hankel determinants with entries of the original coefficients, inverse coefficients, logarithmic coefficients, and inverse logarithmic coefficients for these functions. We obtain the sharp bounds on the new functional for functions in the two classes, and as a consequence, the best results on Hankel determinant for the starlike and convex functions subordinate to secant hyperbolic functions are given. The outcomes include some existing findings as corollaries and may help to deepen the understanding the properties of q-analogue analytic functions. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

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22 pages, 378 KB

Open AccessArticle

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

Cited by 2 | Viewed by 1029

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.

q \to 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

S_{s c} (μ, q)

of functions being analytic in

D = \{z \in C : |z| < 1\}

is introduced. The definition of

S_{s c} (μ, 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

a_{4}

, in which both bounds for

| a_{2} |

and

| a_{3} |

are sharp, while that of

| a_{4} |

is sharp in one case. We also discuss the sharp Fekete–Szegö functional for the said class. In addition, Toeplitz determinant bounds up to

T_{3} (2)

(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 KB

Open AccessArticle

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

Cited by 2 | Viewed by 954

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 KB

Open AccessCommunication

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 7 | Viewed by 1521

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, 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 KB

Open AccessArticle

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 3 | Viewed by 2034

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 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 KB

Open AccessArticle

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 3 | Viewed by 1349

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.

| a_{2} |

and

| a_{3} |

. 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 KB

Open AccessArticle

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 8 | Viewed by 2017

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 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,

| a_{2} |

and

| a_{3} |

, 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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