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Fractional Calculus and Hypergeometric 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: closed (15 March 2023) | Viewed by 14332

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Special Issue Editors

Prof. Dr. Gheorghe 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

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,

Fractional calculus has had a powerful impact on recent research, having many applications in different branches of science and engineering. Various branches of mathematics are also influenced by fractional calculus. Applications in complex analysis research are comprehensive, and interesting new results have been obtained in studies involving univalent functions theory.

This Special Issue aims to gather new research outcomes combining this prolific tool with another that generates exciting results when integrated into studies: hypergeometric functions.

The study of hypergeometric functions dates back 200 years. They appear in the work of Euler, Gauss, Riemann, and Kummer. Interest in hypergeometric functions has grown in the last few decades due to hypergeometric functions’ applications in a large variety of scientific domains and many areas of mathematics. Hypergeometric functions were linked to the theory of univalent functions by L. de Branges’ proof of Bieberbach’s conjecture, published in 1985, which uses the generalized hypergeometric function. After this connection was established, hypergeometric functions was studied intensely using geometric function theory.

Quantum calculus is also involved in studies alongside fractional calculus tools and different hypergeometric functions.

Researchers interested in any of these topics or a combination of them and their applications in different areas concerning complex analysis are welcome to submit their findings and contribute to the success of this Special Issue.

Topics include but are not limited to:

New definitions and applications in fractional calculus operators;
Applications of fractional calculus involving hypergeometric functions in geometric function theory;
Orthogonal polynomials, including Jacobi and their special functions, including 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 hypergeometric functions in differential subordinations and superordinations and their special forms of strong differential subordination and superordination and fuzzy differential subordination and superordination;
Applications of quantum calculus involving fractional calculus in geometric function theory;
Applications of quantum calculus involving hypergeometric functions in complex analysis.

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 operators
differential subordination
differential superordination
quantum calculus

Published Papers (13 papers)

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Editorial

Jump to: Research

5 pages, 172 KiB

Open AccessEditorial

Fractional Calculus and Hypergeometric Functions in Complex Analysis

by Gheorghe Oros and Georgia Irina Oros

Fractal Fract. 2024, 8(4), 233; https://doi.org/10.3390/fractalfract8040233 - 16 Apr 2024

Viewed by 312

Abstract

Fractional calculus has had a powerful impact on recent research, with many applications in different branches of science and engineering [...] Full article

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

Research

Jump to: Editorial

15 pages, 328 KiB

Open AccessArticle

Analytic Functions Related to a Balloon-Shaped Domain

by Adeel Ahmad, Jianhua Gong, Isra Al-Shbeil, Akhter Rasheed, Asad Ali and Saqib Hussain

Fractal Fract. 2023, 7(12), 865; https://doi.org/10.3390/fractalfract7120865 - 05 Dec 2023

Viewed by 1016

Abstract

One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of [...] Read more.

X

defined in the new domain, including the sharp estimates for the coefficients

a_{2}, a_{3}

, and

a_{4}

, and for three second-order and third-order Hankel determinants,

H_{2, 1} X, H_{2, 2} X

, and

H_{3, 1} X

. The optimality of each obtained estimate is given as well. Full article

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

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26 pages, 465 KiB

Open AccessArticle

Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus

by Mohd Idris Qureshi, Tafaz Ul Rahman Shah, Junesang Choi and Aarif Hussain Bhat

Fractal Fract. 2023, 7(10), 700; https://doi.org/10.3390/fractalfract7100700 - 23 Sep 2023

Cited by 3 | Viewed by 665

Abstract

In this article, we introduce three general double-series identities using Whipple transformations for terminating generalized hypergeometric

_{4} F_{3}

and

_{5} F_{4}

functions. Then, by employing the left-sided Riemann–Liouville fractional integral on these identities, we show the ability to derive additional identities [...] Read more.

In this article, we introduce three general double-series identities using Whipple transformations for terminating generalized hypergeometric

_{4} F_{3}

and

_{5} F_{4}

functions. Then, by employing the left-sided Riemann–Liouville fractional integral on these identities, we show the ability to derive additional identities of the same nature successively. These identities are used to derive transformation formulas between the Srivastava–Daoust double hypergeometric function (S–D function) and Kampé de Fériet’s double hypergeometric function (KDF function) with equal arguments. We also demonstrate reduction formulas from the S–D function or KDF function to the generalized hypergeometric function

_{p} F_{q}

. Additionally, we provide general summation formulas for the

_{p} F_{q}

and S–D function (or KDF function) with specific arguments. We further highlight the connections between the results presented here and existing identities. Full article

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

14 pages, 321 KiB

Open AccessArticle

The Properties of Meromorphic Multivalent q-Starlike Functions in the Janowski Domain

by Isra Al-Shbeil, Jianhua Gong, Samrat Ray, Shahid Khan, Nazar Khan and Hala Alaqad

Fractal Fract. 2023, 7(6), 438; https://doi.org/10.3390/fractalfract7060438 - 29 May 2023

Cited by 2 | Viewed by 810

Abstract

Many researchers have defined the q-analogous of differential and integral operators for analytic functions using the concept of quantum calculus in the geometric function theory. In this study, we conduct a comprehensive investigation to identify the uses of the Sălăgean q-differential operator for meromorphic multivalent functions. Many features of functions that belong to geometrically defined classes have been extensively studied using differential operators based on q-calculus operator theory. In this research, we extended the idea of the q-analogous of the Sălăgean differential operator for meromorphic multivalent functions using the fundamental ideas of q-calculus. With the help of this operator, we extend the family of Janowski functions by adding two new subclasses of meromorphic q-starlike and meromorphic multivalent q-starlike functions. We discover significant findings for these new classes, including the radius of starlikeness, partial sums, distortion theorems, and coefficient estimates. Full article

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

23 pages, 362 KiB

Open AccessArticle

Some New Applications of the q-Analogous of Differential and Integral Operators for New Subclasses of q-Starlike and q-Convex Functions

by Suha B. Al-Shaikh, Ahmad A. Abubaker, Khaled Matarneh and Mohammad Faisal Khan

Fractal Fract. 2023, 7(5), 411; https://doi.org/10.3390/fractalfract7050411 - 19 May 2023

Viewed by 846

Abstract

In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of differential equations. Many topics, including complex analysis, hypergeometric series, and particle physics, have been generalized in q-calculus. In this study, first of all, we define the q-analogues of a differential operator (

D R_{λ, q}^{m, n}

) by using the basic idea of q-calculus and the definition of convolution. Additionally, using the newly constructed operator (

D R_{λ, q}^{m, n}

), we establish the q-analogues of two new integral operators (

F_{λ, γ_{1,} γ_{2, \dots} γ_{l}}^{m, n, q}

and

G_{λ, γ_{1,} γ_{2, \dots} γ_{l}}^{m, n, q}

), and by employing these operators, new subclasses of the q-starlike and q-convex functions are defined. Sufficient conditions for the functions (f) that belong to the newly defined classes are investigated. Additionally, certain subordination findings for the differential operator (

D R_{λ, q}^{m, n}

) and novel geometric characteristics of the q-analogues of the integral operators in these classes are also obtained. Our results are generalizations of results that were previously proven in the literature. Full article

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

12 pages, 977 KiB

Open AccessArticle

Application of the Pathway-Type Transform to a New Form of a Fractional Kinetic Equation Involving the Generalized Incomplete Wright Hypergeometric Functions

by Mohammed Z. Alqarni, Ahmed Bakhet and Mohamed Abdalla

Fractal Fract. 2023, 7(5), 348; https://doi.org/10.3390/fractalfract7050348 - 23 Apr 2023

Cited by 3 | Viewed by 782

Abstract

We present in this paper a generalization of the fractional kinetic equation using the generalized incomplete Wright hypergeometric function. The pathway-type transform technique is then used to investigate the solutions to a fractional kinetic equation with specific fractional transforms. Furthermore, exceptional cases of our outcomes are discussed and graphically illustrated using MATLAB software. This work provides a thorough overview for further investigation into these topics in order to gain a better understanding of their implications and applications. Full article

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

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19 pages, 1057 KiB

Open AccessArticle

Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials

by Hari Mohan Srivastava, Waleed Adel, Mohammad Izadi and Adel A. El-Sayed

Fractal Fract. 2023, 7(4), 301; https://doi.org/10.3390/fractalfract7040301 - 29 Mar 2023

Cited by 12 | Viewed by 1562

Abstract

In this research, we present a new computational technique for solving some physics problems involving fractional-order differential equations including the famous Bagley–Torvik method. The model is considered one of the important models to simulate the coupled oscillator and various other applications in science and engineering. We adapt a collocation technique involving a new operational matrix that utilizes the Liouville–Caputo operator of differentiation and Morgan–Voyce polynomials, in combination with the Tau spectral method. We first present the differentiation matrix of fractional order that is used to convert the problem and its conditions into an algebraic system of equations with unknown coefficients, which are then used to find the solutions to the proposed models. An error analysis for the method is proved to verify the convergence of the acquired solutions. To test the effectiveness of the proposed technique, several examples are simulated using the presented technique and these results are compared with other techniques from the literature. In addition, the computational time is computed and tabulated to ensure the efficacy and robustness of the method. The outcomes of the numerical examples support the theoretical results and show the accuracy and applicability of the presented approach. The method is shown to give better results than the other methods using a lower number of bases and with less spent time, and helped in highlighting some of the important features of the model. The technique proves to be a valuable approach that can be extended in the future for other fractional models having real applications such as the fractional partial differential equations and fractional integro-differential equations. Full article

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

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17 pages, 355 KiB

Open AccessArticle

Fekete–Szegö Problem and Second Hankel Determinant for a Class of Bi-Univalent Functions Involving Euler Polynomials

by Sadia Riaz, Timilehin Gideon Shaba, Qin Xin, Fairouz Tchier, Bilal Khan and Sarfraz Nawaz Malik

Fractal Fract. 2023, 7(4), 295; https://doi.org/10.3390/fractalfract7040295 - 29 Mar 2023

Cited by 4 | Viewed by 867

Abstract

Some well-known authors have extensively used orthogonal polynomials in the framework of geometric function theory. We are motivated by the previous research that has been conducted and, in this study, we solve the Fekete–Szegö problem as well as give bound estimates for the [...] Read more.

G_{Σ} (v, σ)

of analytical and bi-univalent functions, implicating the Euler polynomials. Full article

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

18 pages, 348 KiB

Open AccessArticle

Some New Applications of the Faber Polynomial Expansion Method for Generalized Bi-Subordinate Functions of Complex Order γ Defined by q-Calculus

by Mohammad Faisal Khan and Mohammed AbaOud

Fractal Fract. 2023, 7(3), 270; https://doi.org/10.3390/fractalfract7030270 - 18 Mar 2023

Viewed by 775

Abstract

This work examines a new subclass of generalized bi-subordinate functions of complex order

γ

connected to the q-difference operator. We obtain the upper bounds

ρ_{m}

for generalized bi-subordinate functions of complex order

γ

using the Faber polynomial expansion technique. Additionally, we [...] Read more.

This work examines a new subclass of generalized bi-subordinate functions of complex order

γ

connected to the q-difference operator. We obtain the upper bounds

ρ_{m}

for generalized bi-subordinate functions of complex order

γ

using the Faber polynomial expansion technique. Additionally, we find coefficient bounds

|ρ_{2}|

and Feke–Sezgo problems

|ρ_{3} - ρ_{2}^{2}|

for the functions in the newly defined class, subject to gap series conditions. Using the Faber polynomial expansion method, we show some results that illustrate diverse uses of the Ruschewey q differential operator. The findings in this paper generalize those from previous efforts by a number of prior researchers. Full article

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

27 pages, 1577 KiB

Open AccessArticle

Some New Versions of Fractional Inequalities for Exponential Trigonometric Convex Mappings via Ordered Relation on Interval-Valued Settings

by Muhammad Bilal Khan, Adriana Cătaş, Najla Aloraini and Mohamed S. Soliman

Fractal Fract. 2023, 7(3), 223; https://doi.org/10.3390/fractalfract7030223 - 01 Mar 2023

Cited by 4 | Viewed by 1857

Abstract

This paper’s main goal is to introduce left and right exponential trigonometric convex interval-valued mappings and to go over some of their important characteristics. Additionally, we demonstrate the Hermite–Hadamard inequality for interval-valued functions by utilizing fractional integrals with exponential kernels. Moreover, we use the idea of left and right exponential trigonometric convex interval-valued mappings to show various findings for midpoint- and Pachpatte-type inequalities. Additionally, we show that the results provided in this paper are expansions of several of the results already demonstrated in prior publications The suggested research generates variants that are applicable for conducting in-depth analyses of fractal theory, optimization, and research challenges in several practical domains, such as computer science, quantum mechanics, and quantum physics. Full article

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

17 pages, 339 KiB

Open AccessArticle

Subordination Properties of Certain Operators Concerning Fractional Integral and Libera Integral Operator

by Georgia Irina Oros, Gheorghe Oros and Shigeyoshi Owa

Fractal Fract. 2023, 7(1), 42; https://doi.org/10.3390/fractalfract7010042 - 30 Dec 2022

Cited by 5 | Viewed by 1149

Abstract

The results contained in this paper are the result of a study regarding fractional calculus combined with the classical theory of differential subordination established for analytic complex valued functions. A new operator is introduced by applying the Libera integral operator and fractional integral [...] Read more.

λ

for analytic functions. Many subordination properties are obtained for this newly defined operator by using famous lemmas proved by important scientists concerned with geometric function theory, such as Eenigenburg, Hallenbeck, Miller, Mocanu, Nunokawa, Reade, Ruscheweyh and Suffridge. Results regarding strong starlikeness and convexity of order

α

are also discussed, and an example shows how the outcome of the research can be applied. Full article

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

16 pages, 1357 KiB

Open AccessArticle

Sharp Bounds of Hankel Determinant on Logarithmic Coefficients for Functions Starlike with Exponential Function

by Lei Shi, Muhammad Arif, Javed Iqbal, Khalil Ullah and Syed Muhammad Ghufran

Fractal Fract. 2022, 6(11), 645; https://doi.org/10.3390/fractalfract6110645 - 03 Nov 2022

Cited by 11 | Viewed by 1144

Abstract

Using the Lebedev–Milin inequalities, bounds on the logarithmic coefficients of an analytic function can be transferred to estimates on coefficients of the function itself and related functions. From this fact, the study of logarithmic-related problems of a certain subclass of univalent functions has [...] Read more.

S_{e}^{*}

connected with the exponential mapping was considered. The main purpose of this article is to obtain the sharp estimates of the second Hankel determinant with the logarithmic coefficient as entry for this class. Full article

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

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18 pages, 650 KiB

Open AccessArticle

Specific Classes of Analytic Functions Communicated with a Q-Differential Operator Including a Generalized Hypergeometic Function

by Najla M. Alarifi and Rabha W. Ibrahim

Fractal Fract. 2022, 6(10), 545; https://doi.org/10.3390/fractalfract6100545 - 27 Sep 2022

Viewed by 981

Abstract

A special function is a function that is typically entitled after an early scientist who studied its features and has a specific application in mathematical physics or another area of mathematics. There are a few significant examples, including the hypergeometric function and its [...] Read more.

(q, p)

-calculus and k-calculus. By engaging the notion of q-fractional calculus (QFC), we investigate the geometric properties of the generalized Prabhakar fractional differential operator in the open unit disk

\nabla : = {ξ \in C : | ξ | < 1}

. Consequently, we insert the generalized operator in a special class of analytic functions. Our methodology is indicated by the usage of differential subordination and superordination theory. Accordingly, numerous fractional differential inequalities are organized. Additionally, as an application, we study the solution of special kinds of q–fractional differential equation. Full article

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

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