The main subject of this study coincides with the recently adapted methodology in the theory of univalent functions via applications of fractional calculus. The Caputo's definition of fractional derivative of order has not been obviously applied in the field. However, it plays a vital role in other areas like physics and engineering due to its simple restrictions compared to Liouville's integral operator. By modifying the Caputo's derivative operator, a new modified Caputo's operator is defined in this book.
Consequently, new subclasses of analytic functions are introduced, and basic properties, such as distortion theorems, extremal functions, coefficient bounds, the integral transform, superordination and subordination results are discussed. The radii of basic well known properties (Univalency, Starlikeness, Convexity) are also obtained for the new subclasses. Moreover, some results of the modified Hadamard product (convolution) are provided by applying the famous Cauchy-Schwartz inequality.
In order to expand the application of the modified Caputo's operator, two new subclasses of analytic p-valent functions are introduced and investigated through the concepts of neighbourhood and inclusion relations. Further, the definition of Univalency with respect to 1 together with subordination are used to solve a sharp functional of Fekete-Szegö problem. Moreover, some interesting results related to Hankel determinant are obtained.
Contents
1. Preliminaries
1.1 Introduction
1.2 Analytic and Univalent Functions
1.3 Functions with Positive Real Part
1.4 Starlike and Convex Functions
1.5 Growth and Distortion Theorems
1.6 Hadamard Product
1.7 Objectives
1.8 Over View
2. The Modified Caputo’s Operator
2.1 Introduction
2.2 General Properties
2.3 The Concept of Subordination and Superordination
2.4 Applications on Sandwich Theorems
3. The Hankel Determinant Inequalities
3.1 Inequality Results for Starlike and Convex Functions
3.2 Inequality Results for the modified Caputo’s Operator
4. Integral Applications
4.1 The Class TR(η, λ, α)
4.1.1 Characterization property
4.1.2 Results involving convolution
4.1.3 Integral transform of the class TR(η, λ, α)
4.2 The Class Aη,λ(α, β, γ)
4.2.1 Extreme points of the class Aη,λ(α, β, γ)
4.2.2 Radius of Univalency and Starlikeness
5. Further Applications
5.1 The Class Tη,λ (α, β,A,B)
5.1.1 Characterization properties
5.1.2 Distortion Theorem
5.1.3 Properties of the class Tη,λ (α, β,A,B)
5.1.4 Results involving Hadamard product
5.2 The Class T (η, λ, α)
5.2.1 Distortion Theorem
5.2.3 Results involving Hadamard product
6. Neighbourhood and Inclusion Relations
6.1 Introduction
6.2 Coefficient Inequalities
6.3 Inclusion Relations Involving the (n, δ)-neighbourhoods
6.4 Further Neighbourhood Properties
7. Applications on Fekete-Szegö Problem
7.1 Introduction
7.2 Fekete-Szegö Problem
7.3 Applications to Functions defined by Fractional Derivatives
8. Conclusion and Open Problems
Objectives and Research Themes
The main objective of this study is to refine the Caputo fractional derivative operator to define a new modified operator, subsequently applying it to generate and analyze new subclasses of analytic functions, establishing properties like distortion theorems, coefficient bounds, and inclusion relations, and lastly, using these findings to address the Fekete-Szegö problem.
- Modification and analytical characterization of the Caputo fractional derivative operator.
- Introduction of novel subclasses of univalent and p-valent analytic functions based on the new operator.
- Investigation of subordination, superordination, and sandwich theorems.
- Derivation of the second Hankel determinant inequalities for the new classes.
- Study of inclusion relations and (n, δ)-neighbourhood properties of the generated subclasses.
Excerpt from the Book
1.1. Introduction
The theory of univalent functions started to spark since 1907, when a German mathematician Koebe gave a remarkable geometric properties of an analytic function. Later this type of analytic function is known as Koebe function.
Various techniques have been invented by mathematicians to obtain results concerning certain properties of the class of all univalent functions S, inspired by the the Bierberbach conjectures: |a_n| <= n for every univalent function and Bierberbach’s estimate for the second coefficient of normalized univalent functions. Consequently, two important theorems have been proved such as the Distortion and Growth theorems that became basic criterion of investigating of any subclass of S.
The study of univalent functions has been extended by further applications, for example in 1909 Lindelof initiated the concept of subordination, that enriched the study and motivated various researchers to solve differential subordination and superordination equations in order to reduce the study of a whole class to investigating one function which is the dominant of the class. Also, the fractional calculus (that is fractional integral and fractional derivative) has been considered as an asset in the way of applying fractional calculus integral or differential operators to ensure either a beautiful generalization of a known subclass of analytic functions or to provide a new subclass. Moreover, combining a fractional operator tactfully to Hadamard product (convolution) provides many interesting result.
Summary of Chapters
1. Preliminaries: This chapter reviews the fundamental definitions of univalent, starlike, and convex functions, alongside distortion theorems and the Hadamard product necessary for subsequent development.
2. The Modified Caputo’s Operator: This chapter formally introduces the new modified Caputo derivative operator and investigates its general properties, subordination and superordination, and sandwich theorems.
3. The Hankel Determinant Inequalities: This chapter provides results for the second Hankel determinant of the modified Caputo's operator and its inverse for starlike and convex function classes.
4. Integral Applications: This chapter introduces new subclasses of analytic functions defined via the modified Caputo operator, focusing on characterization properties and integral transforms.
5. Further Applications: This chapter extends the study to additional subclasses of analytic functions, providing new distortion theorems and convolution results involving the Hadamard product.
6. Neighbourhood and Inclusion Relations: This chapter establishes the (n, δ)-neighbourhoods for newly defined p-valent subclasses and discusses inclusion relations.
7. Applications on Fekete-Szegö Problem: This chapter applies the modified Caputo operator to solve the Fekete-Szegö problem for specific classes of functions.
8. Conclusion and Open Problems: The final chapter summarizes the thesis results and outlines directions for future research.
Keywords
Modified Caputo derivative, Fractional calculus, Univalent functions, Analytic functions, Subordination, Superordination, Sandwich theorems, Hankel determinant, Hadamard product, Distortion theorems, Geometric function theory, p-valent functions, Fekete-Szegö problem, Neighbourhood relations
Frequently Asked Questions
What is the primary scientific focus of this work?
This thesis examines the field of geometric function theory, specifically by adapting and modifying the Caputo fractional derivative operator to investigate new subclasses of analytic and univalent functions.
What are the core thematic areas covered?
The core themes include fractional calculus operators, the theory of univalent functions, differential subordination and superordination, Hankel determinant inequalities, and neighbourhood properties of function classes.
What is the central research question?
The research seeks to determine how modifying the Caputo operator enables the derivation of new analytic subclasses and their specific geometric properties, such as coefficient bounds and Hankel inequalities.
Which mathematical methodology is primarily utilized?
The work employs techniques from fractional calculus, the theory of subordination and superordination, and established extremal methods for analytic functions, often utilizing the Cauchy-Schwarz inequality.
What does the main body of the text cover?
The main chapters cover the definition of the modified operator, the derivation of second Hankel determinant bounds, the introduction of multiple new analytic function subclasses, and an analysis of their inclusion and neighbourhood relations.
Which keywords best characterize this research?
The work is characterized by terms like fractional calculus, modified Caputo operator, univalent functions, subordination, superordination, Hankel determinant, and geometric function theory.
How is the "modified Caputo operator" defined in this thesis?
It is defined in Chapter 2, building upon the initial value and n-th derivative modifications of the standard Caputo definition, tailored for analytic functions in the unit disc.
What significance do the Hankel determinant results hold?
These results provide essential techniques for approaching boundary inequalities for the coefficients of univalent functions, offering sharp upper bounds for new operators.
What role does the Fekete-Szegö problem play here?
The problem is addressed in Chapter 7 using the modified operator, serving as an application to determine upper bounds on sharp functionals for functions starlike with respect to 1.
What future applications are proposed in the conclusion?
The author suggests that the modified Caputo operator could be utilized in the Cas Wavelet Method for solving nonlinear Fredholm-Differential equations of fractional order.
- Quote paper
- Jamal Salah (Author), 2022, On the Modified Caputo’s Derivative Operator, Munich, GRIN Verlag, https://www.grin.com/document/1292827
