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.
Inhaltsverzeichnis (Table of Contents)
- PREFACE
- CONTENTS
- LIST OF SYMBOLS
- CHAPTER I
- PRELIMINARIES
- Introduction
- Analytic and Univalent Functions
- Functions with Positive Real Part
- Starlike and Convex Functions
- Growth and Distortion Theorems
- Hadamard Product
- Objectives
- Over View
- CHAPTER II
- THE MODIFIED CAPUTO'S OPERATOR
- Introduction
- General Properties
- The Concept of Subordination and Superordination
- Applications on Sandwich Theorems
- CHAPTER III
- THE HANKEL DETERMINANT INEQUALITIES
- Inequality Results for Starlike and Convex Functions
- Inequality Results for the modified Caputo's Operator
- CHAPTER IV
- INTEGRAL APPLICATIONS
- The Class TR(n, λ, a)
- Characterization property
- Results involving convolution
- Integral transform of the class TR(ŋ, λ, α)
- The Class An,λ(a, ß, y)
- Extreme points of the class An, λ(a, ß, y)
- Radius of Univalency and Starlikeness
- CHAPTER V
- FURTHER APPLICATIONS
- The Class Tn,λ (α, ß,A,B)
- Characterization properties
- Distortion Theorem
- Properties of the class Tη,λ (α, ß,A,B)
- Results involving Hadamard product
- The Class T (n, λ, α)
- Distortion Theorem
- Results involving Hadamard product
- CHAPTER VI
- NEIGHBOURHOOD AND INCLUSION RELATIONS
- Introduction
- Coefficient Inequalities
- Inclusion Relations Involving the (n, d)-neighbourhoods
- Further Neighbourhood Properties
- CHAPTER VII
- APPLICATIONS ON FEKETE-SZEGÖ PROBLEM
- Introduction
- Fekete-Szegö Problem
- Applications to Functions defined by Fractional Derivatives
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This study focuses on the application of fractional calculus to the theory of univalent functions, specifically by exploring a modified Caputo's derivative operator. The book aims to introduce and investigate new subclasses of analytic functions based on this modified operator, examining their properties and applications.
- Modified Caputo's derivative operator and its applications in univalent function theory
- New subclasses of analytic functions defined using the modified Caputo's operator
- Basic properties of these subclasses, including distortion theorems, coefficient bounds, and integral transforms
- Applications of the modified Caputo's operator in areas such as Hankel determinants, neighbourhood and inclusion relations, and the Fekete-Szegö problem
- Exploration of the relationship between subordination, superordination, and the modified Caputo's operator
Zusammenfassung der Kapitel (Chapter Summaries)
- Chapter I: PRELIMINARIES This chapter provides an overview of the fundamental concepts and definitions related to univalent function theory, including analytic functions, starlike and convex functions, growth and distortion theorems, and the Hadamard product. It also outlines the objectives and scope of the book.
- Chapter II: THE MODIFIED CAPUTO'S OPERATOR This chapter introduces the modified Caputo's derivative operator and discusses its general properties. It also explores the concepts of subordination and superordination and their applications in sandwich theorems.
- Chapter III: THE HANKEL DETERMINANT INEQUALITIES This chapter delves into inequalities related to Hankel determinants, specifically examining their connection to starlike and convex functions as well as the modified Caputo's operator.
- Chapter IV: INTEGRAL APPLICATIONS This chapter explores integral applications of the modified Caputo's operator. It introduces two new classes of analytic functions, TR(n, λ, a) and An,λ(a, ß, y), and investigates their properties, including characterization, convolution results, integral transforms, extreme points, and radii of univalency and starlikeness.
- Chapter V: FURTHER APPLICATIONS This chapter expands on the applications of the modified Caputo's operator by introducing two additional classes of analytic functions, Tn,λ (α, ß,A,B) and T (n, λ, α). It investigates their properties, including characterization, distortion theorems, Hadamard product results, and other relevant properties.
- Chapter VI: NEIGHBOURHOOD AND INCLUSION RELATIONS This chapter focuses on the concepts of neighbourhood and inclusion relations, exploring their applications in relation to the modified Caputo's operator and the classes of analytic functions introduced in previous chapters.
- Chapter VII: APPLICATIONS ON FEKETE-SZEGÖ PROBLEM This chapter examines the Fekete-Szegö problem in the context of univalent function theory and applies the modified Caputo's operator to obtain solutions and related results.
Schlüsselwörter (Keywords)
This book primarily focuses on the concepts of fractional calculus, univalent function theory, modified Caputo's derivative operator, analytic functions, subclasses of analytic functions, distortion theorems, coefficient bounds, integral transforms, Hankel determinants, neighbourhood and inclusion relations, Fekete-Szegö problem, subordination and superordination, and sandwich theorems.
- Quote paper
- Jamal Salah (Author), 2022, On the Modified Caputo’s Derivative Operator, Munich, GRIN Verlag, https://www.grin.com/document/1292827
