Integral Inequalities and Generalized Convexity

✍ Scribed by Shashi Kant Mishra, Nidhi Sharma, Jaya Bisht

• Publisher: CRC Press/Chapman & Hall
• Year: 2023
• Tongue: English
• Leaves: 277
• Category: Library

✦ Synopsis

The book covers several new research findings in the area of generalized convexity and integral inequalities. Integral inequalities using various type of generalized convex functions are applicable in many branches of mathematics such as mathematical analysis, fractional calculus, and discrete fractional calculus.

The book contains integral inequalities of Hermite-Hadamard type, Hermite- Hadamard-Fejer type and majorization type for the generalized strongly convex functions. It presents Hermite-Hadamard type inequalities for functions defined on Time scales. Further, it provides the generalization and extensions of the concept of preinvexity for interval-valued functions and stochastic processes, and give Hermite-Hadamard type and Ostrowski type inequalities for these functions. These integral inequalities are utilized in numerous areas for the boundedness of generalized convex functions.

Features:

• Covers Interval-valued calculus, Time scale calculus, Stochastic processes – all in one single book.

• Numerous examples to validate results

• Provides an overview of the current state of integral inequalities and convexity for a much wider audience, including practitioners.

• Applications of some special means of real numbers are also discussed.

The book is ideal for anyone teaching or attending courses in integral inequalities along with researchers in this area.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
Foreword
Author Biographies
Preface
Symbol Description
1. Introduction
1.1. Generalized Convexity
1.2. Invexity
1.3. Integral Inequalities
1.4. Fractional Calculus
1.5. Majorization Inequalities
1.6. Time Scale Calculus
1.7. Interval Analysis
1.7.1. Interval arithmetic
1.7.2. Integral of interval-valued functions
1.8. Stochastic Processes
2. Integral Inequalities for Strongly Generalized Convex Functions
2.1. Introduction
2.2. Preliminaries
2.3. Hermite–Hadamard Type Inequalities for Functions Whose Derivatives are Strongly n-Convex
2.3.1. Application to special means
2.4. Weighted Version of Hermite–Hadamard Type Inequalities for Strongly GA-Convex Functions
2.5. Hermite–Hadamard Type Integral Inequalities for the Class of Strongly Convex Functions on Time Scales
3. Integral Inequalities for Strongly Generalized Convex Functions of Higher Order
3.1. Introduction
3.2. Preliminaries
3.3. Strongly Generalized Convex Functions of Higher Order
3.4. Integral Inequalities for Higher Order Strongly Exponentially Convex Functions
4. Integral Inequalities for Generalized Preinvex Functions
4.1. Introduction
4.2. Preliminaries
4.3. Hermite–Hadamard Type Inequalities via Preinvex Functions
4.4. Application to Special Means
4.5. Generalized (m,h)-Preunivex Mappings via k-Fractional Integrals
5. Some Majorization Integral Inequalities for Functions Defined on Rectangles via Strong Convexity
5.1. Introduction
5.2. Preliminaries
5.3. Majorization Integral Inequalities for Strong Convexity
6. Hermite–Hadamard Type Inclusions for Interval-Valued Generalized Preinvex Functions
6.1. Introduction
6.2. Preliminaries
6.3. Hermite–Hadamard Type Inclusions for Interval-Valued Preinvex Functions
6.4. Hermite–Hadamard Type Inclusions for Interval-Valued Coordinated Preinvex Functions
6.5. Hermite–Hadamard Type Fractional Inclusions for Harmonically h-Preinvex Interval-Valued Functions
7. Some Inequalities for Multidimensional General h-Harmonic Preinvex and Strongly Generalized Convex Stochastic Processes
7.1. Introduction
7.2. Preliminaries
7.3. General h−Harmonically Preinvex Stochastic Process (Gh − HPnφSP)
7.4. Multidimensional General h−Harmonic Preinvex Stochastic Processes (MGh − HPnφSP)
7.5. Strongly Generalized Convex Stochastic Processes
8. Applications
8.1. Hermite–Hadamard Inequality
8.1.1. Higher dimensional Hermite–Hadamard inequality
8.1.2. Mass transportation and higher dimensional Hermite–Hadamard inequality
8.2. Jensen’s Inequality
8.3. Time Scales
8.4. Interval-Valued Functions
8.4.1. Error estimation to quadrature rules Using Ostrowski type inequality for interval-valued functions
Bibliography
Index