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Title of Thesis

On Jensen’s and Related Inequalities

Author(s)

SABIR HUSSAIN

Institute/University/Department Details
Abdus Salam School of Mathematical Sciences / GC University, Lahore
Session
2009
Subject
Mathematics
Number of Pages
99
Keywords (Extracted from title, table of contents and abstract of thesis)
Jensen’s, Related, functional, analysis, theory, integral, equations, interpolation, convex functions, Hardy-Littlewood averages

Abstract
Inequalities are one of the most important instruments in many branches of mathematics such as functional analysis, theory of differential and integral equations, interpolation theory, harmonic analysis, probability theory, etc. They are also useful in mechanics, physics and other sciences. A systematic study of inequalities was started in the classical book [31] and continued in [54, 55]. In the eighties and nineties of the last century an impetuous increase of interest in inequalities took place. One result of this fact was a great number of published books on inequalities (see e.g. [4, 5, 37, 39, 38]) and on their applications (see e.g. [2, 11]). Nowadays the theory of inequalities is still being intensively developed. This fact is confirmed by a great number of recent published books (see e.g. [6, 56]) and a huge number of articles on inequalities. Thus, the theory of inequalities may be regarded as an independent area of mathematics. This PhD thesis is devoted to special kind of inequalities, namely Jensen's and some its related inequalities involving Hermite-Hadamard inequality, Hardy and its limit Polya-Knopp inequality.
In the first chapter, called Introduction, some basic notions and results from theory of convex functions and theory of inequalities are being introduced along with classical results of convex functions.
In the second chapter, The weighted Jensen's Inequality for convex-concave antisymmetric functions is proved and some applications are given.
In the third chapter we have discussed the generalized form of Hermite-Hadamard inequality for integrable Convex functions.
In the fourth chapter Some estimates of Hardy, strengthened Hardy-Knopp and multidimensional Hardy-Polya-Knopp type di erences for p < 0 and 0 < p < 1 are calculated.
In the fifth chapter we prove a new general one-dimensional inequality for convex functions and Hardy-Littlewood averages. Furthermore, we apply this result to unify and refine the so-called Boas's inequality and the strengthened inequalities of the Hardy-Knopp-type, deriving their new refinements as special cases of the obtained general relation. In particular, we get new refinements of strengthened versions of the well-known Hardy and Polya-Knopp's inequalities, while in the last chapter some measures of divergences between vectors in a convex set of n-dimensional real vector space are defined in terms of certain types of entropy functions, and their log-convexity properties with some applications in Information theory are discussed.

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

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2

1

INTRODUCTION

1.1 Convex Function

1.2 Hardy And Polya-Knopp's Inequalities

1
138 KB
3 2 JENSEN'S INEQUALITY FOR CONVEX-CONCAVE ANTI-SYMMETRIC FUNCTIONS AND APPLICATIONS

2.1 Introduction

2.2 Main results

2.3 Applications

11
133 KB
4 3 ON CERTAIN INEQUALITIES IMPROVING THE HERMITE-HADAMARD INEQUALITY

3.1 Introduction

3.2 Main results

3.3 Applications

17
118 KB
5 4 BOUNDS FOR HARDY'S AND POLYA-KNOPP'S DIFFERENCES

4.1 Introduction

4.2 Bounds For Hardy's Differences

4.3 Bounds For Strengthened Hardy and Polya-Knopp's Differences

4.4 Bounds For Multidimensional Hardy Type Polya-Knopp Difference

23
212 KB
6 5 SOME NEW REFINEMENTS OF STRENGTHENED HARDY AND POLYA-KNOPP'S INEQUALITIES

5.1 Introduction

5.2 The main results

5.3 Refinements of strengthened Hardy and Polya-Knopp's inequalities

50
167 KB
7 6 APPLICATIONS IN INFORMATION THEORY

6.1 Introduction

6.2 log-convexity of J-divergence

6.3 log-convexity of L-divergence

6.4 log-convexity of K-divergence

6.5 log-convexity of B-divergence

6.6 Applications

67
170 KB
8 7 BIBLIOGRAPHY

81


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