Title of Thesis

Majoriztion And Its Applications

Author(s)

Naveed Latif

Abstract
The notion of majorization arose as a measure of the diversity of the components of an n-dimensional vector (an n-tuple) and is closely related to convexity. Many of the key ideas relating to majorization were discussed in the volume entitled Inequalities by Hardy, Littlewood and Polya (1934). Only a relatively small number of researchers were inspired by it to work on questions relating to majorization.After the volume entitled Theory of Majorization and its Applications (Marshall and Olkin, 1979), they heroically had shifted the literature and endeavored to rearrange ideas in order, often provided references to multiple proofs and multiple viewpoints on key results, with reference to a variety of applied elds.For certain kinds of inequalities, the notion of majorization leads to such a theory that is sometime extremely useful and powerful for deriving inequalities.Moreover, the derivation of an inequality by methods of majorization is often very helpful both for providing a deeper understanding and for suggesting natural generalizations. Majorization theory is a key tool that allows us to transform complicated non-convex constrained optimization problems that involve matrix-valued variables into simple problems with scalar variables that can be easily solved.
In this PhD thesis, we restrict our attention to results in majorization that directly involve convex functions. The theory of convex functions is a part of the general subject of convexity, since a convex function is one whose epigraph is a convex set. Nonetheless it is an important theory, which touches almost all branches of mathematics.In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the "average" derivative of the section.It is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval.In the rst chapter some basic results about convex functions, some other classes of convex functions and majorization theory are given.In the second chapter we prove positive semi-de nite matrices which imply exponential convexity and log-convexity for di erences of majorization type results in discrete case as well as integral case. We also obtain Lypunov's and Dresher's type inequalities for these di erences. In this chapter both sequences and functions are monotonic and positive.We give some mean value theorems and related Cauchy means. We also show that these means are monotonic.
In the third chapter we prove positive semi-de nite matrices which imply a surprising property of exponential convexity and log-convexity for di erences of additive and multiplicative majorization type results in discrete case.We also obtain Lypunov's and Dresher's type inequalities for these di erences. In this chapter we use monotonic non-negative as well as real sequences in our results. We give some applications of majorization. Related Cauchy means are de ned and prove that these means are monotonic.
In the fourth chapter we obtain an extension of majorization type results and extensions of weighted Favard's and Berwald's inequality when only one of function is monotonic. We prove positive semi-de niteness of matrices generated by di erences deduced from majorization type results and di erences deduced from weighted Favard's and Berwald's inequality.This implies a surprising property of exponential convexity and log-convexity of these di erences which allows us to deduce Lyapunov's and Dresher's type inequalities for these di erences, which are improvements of majorization type results and weighted Favard's and Berwald's inequalities. Analogous Cauchy's type means, as equivalent forms of exponentially convexity and logconvexity, are also studied and the monotonicity properties are proved.
In the fth chapter we obtain all results in discrete case from chapter four. We give majorization type results in the case when only one sequence is monotonic.We also give generalization of Favard's inequality, generalization of Berwald's inequality and related results. We prove positive semi-de niteness of matrices generated by di erences deduced from majorization type results and di erences deduced from weighted Favard's and Berwald's inequality which implies exponential convexity and log-convexity of these di erences which allow us to deduce Lyapunov's and Dresher's type inequalities for these di erences. We introduce new Cauchy's means as equivalent form of exponential convexity and log-convexity.
In the sixth chapter we prove positive semi-de niteness of matrices generated by differences deduced from Popoviciu's inequalities which implies a surprising property of exponential convexity and log-convexity of these di erences which allows us to deduce Gram's, Lyapunov's and Dresher's type inequalities for these di erences. We introduce some mean value theorems.Also we give the Cauchy means of the Popoviciu type and we show that these means are monotonic.

1

1.1 Convex functions
1.2 Some other classes of convex functions
1.3 Majorization theory

2.1 Introduction
2.2 Main Results
2.3 Cauchy Means
2.4 Some Related Results

3.1 Introduction
3.2 The case of non-negative sequences and functions
3.3 Multiplicative Majorization

4.1 Introduction
4.2 Main Results
4.3 Exponentially Convexity, Lyapunov's And Dresher's Type Of Inequalities
4.4 Mean Value Theorems

5.1 Introduction
5.2 Main Results
5.3 Exponential Convexity, Lyapunov's and Dresher's type of inequalities
5.4 Mean Value Theorems

6.1 Introduction
6.2 Main Results
6.3 Cauchy Means