

Title of Thesis
Majoriztion And Its Applications 
Author(s)
Naveed Latif 
Institute/University/Department
Details Abdus Salam School of Mathematical Sciences / GC
University, Lahore 
Session 2011 
Subject Mathematics 
Number of Pages 123 
Keywords (Extracted from title, table of contents and
abstract of thesis) Applications, Theorems, Majorization,
Results, Discrete, Cauchy, Surprising, Introduce, Majoriztion,
Section, Positive 
Abstract The notion of
majorization arose as a measure of the diversity of the components
of an ndimensional vector (an ntuple) 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 nonconvex constrained optimization problems that
involve matrixvalued 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 semide
nite matrices which imply exponential convexity and logconvexity
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 semide nite matrices which
imply a surprising property of exponential convexity and
logconvexity 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 nonnegative 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 semide
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 logconvexity 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 semide 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 logconvexity 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 logconvexity.
In the sixth chapter we prove positive semide niteness of matrices
generated by differences deduced from Popoviciu's inequalities which
implies a surprising property of exponential convexity and
logconvexity 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. 
