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 fields. 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 mathe-
matics. 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
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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 first 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-definite matrices which imply exponen-
tial convexity and log-convexity for differences of majorization type results in discrete
case as well as integral case. We also obtain Lypunov’s and Dresher’s type inequalities
for these differences. 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-definite matrices which imply a surprising
property of exponential convexity and log-convexity for differences of additive and
multiplicative majorization type results in discrete case. We also obtain Lypunov’s
and Dresher’s type inequalities for these differences. In this chapter we use mono-
tonic non-negative as well as real sequences in our results. We give some applications
of majorization. Related Cauchy means are defined and prove that these means are
monotonic.
In the fourth chapter we obtain an extension of majorization type results and ex-
tensions of weighted Favard’s and Berwald’s inequality when only one of function
is monotonic. We prove positive semi-definiteness of matrices generated by differ-
ences deduced from majorization type results and differences deduced from weighted
Favard’s and Berwald’s inequality. This implies a surprising property of exponen-
tial convexity and log-convexity of these differences which allows us to deduce Lya-
punov’s and Dresher’s type inequalities for these differences, which are improvements
of majorization type results and weighted Favard’s and Berwald’s inequalities. Anal-
ogous Cauchy’s type means, as equivalent forms of exponentially convexity and log-
convexity, are also studied and the monotonicity properties are proved.
In the fifth chapter we obtain all results in discrete case from chapter four. Weix
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 inequal-
ity and related results. We prove positive semi-definiteness of matrices generated
by differences deduced from majorization type results and differences deduced from
weighted Favard’s and Berwald’s inequality which implies exponential convexity and
log-convexity of these differences which allow us to deduce Lyapunov’s and Dresher’s
type inequalities for these differences. We introduce new Cauchy’s means as equiva-
lent form of exponential convexity and log-convexity.
In the sixth chapter we prove positive semi-definiteness of matrices generated by dif-
ferences deduced from Popoviciu’s inequalities which implies a surprising property of
exponential convexity and log-convexity of these differences which allows us to deduce
Gram’s, Lyapunov’s and Dresher’s type inequalities for these differences. We intro-
duce some mean value theorems. Also we give the Cauchy means of the Popoviciu
type and we show that these means are monotonic.