Keywords (Extracted from title, table of contents and
abstract of thesis) Jensen’s, Related, functional,
analysis, theory, integral, equations, interpolation, convex
functions, HardyLittlewood 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
HermiteHadamard inequality, Hardy and its limit PolyaKnopp
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
convexconcave antisymmetric functions is proved and some
applications are given.
In the third chapter we have discussed the generalized form of
HermiteHadamard inequality for integrable Convex functions.
In the fourth chapter Some estimates of Hardy, strengthened HardyKnopp
and multidimensional HardyPolyaKnopp type di erences for p < 0 and
0 < p < 1 are calculated.
In the fifth chapter we prove a new general onedimensional inequality
for convex functions and HardyLittlewood averages. Furthermore, we
apply this result to unify and refine the socalled Boas's inequality
and the strengthened inequalities of the HardyKnopptype, deriving
their new refinements as special cases of the obtained general
relation. In particular, we get new refinements of strengthened
versions of the wellknown Hardy and PolyaKnopp's inequalities,
while in the last chapter some measures of divergences between
vectors in a convex set of ndimensional real vector
space are defined in terms of certain types of entropy functions, and their logconvexity
properties with some applications in Information theory are discussed.
