Abstract:
Inequalities lie at the heart of a great deal of mathematics. G. H. Hardy reported
Harald Bohr as saying ‘all analysts spend half their time hunting through the literature
for inequalities which they want to use but cannot prove’. Inequalities involving
means open many doors for analysts e.g generalization of mixed means fallouts the
refinements to the important inequalities of Holder and Minkowski. The well known
Jensen’s inequality asserts a remarkable relation among the mean and the mean of
function values and any improvement or refinements of Jensen’s inequality is a source
to enrichment of monotone property of mixed means.
our aim is to utilize all known refinements of Jensen’s inequality to give the re-
finements of inequality among the power means by newly defined mixed symmetric
means. In this context, our results not only ensures the generalization of classical but
also speak about the most recent notions (e.g n-exponential convexity) of this era.
In first chapter we start with few basic notions about means and convex functions.
Then the classical Jensen’s inequality and the historical results about refinements of
Jensen’s inequality are given from the literature together with their applications to
the mixed symmetric means.
In second chapter we consider recent refinements of Jensen’s inequality to refine
inequality between power means by mixed symmetric means with positive weights
under more comprehensive settings of index set. A new refinement of the classical
Jensen’s inequality is also established. The Popovicui type inequality is generalized
using green function. Using these refinements we define various versions of linear
functionals that are positive on convex functions. This step ultimately leads us to
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the important and recently revitalized area of exponential convexity. Mean value
theorems are proved for these functionals. Some non-trivial examples of exponential
convexity and some classes of Cauchy means are given. These examples are further
used to show monotonicity in defining parameters of constructed Cauchy means.
In third chapter we develop the refinements of discrete Jensen’s inequality for con-
vex functions of several variables which causes the generalizations of Beck’s results.
The consequences of Beck’s results are given in more general settings. We also gen-
eralize the inequalities of H ̈older and Minkowski by using the Quasiarithmetic mean
function.
In forth chapter we investigate the class of self-adjoint operators defined on a
Hilbert space, whose spectra are contained in an interval. We extend several re-
finements of the discrete Jensen’s inequality for convex functions to operator convex
functions. The mixed symmetric operator means are defined for a subclass of positive
self-adjoint operators to give the refinements of inequality between power means of
strictly positive operators.
In last chapter, some new refinements are given for Jensen’s type inequalities in-
volving the determinants of positive definite matrices. Bellman-Bergstrom-Fan func-
tionals are considered. These functionals are not only concave, but superlinear which
is a stronger condition. The results take advantage of this property.