Abstract:
The role of mathematical inequalities in the growth of different branches of mathematics
as well as in other areas of science is well recognized in the past several years. The uses of
contributions of Newton and Euler in mathematical analysis have resulted in a numerous
applications of modern mathematics in physical sciences, engineering and other areas
sciences and hence have employed a dominat effect on mathematical inequalities.
Mathematical inequalities play a dynamic role in numerical analysis for approximation of
errors in some quadrature rules. Speaking more specifically, the error approximation in
quadrature rules such as the mid-point rule, trapezoidal rule and Simpson rule etc. have
been investigated extensively and hence, a number of bounds for these quadrature rules in
terms of at most second derivative are proven by a number of researchers during the past
few years.
The theorey of mathematical inequalities heavily based on theory of convex functions.
Actually, the theory of convex functions is very old and its commencement is found to be
the end of the nineteenth century. The fundamental contributions of the theory of convex
functions can be found in the in the works of O. HΓΆlder [50], O. Stolz [151] and J.
Hadamard [48]. At the beginning of the last century J. L. W. V. Jensen [72] first realized
the importance convex functions and commenced the symmetric study of the convex
functions. In years thereafter this research resulted in the appearance of the theory of
convex functions as an independent domain of mathematical analysis.
Although, there are a number of results based on convex function but the most celebrated
results about convex functions is the Hermite-Hadamard inequality, due to its rich
geometrical significance and many applications in the theory of means and in numerical
analysis. A huge number of research articles have been written during the last decade by a
number of mathematicians which give new proofs, generalizations, extensions and
refitments of the Hermite-Hadamard inequality.
The main intention of this dissertation is to present several Hermite-Hadamard type
inequalities when |𝑓(𝑛)|
𝑞
, 𝑛 β β, 𝑞 β₯ 1 belongs to different classes of convex functions
such as convex, 𝑠-convex, 𝑚-convex, (𝛼, 𝑚)-convex, quasi-convex, β-convex,
logarithmically convex, 𝑠-logarithmically convex, (𝛼, 𝑚)-logarithmically convex and
geometrically-arithmetically convex functions.
The given results in this dissertation not only generalize and refine a number of results
proved for 𝑛-times differentiable convex functions but also many interesting refinements
inequalities of Hermite-Hadamard type are deduced when |𝑓β²|𝑞 or |𝑓β²β²|𝑞, 𝑞 β₯ 1 is convex
functions, 𝑠-convex functions, 𝑚-convex functions, (𝛼, 𝑚)-convex functions, quasiconvex
functions, β-convex functions, logarithmically convex functions, 𝑠-logarithmically
convex functions, (𝛼, 𝑚)-logarithmically convex functions. Applications of the obtained
results to special means and new improved error bounds of the quadrature rules such as for
the mid-point rule and trapezoidal rule are presented.
Inequalities of Hermite-Hadamard type are also investigated for preinvex, 𝑠-preinvex, β-
preinvex, quasipreiinvex, logarithmically preinvex functions which are more general
phenomena than that of convex, 𝑠-convex, β-convex, quasi-convex and logarithmically
convex functions respectively. Applications of the results for these classes of functions are
given. The research upshots of this thesis make significant contributions in the theory of
means and the theory of inequalities.
Description:
A thesis submitted to the Faculty of Science, University of the
Witwatersrand, Johannesburg, South Africa, in fulfilment of the
requirements for the degree of Doctor of Philosophy. Johannesburg, 17 October 2016.