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1 
Integral inequalities of hermitehadamard type and their applicationsLatif, Muhammad Amer January 2017 (has links)
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. / 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 midpoint 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. Hölder [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 HermiteHadamard 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 HermiteHadamard inequality.
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. / MT 2017

2 
Integral inequalities and solvability of boundary value problems with p(t)Laplacian operatorsZhao, Dandan., 趙丹丹. January 2009 (has links)
published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

3 
Integral inequalities and solvability of boundary value problems with p(t)Laplacian operatorsZhao, Dandan. January 2009 (has links)
Thesis (Ph. D.)University of Hong Kong, 2009. / Includes bibliographical references (leaves 8091). Also available in print.

4 
Some New Contributions in the Theory of Hardy Type InequalitiesYimer, Markos Fisseha January 2023 (has links)
In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, PólyaKnopp, Jensen, Minkowski and BeckenbachDresher inequalities. The main focus is put on the Hardy and their limit PólyaKnopp inequalities. Indeed, we derive such inequalities even in a general Banach functionsetting. The thesis consists of three papers (A, B and C) and an introduction, which put these papers into a more general frame. This introduction has also independent interest since it shortly describe the dramatic more than 100 years of development of Hardytype inequalities. It contains both wellknown and very new ideas and results. In paper A we prove and discuss some new Hardytype inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator, which is introduced in this paper (this operator generalizes the usualHardy kernel operator). These results generalize and unify several classical Hardytype inequalities. In paper B we prove some new refined Hardytype inequalities again in Banach function space settings. The used (super quadraticity) technique is also illustrated by making refinements of some generalized forms of the Jensen, Minkowski and BeckenbachDresher inequalities. These results both generalize and unify several results of this type. In paper C for the case 0<p≤q<∞ we prove some new PólyaKnopp inequalities in two and higher dimensions with good twosided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted PólyaKnopp inequality with sharp constant. / In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, PólyaKnopp, Jensen, Minkowski and BeckenbachDresher inequalities. The main focus is put on the Hardy and their limit, PólyaKnopp inequalities. Indeed, we derive such inequalities even in a general Banach function setting. We prove and discuss some new Hardytype inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator. These results generalize and unify several classical Hardytype inequalities. Next, we prove some new refined Hardytype inequalities again in Banach function space settings. We used superquadraticity technique to prove refinements of some classical inequalities. Finally, for the case 0<p≤q<∞, we prove some new PólyaKnopp inequalities in two and higher dimensions with good twosided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted PólyaKnopp inequality with sharp constant.

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