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 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. 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 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. 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

Bibliography: leaves 108-113. / ii, 113 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1998?

Thesis (Ph. D.)--University of Wisconsin--Madison, 1951. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 46-47).

We study two related inequalities that arise in Harmonic Analysis: restriction type inequalities and isoperimetric inequalities. The (Lp, Lq) Restriction type inequalities have been the subject of much interest since they were first conceived in the 1960s. The classical restriction type inequality involving surfaces of non-vanishing curvature is only fully resolved in two dimensions and there have been a lot of recent developments to establish the conjectured (p,q) range in higher dimensions. However, it also interesting to consider what can be said for curves where the curvature does vanish. In particular we build upon a restriction result for homogeneous polynomial surfaces, using what is considered the natural weight - the one induced by the affine curvature of the surface. This is known to hold with a non-universal constant which depends in some way on the coefficients of the polynomial. In this dissertation we shall quantify that relationship. Restriction estimates (for curves or surfaces) using the affine curvature weight can be shown to lead to an affine isoperimetric inequality for such curves or surfaces. We first prove, directly, this inequality for polynomial curves, where the constant depends only on the degree of the underlying polynomials. We then adapt this method, to prove an isoperimetric inequality for a wide class of curves, which includes curves for which a restriction estimate is not yet known. Next we state and prove an analogous result of the relative affine isoperimetric inequality, which applies to unbounded convex sets. Lastly we demonstrate that this relative affine isoperimetric inequality for unbounded sets is in fact equivalent to the classical affine isoperimetric inequality.

Yes / Various inequalities (Boole inequality, Chung–Erdös inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra 'quantum correction' term, and which hold for all quantum states. When certain sufficient conditions are satisfied, the quantum correction term is zero, and the classical version of these inequalities holds for all states. But in general, the classical version of these inequalities is violated by some of the quantum states. For example in bipartite systems, classical Boole inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. A logical approach to CHSH inequalities (which are related to the Frechet inequalities), is studied in this context. It is shown that CHSH inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. The reduction of the rank of a pure state by a quantum measurement with both orthogonal and coherent projectors, is studied. Bounds for the average rank reduction are given.

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by Chu Chiu Wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Bibliography: leaves 31-32. / Introduction / Chapter Section 1. --- A Minimization Problem / Chapter Section 2. --- Radial Symmetry of The Solution / Chapter Section 3. --- Proof of The Main Theorem / References

by Chan Ho Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 128-[131]). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Information Theory --- p.1 / Chapter 1.2 --- Approaches for characterizing Γ*n --- p.4 / Chapter 1.3 --- Outline of the thesis --- p.7 / Chapter 2 --- Quasi-Uniformity --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- Box Assignment --- p.9 / Chapter 2.2.1 --- Box Assignment --- p.9 / Chapter 2.2.2 --- Conditional Box Assignment --- p.17 / Chapter 2.3 --- Quasi-Uniform Random Variables --- p.18 / Chapter 2.4 --- Main Theorems --- p.20 / Chapter 2.4.1 --- Preliminaries --- p.20 / Chapter 2.4.2 --- Main Theorems --- p.25 / Chapter 2.5 --- Quasi-Uniformity and Inequality --- p.32 / Chapter 2.5.1 --- Quasi-Uniformity and Inequality --- p.32 / Chapter 2.6 --- A New Perspective of Information Inequality --- p.34 / Chapter 2.6.1 --- Combinatorial Inequality --- p.34 / Chapter 2.6.2 --- Relations between combinatorial inequalities and informa- tion inequalities --- p.36 / Chapter 2.7 --- Summary --- p.40 / Chapter 3 --- Groups and Quasi Uniformity --- p.41 / Chapter 3.1 --- Introduction --- p.41 / Chapter 3.2 --- Group --- p.42 / Chapter 3.3 --- Group Represent ability --- p.47 / Chapter 3.4 --- Tightness of Group Represent ability --- p.54 / Chapter 3.4.1 --- Tightness of γn --- p.54 / Chapter 3.5 --- Abelian group represent able --- p.58 / Chapter 3.5.1 --- Δ operator and sub cone con(γαb) --- p.63 / Chapter 3.5.2 --- Decomposition of con(γαb) --- p.67 / Chapter 3.6 --- Summary --- p.73 / Chapter 4 --- Linear Representability --- p.74 / Chapter 4.1 --- Introduction --- p.74 / Chapter 4.2 --- Preliminaries of Vector Space --- p.75 / Chapter 4.3 --- Linear Representability --- p.80 / Chapter 4.3.1 --- Orthogonal Space --- p.80 / Chapter 4.3.2 --- Linear Representability --- p.81 / Chapter 4.3.3 --- Direct Sum --- p.90 / Chapter 4.3.4 --- Conditional Entropy --- p.93 / Chapter 4.4 --- Tightness of γαb --- p.95 / Chapter 4.5 --- Reverse Representation --- p.98 / Chapter 4.6 --- Summary --- p.106 / Chapter A --- AEP and BOX ASSIGNMENT --- p.107 / Chapter B --- Proof of Chapter 4's lemma --- p.110 / Chapter C --- Tightness of and Ψαb --- p.118

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the Degree of Master of Science. Johannesburg 2015. / This Dissertation focuses on the Jacobi polynomial. Specifically, it discusses certain aspects of the zeros of the Jacobi polynomial such as the interlacing property and quasiorthogonality. Also found in the Dissertation is a chapter on the inequalities of the zeros of the Jacobi polynomial, mainly those developed by Walter Gautschi.

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