Integral inequalities of hermite-hadamard type and their applications

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dc.contributor.author Latif, Muhammad Amer
dc.date.accessioned 2017-12-06T13:08:28Z
dc.date.available 2017-12-06T13:08:28Z
dc.date.issued 2017
dc.identifier.citation Latif, Muhammad Amer (2017) Integral inequalities of hermite-hadamard type and their applications, University of the Witwatersrand, Johannesburg, <http://hdl.handle.net/10539/23457>
dc.identifier.uri http://hdl.handle.net/10539/23457
dc.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. en_ZA
dc.description.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„•, 𝑞 β‰₯ 1 belongs to different classes of convex functions such as convex, 𝑠-convex, 𝑚-convex, (𝛼, 𝑚)-convex, quasi-convex, β„Ž-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 |𝑓β€²|𝑞 or |𝑓β€²β€²|𝑞, 𝑞 β‰₯ 1 is convex functions, 𝑠-convex functions, 𝑚-convex functions, (𝛼, 𝑚)-convex functions, quasiconvex functions, β„Ž-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„Ž- preinvex, quasipreiinvex, logarithmically preinvex functions which are more general phenomena than that of convex, 𝑠-convex, β„Ž-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. en_ZA
dc.format.extent Online resource ([13], 265 leaves)
dc.language.iso en en_ZA
dc.subject.lcsh Integral inequalities
dc.subject.lcsh Inequalities (Mathematics)
dc.title Integral inequalities of hermite-hadamard type and their applications en_ZA
dc.type Thesis en_ZA
dc.description.librarian MT 2017 en_ZA
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