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## Integral inequalities of hermite-hadamard type and their applications

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. HÃ¶lder [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

Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/23457 |

Date | January 2017 |

Creators | Latif, Muhammad Amer |

Source Sets | South African National ETD Portal |

Language | English |

Detected Language | English |

Type | Thesis |

Format | Online resource ([13], 265 leaves), application/pdf, application/pdf |

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