College of Science

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Superquadratic functions and the refinement of some classical inequalities
(MARCH 2016) Asare-Tuah, Anton
Convexity and inequalities have proved to be useful in many branches of mathematics such as functional analysis, theory of di fferential and integral equations, optimisation, interpolation, harmonic analysis and probability theory. They have useful applications also in mechanics, physics, economics and other sciences. Over the last 80 years, inequalities have seen a lot of interest from researchers, which has led to a great number of recent published books and articles. Thus, the theory of inequalities has developed into an independent branch or area of mathematics. This thesis is dedicated to a new refi nement of Jensen's inequality, which permits the use of non-continuous functions considered to be superquadratic. A re finement of Minkowski's inequality is established as a direct consequence of our refi ned Jensen's inequality. A lower bound for Young's inequality is also established as an application of superquadratic functions. Finally a re finement of the local submean inequality for subharmonic functions is also presented as an application of superquadratic functions.