On Sherman Method to Deriving Inequalities for Some Classes of Functions Related to Convexity
Introduction, Notation, and Summary
In this chapter, we show the usefulness of Sherman method in deriving inequalities for convex, strongly convex, uniformly convex, and superquadratic functions. The inequality due to Sherman  generalizes the well-known inequality by Hardy, Littlewood, and Pólya in majorization theory [17, 21]. In addition, the HLP inequality includes the celebrated Jensen’s inequality. These results have been extensively studied by many researchers.
The theory of majorization has many applications in linear algebra, convex analysis, probability, statistics, geometry, optimization, approximation, numerical analysis, statistical mechanics, econometrics, etc . So, Sherman method gives further perspectives to find some nice applications. Therefore, this research topic is important and intriguing.
In this work, we provide a unified framework for generalizations of some classical results. First, we demonstrate the method by giving alternative unified proofs for some...
KeywordsConvex function Uniformly convex function Strongly convex function Superquadratic function Jensen’s inequality Sherman’s inequality Jensen’s functional
Mathematics Subject Classification (2010):26A51 26D15 15A45
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