# On Sherman Method to Deriving Inequalities for Some Classes of Functions Related to Convexity

• Marek Niezgoda
Chapter
Part of the Trends in Mathematics book series (TM)

## 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 [32] 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 [21]. 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...

## Keywords

Convex 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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