The subject of this monograph is the fusion of two theories: mathematical inequalities and Stieltjes calculus, presented in a comprehensive and systematic manner. The primary aim is to extend a variety of fundamental inequalities based on the Stieltjes calculus approach. Research on Stieltjes differential and integral calculus has recently attracted increased interest among researchers, as Stieltjes calculus provides the advantage of allowing the study of many classical problems within a unique framework.This book offers a broad, comprehensive, and systematic approach to the foundations and recent developments of various inequalities in the context of Stieltjes. Written in a self-contained manner, this book comprises six pedagogically organized chapters, each divided into multiple sections covering a wide range of Stieltjes inequalities. For each new concept introduced, a sufficient number of solved examples and exercises are provided to enrich the reader's understanding.The book is primarily designed for researchers seeking to understand a new mathematical theory, Stieltjes inequalities, and apply them to the study of the qualitative properties of differential equations. Here, the differential equations can include ordinary differential equations, difference equations, quantum equations, generalized ordinary differential equations, and other related types. Graduate students at various levels will consider this text a valuable resource for both introductory and advanced material.
