Refinements of Young’s Integral Inequality via Fundamental Inequalities and Mean Value Theorems for Derivatives1

In the paper, the authors review several refinements of Young's integral inequality via several mean value theorems, such as Lagrange's and Taylor's mean value theorems of Lagrange's and Cauchy's type remainders, and via several fundamental inequalities, such as Cebysev's integral inequality, Hermite--Hadamard's type integral inequalities, Holder's integral inequality, and Jensen's discrete and integral inequalities, in terms of higher order derivatives and their norms, survey several applications of several refinements of Young's integral inequality, and further refine Young's integral inequality via Polya's type integral inequalities.